Siyavula textbooks: Grade 12 Maths

Collection Editor:

Free High School Science Texts Project

Siyavula textbooks: Grade 12 Maths

Collection Editor:

Free High School Science Texts Project

Authors:

Free High School Science Texts Project

Rory Adams

Mark Horner

Heather Williams

Online:

< http://siyavula.cnx.org/content/col11242/1.2/ >

C O N N E X I O N S

Rice University, Houston, Texas

❚❤✐s s❡❧❡❝t✐♦♥ ❛♥❞ ❛rr❛♥❣❡♠❡♥t ♦❢ ❝♦♥t❡♥t ❛s ❛ ❝♦❧❧❡❝t✐♦♥ ✐s ❝♦♣②r✐❣❤t❡❞ ❜② ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t✳

■t ✐s ❧✐❝❡♥s❡❞ ✉♥❞❡r t❤❡ ❈r❡❛t✐✈❡ ❈♦♠♠♦♥s ❆ttr✐❜✉t✐♦♥ ✸✳✵ ❧✐❝❡♥s❡ ✭❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴✮✳

❈♦❧❧❡❝t✐♦♥ str✉❝t✉r❡ r❡✈✐s❡❞✿ ❆✉❣✉st ✸✱ ✷✵✶✶

P❉❋ ❣❡♥❡r❛t❡❞✿ ❆✉❣✉st ✸✱ ✷✵✶✶

❋♦r ❝♦♣②r✐❣❤t ❛♥❞ ❛ttr✐❜✉t✐♦♥ ✐♥❢♦r♠❛t✐♦♥ ❢♦r t❤❡ ♠♦❞✉❧❡s ❝♦♥t❛✐♥❡❞ ✐♥ t❤✐s ❝♦❧❧❡❝t✐♦♥✱ s❡❡ ♣✳ ✶✺✼✳

❚❛❜❧❡ ♦❢ ❈♦♥t❡♥ts

▲♦❣❛r✐t❤♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶

✶ ❙❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s

✶✳✶ ❆r✐t❤♠❡t✐❝ ✫ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s✱ ❘❡❝✉rs✐✈❡ ❋♦r♠✉❧❛❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✶✳✷ ❙✐❣♠❛ ♥♦t❛t✐♦♥✱ ❋✐♥✐t❡ ✫ ■♥✜♥✐t❡ ❙❡r✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✷ ❋✐♥❛♥❝❡

✷✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❙❡q✉❡♥❝❡s ✫ ❙❡r✐❡s✱ ❋✉t✉r❡ ❱❛❧✉❡ ♦❢ P❛②♠❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

✷✳✷ ■♥✈❡st♠❡♥ts ✫ ▲♦❛♥s✱ ▲♦❛♥ ❙❝❤❡❞✉❧❡s✱ ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣✱ ❋♦r♠✉❧❛❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✸ ❋❛❝t♦r✐s✐♥❣ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❋❛❝t♦r ❚❤❡♦r❡♠✱ ❋❛❝t♦r✐s✐♥❣ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺

✸✳✷ ❋❛❝t♦r ❚❤❡♦r❡♠ ❊①❡r❝✐s❡s✱ ❙♦❧✈✐♥❣ ❈✉❜✐❝ ❊q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✹ ❋✉♥❝t✐♦♥s ❛♥❞ ❣r❛♣❤s

✹✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❉❡✜♥✐t✐♦♥✱ ◆♦t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸

✹✳✷ ●r❛♣❤s ♦❢ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹

✺ ❉✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s

✺✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ▲✐♠✐ts✱ ❆✈❡r❛❣❡ ●r❛❞✐❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶

✺✳✷ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✭❋✐rst Pr✐♥❝✐♣❧❡s✱ ❘✉❧❡s✮ ❛♥❞ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

✺✳✸ ❙♦❧✈✐♥❣ Pr♦❜❧❡♠s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼

✻ ▲✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣

✻✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❚❡r♠✐♥♦❧♦❣②✱ ❚❤❡ ❋❡❛s✐❜❧❡ ❘❡❣✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺

✻✳✷ ▼❡t❤♦❞✱ ❊①❡r❝✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

✼ ●❡♦♠❡tr②

✼✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❈✐r❝❧❡ ●❡♦♠❡tr② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

✼✳✷ ❈♦♦r❞✐♥❛t❡ ●❡♦♠❡tr②✱ ❊q✉❛t✐♦♥ ♦❢ ❚❛♥❣❡♥t✱ ❚r❛♥s❢♦r♠❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✶

✽ ❚r✐❣♦♥♦♠❡tr②

✽✳✶ ❈♦♠♣♦✉♥❞ ■❞❡♥t✐t✐❡s✱ Pr♦❜❧❡♠ ❙♦❧✈✐♥❣ ❙tr❛t❡❣✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✺

✽✳✷ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❚r✐❣ ❋✉♥❝t✐♦♥s ✭✷❉ ✫ ✸❉✮✱ ❖t❤❡r ●❡♦♠❡tr✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✺

✾ ❙t❛t✐st✐❝s

✾✳✶ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥✱ ❙❛♠♣❧✐♥❣✱ ❋✉♥❝t✐♦♥ ❋✐tt✐♥❣ ✫ ❘❡❣r❡ss✐♦♥ ❆♥❛❧②s✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✾

✾✳✷ ▲❡❛st ❙q✉❛r❡s✱ ❈❛❧❝✉❧❛t♦r ❲♦r❦✱ ❈♦rr❡❧❛t✐♦♥ ❈♦❡✣❝✐❡♥ts ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✹

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✶

✶✵ ❈♦♠❜✐♥❛t✐♦♥s ❛♥❞ ♣❡r♠✉t❛t✐♦♥s

✶✵✳✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ◆♦t❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✺

✶✵✳✷ ❋✉♥❞❛♠❡♥t❛❧ ❈♦✉♥t✐♥❣ Pr✐♥❝✐♣❧❡ ❛♥❞ ❈♦♠❜✐♥❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✼

✶✵✳✸ P❡r♠✉t❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✾

❙♦❧✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✸

●❧♦ss❛r② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✺

■♥❞❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻

❆ttr✐❜✉t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✺✼

✐✈

▲♦❣❛r✐t❤♠s✶

■♥tr♦❞✉❝t✐♦♥

■♥ ♠❛t❤❡♠❛t✐❝s ♠❛♥② ✐❞❡❛s ❛r❡ r❡❧❛t❡❞✳ ❲❡ s❛✇ t❤❛t ❛❞❞✐t✐♦♥ ❛♥❞ s✉❜tr❛❝t✐♦♥ ❛r❡ r❡❧❛t❡❞ ❛♥❞ t❤❛t ♠✉❧t✐✲

♣❧✐❝❛t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ❛r❡ r❡❧❛t❡❞✳ ❙✐♠✐❧❛r❧②✱ ❡①♣♦♥❡♥t✐❛❧s ❛♥❞ ❧♦❣❛r✐t❤♠s ❛r❡ r❡❧❛t❡❞✳

▲♦❣❛r✐t❤♠s ❛r❡ ❝♦♠♠♦♥❧② r❡❢❡r❡❞ t♦ ❛s ❧♦❣s✱ ❛r❡ t❤❡ ✧♦♣♣♦s✐t❡✧ ♦❢ ❡①♣♦♥❡♥t✐❛❧s✱ ❥✉st ❛s s✉❜tr❛❝t✐♦♥ ✐s t❤❡

♦♣♣♦s✐t❡ ♦❢ ❛❞❞✐t✐♦♥ ❛♥❞ ❞✐✈✐s✐♦♥ ✐s t❤❡ ♦♣♣♦s✐t❡ ♦❢ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ▲♦❣s ✧✉♥❞♦✧ ❡①♣♦♥❡♥t✐❛❧s✳ ❚❡❝❤♥✐❝❛❧❧②

s♣❡❛❦✐♥❣✱ ❧♦❣s ❛r❡ t❤❡ ✐♥✈❡rs❡s ♦❢ ❡①♣♦♥❡♥t✐❛❧s✳ ❚❤❡ ❧♦❣❛r✐t❤♠ ♦❢ ❛ ♥✉♠❜❡r x ✐♥ t❤❡ ❜❛s❡ a ✐s ❞❡✜♥❡❞ ❛s t❤❡

♥✉♠❜❡r n s✉❝❤ t❤❛t an = x✳

❙♦✱ ✐❢ an = x✱ t❤❡♥✿

log (x) = n

✭✶✮

a

❛s✐❞❡✿ ❲❤❡♥ ✇❡ s❛② ✏✐♥✈❡rs❡ ❢✉♥❝t✐♦♥✑ ✇❡ ♠❡❛♥ t❤❛t t❤❡ ❛♥s✇❡r ❜❡❝♦♠❡s t❤❡ q✉❡st✐♦♥ ❛♥❞ t❤❡

q✉❡st✐♦♥ ❜❡❝♦♠❡s t❤❡ ❛♥s✇❡r✳ ❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ab = x t❤❡ ✏q✉❡st✐♦♥✑ ✐s ✏✇❤❛t ✐s a

r❛✐s❡❞ t♦ t❤❡ ♣♦✇❡r b❄✑ ❚❤❡ ❛♥s✇❡r ✐s ✏x✳✑ ❚❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✇♦✉❧❞ ❜❡ logax = b ♦r ✏❜② ✇❤❛t

♣♦✇❡r ♠✉st ✇❡ r❛✐s❡ a t♦ ♦❜t❛✐♥ x❄✑ ❚❤❡ ❛♥s✇❡r ✐s ✏b✳✑

❚❤❡ ♠❛t❤❡♠❛t✐❝❛❧ s②♠❜♦❧ ❢♦r ❧♦❣❛r✐t❤♠ ✐s log (x) ❛♥❞ ✐t ✐s r❡❛❞ ✏❧♦❣ t♦ t❤❡ ❜❛s❡ a ♦❢ x✑✳ ❋♦r ❡①❛♠♣❧❡✱

a

log

(100) ✐s ✏❧♦❣ t♦ t❤❡ ❜❛s❡ ✶✵ ♦❢ ✶✵✵✳✑

10

▲♦❣❛r✐t❤♠ ❙②♠❜♦❧s ✿

❲r✐t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦✉t ✐♥ ✇♦r❞s✳ ❚❤❡ ✜rst ♦♥❡ ✐s ❞♦♥❡ ❢♦r ②♦✉✳

✶✳ log (4) ✐s ❧♦❣ t♦ t❤❡ ❜❛s❡ ✷ ♦❢ ✹

2

✷✳ log (14)

10

✸✳ log (4)

16

✹✳ log (8)

x

✺✳ log (x)

y

❉❡✜♥✐t✐♦♥ ♦❢ ▲♦❣❛r✐t❤♠s

❚❤❡ ❧♦❣❛r✐t❤♠ ♦❢ ❛ ♥✉♠❜❡r ✐s t❤❡ ✈❛❧✉❡ t♦ ✇❤✐❝❤ t❤❡ ❜❛s❡ ♠✉st ❜❡ r❛✐s❡❞ t♦ ❣✐✈❡ t❤❛t ♥✉♠❜❡r ✐✳❡✳ t❤❡

❡①♣♦♥❡♥t✳ ❋r♦♠ t❤❡ ✜rst ❡①❛♠♣❧❡ ♦❢ t❤❡ ❛❝t✐✈✐t② log (4) ♠❡❛♥s t❤❡ ♣♦✇❡r ♦❢ ✷ t❤❛t ✇✐❧❧ ❣✐✈❡ ✹✳ ❆s 22 = 4✱

2

✇❡ s❡❡ t❤❛t

log (4) = 2

✭✷✮

2

❚❤❡ ❡①♣♦♥❡♥t✐❛❧✲❢♦r♠ ✐s t❤❡♥ 22 = 4 ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠✐❝✲❢♦r♠ ✐s log 4 = 2✳

2

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✶✽✽✸✴✶✳✹✴❃✳

✶

✷

❉❡✜♥✐t✐♦♥ ✶✿ ▲♦❣❛r✐t❤♠s

■❢ an = x✱ t❤❡♥✿ log (x) = n✱ ✇❤❡r❡ a > 0❀ a = 1 ❛♥❞ x > 0✳

a

❆♣♣❧②✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ ✿

❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢✿

✶✳ log 343

7

Reasoning :

73 = 343

✭✸✮

therefore, log 343 = 3

7

✷✳ log 8

2

✸✳ log 1

4 64

✹✳ log 1 000

10

▲♦❣❛r✐t❤♠ ❇❛s❡s

▲♦❣❛r✐t❤♠s✱ ❧✐❦❡ ❡①♣♦♥❡♥t✐❛❧s✱ ❛❧s♦ ❤❛✈❡ ❛ ❜❛s❡ ❛♥❞ log (2) ✐s ♥♦t t❤❡ s❛♠❡ ❛s log (2)✳

2

10

❲❡ ❣❡♥❡r❛❧❧② ✉s❡ t❤❡ ✏❝♦♠♠♦♥✑ ❜❛s❡✱ ✶✵✱ ♦r t❤❡ ♥❛t✉r❛❧ ❜❛s❡✱ e✳

❚❤❡ ♥✉♠❜❡r e ✐s ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r ❜❡t✇❡❡♥ 2.71 ❛♥❞ 2.72✳ ■t ❝♦♠❡s ✉♣ s✉r♣r✐s✐♥❣❧② ♦❢t❡♥ ✐♥ ▼❛t❤❡✲

♠❛t✐❝s✱ ❜✉t ❢♦r ♥♦✇ s✉✣❝❡ ✐t t♦ s❛② t❤❛t ✐t ✐s ♦♥❡ ♦❢ t❤❡ t✇♦ ❝♦♠♠♦♥ ❜❛s❡s✳

◆❛t✉r❛❧ ▲♦❣❛r✐t❤♠

❚❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ✭s②♠❜♦❧ ln✮ ✐s ✇✐❞❡❧② ✉s❡❞ ✐♥ t❤❡ s❝✐❡♥❝❡s✳ ❚❤❡ ♥❛t✉r❛❧ ❧♦❣❛r✐t❤♠ ✐s t♦ t❤❡ ❜❛s❡ e

✇❤✐❝❤ ✐s ❛♣♣r♦①✐♠❛t❡❧② 2.71828183...✳ e ✐s ❧✐❦❡ π ❛♥❞ ✐s ❛♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ ❛♥ ✐rr❛t✐♦♥❛❧ ♥✉♠❜❡r✳

❲❤✐❧❡ t❤❡ ♥♦t❛t✐♦♥ log (x) ❛♥❞ log (x) ♠❛② ❜❡ ✉s❡❞✱ log (x) ✐s ♦❢t❡♥ ✇r✐tt❡♥ log (x) ✐♥ ❙❝✐❡♥❝❡ ❛♥❞

10

e

10

log (x) ✐s ♥♦r♠❛❧❧② ✇r✐tt❡♥ ❛s ln (x) ✐♥ ❜♦t❤ ❙❝✐❡♥❝❡ ❛♥❞ ▼❛t❤❡♠❛t✐❝s✳ ❙♦✱ ✐❢ ②♦✉ s❡❡ t❤❡ log s②♠❜♦❧ ✇✐t❤♦✉t

e

❛ ❜❛s❡✱ ✐t ♠❡❛♥s log ✳

10

■t ✐s ♦❢t❡♥ ♥❡❝❡ss❛r② ♦r ❝♦♥✈❡♥✐❡♥t t♦ ❝♦♥✈❡rt ❛ ❧♦❣ ❢r♦♠ ♦♥❡ ❜❛s❡ t♦ ❛♥♦t❤❡r✳ ❆♥ ❡♥❣✐♥❡❡r ♠✐❣❤t ♥❡❡❞ ❛♥

❛♣♣r♦①✐♠❛t❡ s♦❧✉t✐♦♥ t♦ ❛ ❧♦❣ ✐♥ ❛ ❜❛s❡ ❢♦r ✇❤✐❝❤ ❤❡ ❞♦❡s ♥♦t ❤❛✈❡ ❛ t❛❜❧❡ ♦r ❝❛❧❝✉❧❛t♦r ❢✉♥❝t✐♦♥✱ ♦r ✐t ♠❛②

❜❡ ❛❧❣❡❜r❛✐❝❛❧❧② ❝♦♥✈❡♥✐❡♥t t♦ ❤❛✈❡ t✇♦ ❧♦❣s ✐♥ t❤❡ s❛♠❡ ❜❛s❡✳

▲♦❣❛r✐t❤♠s ❝❛♥ ❜❡ ❝❤❛♥❣❡❞ ❢r♦♠ ♦♥❡ ❜❛s❡ t♦ ❛♥♦t❤❡r✱ ❜② ✉s✐♥❣ t❤❡ ❝❤❛♥❣❡ ♦❢ ❜❛s❡ ❢♦r♠✉❧❛✿

log x

log x =

b

✭✹✮

a

log a

b

✇❤❡r❡ b ✐s ❛♥② ❜❛s❡ ②♦✉ ✜♥❞ ❝♦♥✈❡♥✐❡♥t✳ ◆♦r♠❛❧❧② a ❛♥❞ b ❛r❡ ❦♥♦✇♥✱ t❤❡r❡❢♦r❡ log a ✐s ♥♦r♠❛❧❧② ❛ ❦♥♦✇♥✱

b

✐❢ ✐rr❛t✐♦♥❛❧✱ ♥✉♠❜❡r✳

❋♦r ❡①❛♠♣❧❡✱ ❝❤❛♥❣❡ log 12 ✐♥ ❜❛s❡ ✶✵ ✐s✿

2

log 12

log 12 =

10

✭✺✮

2

log 2

10

❈❤❛♥❣❡ ♦❢ ❇❛s❡ ✿ ❈❤❛♥❣❡ t❤❡ ❢♦❧❧♦✇✐♥❣ t♦ t❤❡ ✐♥❞✐❝❛t❡❞ ❜❛s❡✿

✶✳ log (4) t♦ ❜❛s❡ ✽

2

✷✳ log (14) t♦ ❜❛s❡ ✷

10

✸✳ log (4) t♦ ❜❛s❡ ✶✵

16

✹✳ log (8) t♦ ❜❛s❡ y

x

✸

✺✳ log (x) t♦ ❜❛s❡ x

y

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ❧♦❣❛r✐t❤♠s ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴♠◗❚❲③▲♣❈❝❲✵✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✶

▲❛✇s ♦❢ ▲♦❣❛r✐t❤♠s

❏✉st ❛s ❢♦r t❤❡ ❡①♣♦♥❡♥ts✱ ❧♦❣❛r✐t❤♠s ❤❛✈❡ s♦♠❡ ❧❛✇s ✇❤✐❝❤ ♠❛❦❡ ✇♦r❦✐♥❣ ✇✐t❤ t❤❡♠ ❡❛s✐❡r✳ ❚❤❡s❡ ❧❛✇s ❛r❡

❜❛s❡❞ ♦♥ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❧❛✇s ❛♥❞ ❛r❡ s✉♠♠❛r✐s❡❞ ✜rst ❛♥❞ t❤❡♥ ❡①♣❧❛✐♥❡❞ ✐♥ ❞❡t❛✐❧✳

log (1)

=

0

a

log (a)

=

1

a

log (x · y)

=

log (x) + log (y)

a

a

a

✭✻✮

log

x

=

log (x) − log (y)

a

y

a

a

log

xb

=

blog (x)

a

a

√

log ( b x)

=

log (x)

a

a

b

▲♦❣❛r✐t❤♠ ▲❛✇ ✶✿ loga1 = 0

Since

a0

=

1

Then,

log (1)

=

log

a0

✭✼✮

a

a

=

0

by definition of logarithm

❋♦r ❡①❛♠♣❧❡✱

log 1 = 0

✭✽✮

2

❛♥❞

log 1 = 0

✭✾✮

25

▲♦❣❛r✐t❤♠ ▲❛✇ ✶✿ log 1 = 0 ✿

a

❙✐♠♣❧✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✳ log (1) + 5

2

✷✳ log (1) × 100

10

✹

✸✳ 3 × log (1)

16

✹✳ log (1) + 2xy

x

✺✳ log (1)

y

x

▲♦❣❛r✐t❤♠ ▲❛✇ ✷✿ loga (a) = 1

Since

a1

=

a

Then,

log (a)

=

log

a1

✭✶✵✮

a

a

=

1

by definition of logarithm

❋♦r ❡①❛♠♣❧❡✱

log 2 = 1

✭✶✶✮

2

❛♥❞

log 25 = 1

✭✶✷✮

25

▲♦❣❛r✐t❤♠ ▲❛✇ ✷✿ log (a) = 1 ✿

a

❙✐♠♣❧✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✳ log (2) + 5

2

✷✳ log (10) × 100

10

✸✳ 3 × log (16)

16

✹✳ log (x) + 2xy

x

✺✳ log (y)

y

x

t✐♣✿ ❯s❡❢✉❧ t♦ ❦♥♦✇ ❛♥❞ r❡♠❡♠❜❡r

❲❤❡♥ t❤❡ ❜❛s❡ ✐s ✶✵✱ ✇❡ ❞♦ ♥♦t ♥❡❡❞ t♦ st❛t❡ ✐t✳ ❋r♦♠ t❤❡ ✇♦r❦ ❞♦♥❡ ✉♣ t♦ ♥♦✇✱ ✐t ✐s ❛❧s♦ ✉s❡❢✉❧ t♦ s✉♠♠❛r✐s❡

t❤❡ ❢♦❧❧♦✇✐♥❣ ❢❛❝ts✿

✶✳ log1 = 0

✷✳ log10 = 1

✸✳ log100 = 2

✹✳ log1000 = 3

▲♦❣❛r✐t❤♠ ▲❛✇ ✸✿ loga (x · y) = loga (x) + loga (y)

❚❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤✐s ❧❛✇ ✐s ❛ ❜✐t tr✐❝❦✐❡r t❤❛♥ t❤❡ ✜rst t✇♦✳ ❋✐rst❧②✱ ✇❡ ♥❡❡❞ t♦ r❡❧❛t❡ x ❛♥❞ y t♦ t❤❡ ❜❛s❡

a✳ ❙♦✱ ❛ss✉♠❡ t❤❛t x = am ❛♥❞ y = an✳ ❚❤❡♥ ❢r♦♠ ❊q✉❛t✐♦♥ ✭✶✮✱ ✇❡ ❤❛✈❡ t❤❛t✿

log (x)

=

m

a

✭✶✸✮

and

log (y)

=

n

a

✺

❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❝❛♥ ✇r✐t❡✿

log (x · y)

=

log (am · an)

a

a

=

log (am+n)

Exponential laws

a

✭✶✹✮

=

log

alog (x)+log (y)

a

a

a

=

log (x) + log (y)

a

a

❋♦r ❡①❛♠♣❧❡✱ s❤♦✇ t❤❛t log (10 · 100) = log10 + log100✳ ❙t❛rt ✇✐t❤ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡✿

log (10 · 100)

=

log (1000)

=

log 103

✭✶✺✮

=

3

❚❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✿

log10 + log100

=

1 + 2

✭✶✻✮

=

3

❇♦t❤ s✐❞❡s ❛r❡ ❡q✉❛❧✳ ❚❤❡r❡❢♦r❡✱ log (10 · 100) = log10 + log100✳

▲♦❣❛r✐t❤♠ ▲❛✇ ✸✿ log (x · y) = log (x) + log (y) ✿

a

a

a

❲r✐t❡ ❛s s❡♣❡r❛t❡ ❧♦❣s✿

✶✳ log (8 × 4)

2

✷✳ log (10 × 10)

8

✸✳ log (xy)

16

✹✳ log (2xy)

z

✺✳ log y2

x

▲♦❣❛r✐t❤♠ ▲❛✇ ✹✿ log x

a

= log

y

a (x) − loga (y)

❚❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤✐s ❧❛✇ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ ▲♦❣❛r✐t❤♠ ▲❛✇ ✸ ❛♥❞ ✐s ❧❡❢t ❛s ❛♥ ❡①❡r❝✐s❡✳

❋♦r ❡①❛♠♣❧❡✱ s❤♦✇ t❤❛t log 10 = log10 − log100✳ ❙t❛rt ✇✐t❤ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡✿

100

log

10

=

log

1

100

10

=

log 10−1

✭✶✼✮

=

−1

❚❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✿

log10 − log100

=

1 − 2

✭✶✽✮

=

−1

❇♦t❤ s✐❞❡s ❛r❡ ❡q✉❛❧✳ ❚❤❡r❡❢♦r❡✱ log 10 = log10 − log100✳

100

✻

▲♦❣❛r✐t❤♠ ▲❛✇ ✹✿ log

x

= log (x) − log (y) ✿

a

y

a

a

❲r✐t❡ ❛s s❡♣❡r❛t❡ ❧♦❣s✿

✶✳ log 8

2

5

✷✳ log 100

8

3

✸✳ log

x

16

y

✹✳ log

2

z

y

✺✳ log y

x

2

▲♦❣❛r✐t❤♠ ▲❛✇ ✺✿ loga xb = bloga (x)

❖♥❝❡ ❛❣❛✐♥✱ ✇❡ ♥❡❡❞ t♦ r❡❧❛t❡ x t♦ t❤❡ ❜❛s❡ a✳ ❙♦✱ ✇❡ ❧❡t x = am✳ ❚❤❡♥✱

log

xb

=

log

(am)b

a

a

=

log

am·b

(exponential laws

a

But,

m

=

log (x)

(Assumption that x = am)

✭✶✾✮

a

∴

log

xb

=

log

ab·log (x)

a

a

a

=

b · log (x)

(Definition of logarithm)

a

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t log 53 = 3log (5)✳

2

2

log

53

=

log (5 · 5 · 5)

2

2

=

log 5 + log 5 + log 5

(

(x · y) = log (am · an))

✭✷✵✮

2

2

2

∵ loga

a

=

3log 5

2

❚❤❡r❡❢♦r❡✱ log 53 = 3log (5)✳

2

2

▲♦❣❛r✐t❤♠ ▲❛✇ ✺✿ log xb = blog (x) ✿

a

a

❙✐♠♣❧✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✳ log 84

2

✷✳ log 1010

8

✸✳ log (xy)

16

✹✳ log (yx)

z

✺✳ log y2x

x

√

▲♦❣❛r✐t❤♠ ▲❛✇ ✻✿ log

(x)

a ( b x) = logab

❚❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ t❤✐s ❧❛✇ ✐s ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❞❡r✐✈❛t✐♦♥ ♦❢ ▲♦❣❛r✐t❤♠ ▲❛✇ ✺ ❛♥❞ ✐s ❧❡❢t ❛s ❛♥ ❡①❡r❝✐s❡✳

√

❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ s❤♦✇ t❤❛t log

3 5 = log 5

2

✳

2

3

√

1

log

3 5

=

log

5 3

2

2

=

1 log 5

∵ log

xb = blog (x)

✭✷✶✮

3

2

a

a

=

log 5

2

3

√

❚❤❡r❡❢♦r❡✱ log

3 5 = log 5

2

✳

2

3

✼

√

▲♦❣❛r✐t❤♠ ▲❛✇ ✻✿ log ( b x) = loga(x) ✿

a

b

❙✐♠♣❧✐❢② t❤❡ ❢♦❧❧♦✇✐♥❣✿

√

✶✳ log

4 8

2

√

✷✳ log

10 10

8

√

✸✳ log ( y x)

16

✹✳

√

log

x y

z

✺✳

√

log

2x y

x

t✐♣✿ ❚❤❡ ✜♥❛❧ ❛♥s✇❡r ❞♦❡s♥✬t ❤❛✈❡ t♦ ❧♦♦❦ s✐♠♣❧❡✳

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ❧♦❣❛r✐t❤♠s ✲ ✷

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴P✉♣◆❣✈✹✾❴❲❨✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✷

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ❧♦❣❛r✐t❤♠s ✲ ✸

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❚▼♠①❑❩❛❈q❡✵✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✸

❊①❡r❝✐s❡ ✶✿ ❙✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ ▲♦❣s

❙✐♠♣❧✐❢②✱ ✇✐t❤♦✉t ✉s❡ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✿

3log3 + log125

✭✷✷✮

❊①❡r❝✐s❡ ✷✿ ❙✐♠♣❧✐✜❝❛t✐♦♥ ♦❢ ▲♦❣s

❙✐♠♣❧✐❢②✱ ✇✐t❤♦✉t ✉s❡ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✿

2

8 3 + log 32

✭✷✸✮

2

❊①❡r❝✐s❡ ✸✿ ❙✐♠♣❧✐❢② t♦ ♦♥❡ ❧♦❣

❲r✐t❡ 2log3 + log2 − log5 ❛s t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ ❛ s✐♥❣❧❡ ♥✉♠❜❡r✳

t✐♣✿ ❊①♣♦♥❡♥t r✉❧❡✿ xb a = xab

✽

❙♦❧✈✐♥❣ s✐♠♣❧❡ ❧♦❣ ❡q✉❛t✐♦♥s

■♥ ❣r❛❞❡ ✶✵ ②♦✉ s♦❧✈❡❞ s♦♠❡ ❡①♣♦♥❡♥t✐❛❧ ❡q✉❛t✐♦♥s ❜② tr✐❛❧ ❛♥❞ ❡rr♦r✱ ❜❡❝❛✉s❡ ②♦✉ ❞✐❞ ♥♦t ❦♥♦✇ t❤❡ ❣r❡❛t

♣♦✇❡r ♦❢ ❧♦❣❛r✐t❤♠s ②❡t✳ ◆♦✇ ✐t ✐s ♠✉❝❤ ❡❛s✐❡r t♦ s♦❧✈❡ t❤❡s❡ ❡q✉❛t✐♦♥s ❜② ✉s✐♥❣ ❧♦❣❛r✐t❤♠s✳

❋♦r ❡①❛♠♣❧❡ t♦ s♦❧✈❡ x ✐♥ 25x = 50 ❝♦rr❡❝t t♦ t✇♦ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ②♦✉ s✐♠♣❧② ❛♣♣❧② t❤❡ ❢♦❧❧♦✇✐♥❣ r❡❛s♦♥✐♥❣✳

■❢ t❤❡ ▲❍❙ ❂ ❘❍❙ t❤❡♥ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ t❤❡ ▲❍❙ ♠✉st ❜❡ ❡q✉❛❧ t♦ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ t❤❡ ❘❍❙✳ ❇② ❛♣♣❧②✐♥❣

▲❛✇ ✺✱ ②♦✉ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ✉s❡ ②♦✉r ❝❛❧❝✉❧❛t♦r t♦ s♦❧✈❡ ❢♦r x✳

❊①❡r❝✐s❡ ✹✿ ❙♦❧✈✐♥❣ ▲♦❣ ❡q✉❛t✐♦♥s

❙♦❧✈❡ ❢♦r x✿ 25x = 50 ❝♦rr❡❝t t♦ t✇♦ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳

■♥ ❣❡♥❡r❛❧✱ t❤❡ ❡①♣♦♥❡♥t✐❛❧ ❡q✉❛t✐♦♥ s❤♦✉❧❞ ❜❡ s✐♠♣❧✐✜❡❞ ❛s ♠✉❝❤ ❛s ♣♦ss✐❜❧❡✳ ❚❤❡♥ t❤❡ ❛✐♠ ✐s t♦ ♠❛❦❡ t❤❡

✉♥❦♥♦✇♥ q✉❛♥t✐t② ✭✐✳❡✳ x✮ t❤❡ s✉❜❥❡❝t ♦❢ t❤❡ ❡q✉❛t✐♦♥✳

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡q✉❛t✐♦♥

2(x+2) = 1

✭✷✹✮

✐s s♦❧✈❡❞ ❜② ♠♦✈✐♥❣ ❛❧❧ t❡r♠s ✇✐t❤ t❤❡ ✉♥❦♥♦✇♥ t♦ ♦♥❡ s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥ ❛♥❞ t❛❦✐♥❣ ❛❧❧ ❝♦♥st❛♥ts t♦ t❤❡

♦t❤❡r s✐❞❡ ♦❢ t❤❡ ❡q✉❛t✐♦♥

2x · 22

=

1

✭✷✺✮

2x

=

1

22

❚❤❡♥✱ t❛❦❡ t❤❡ ❧♦❣❛r✐t❤♠ ♦❢ ❡❛❝❤ s✐❞❡✳

log (2x)

=

log

1

22

xlog (2)

=

−log 22

✭✷✻✮

xlog (2)

=

−2log (2)

Divide both sides by log (2)

∴

x

=

−2

❙✉❜st✐t✉t✐♥❣ ✐♥t♦ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✱ ②✐❡❧❞s

2−2+2 = 20 = 1

✭✷✼✮

❙✐♠✐❧❛r❧②✱ 9(1−2x) = 34 ✐s s♦❧✈❡❞ ❛s ❢♦❧❧♦✇s✿

9(1−2x)

=

34

32(1−2x)

=

34

32−4x

=

34

take the logarithm of both sides

log 32−4x

=

log 34

✭✷✽✮

(2 − 4x) log (3)

=

4log (3)

divide both sides by log (3)

2 − 4x

=

4

−4x

=

2

∴ x =

− 12

❙✉❜st✐t✉t✐♥❣ ✐♥t♦ t❤❡ ♦r✐❣✐♥❛❧ ❡q✉❛t✐♦♥✱ ②✐❡❧❞s

9(1−2( −1 ))

2

= 9(1+1) = 32(2) = 34

✭✷✾✮

❊①❡r❝✐s❡ ✺✿ ❊①♣♦♥❡♥t✐❛❧ ❊q✉❛t✐♦♥

❙♦❧✈❡ ❢♦r x ✐♥ 7 · 5(3x+3) = 35

✾

❊①❡r❝✐s❡s

❙♦❧✈❡ ❢♦r x✿

✶✳ log x = 2

3

✷✳ 10log27 = x

✸✳ 32x−1 = 272x−1

▲♦❣❛r✐t❤♠✐❝ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ t❤❡ ❘❡❛❧ ❲♦r❧❞

▲♦❣❛r✐t❤♠s ❛r❡ ♣❛rt ♦❢ ❛ ♥✉♠❜❡r ♦❢ ❢♦r♠✉❧❛❡ ✉s❡❞ ✐♥ t❤❡ P❤②s✐❝❛❧ ❙❝✐❡♥❝❡s✳ ❚❤❡r❡ ❛r❡ ❢♦r♠✉❧❛❡ t❤❛t ❞❡❛❧

✇✐t❤ ❡❛rt❤q✉❛❦❡s✱ ✇✐t❤ s♦✉♥❞✱ ❛♥❞ ♣❍✲❧❡✈❡❧s t♦ ♠❡♥t✐♦♥ ❛ ❢❡✇✳ ❚♦ ✇♦r❦ ♦✉t t✐♠❡ ♣❡r✐♦❞s ✐s ❣r♦✇t❤ ♦r ❞❡❝❛②✱

❧♦❣s ❛r❡ ✉s❡❞ t♦ s♦❧✈❡ t❤❡ ♣❛rt✐❝✉❧❛r ❡q✉❛t✐♦♥✳

❊①❡r❝✐s❡ ✻✿ ❯s✐♥❣ t❤❡ ❣r♦✇t❤ ❢♦r♠✉❧❛

❆ ❝✐t② ❣r♦✇s ✺✪ ❡✈❡r② ✷ ②❡❛rs✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ❝✐t② t♦ tr✐♣❧❡ ✐ts s✐③❡❄

❊①❡r❝✐s❡ ✼✿ ▲♦❣s ✐♥ ❈♦♠♣♦✉♥❞ ■♥t❡r❡st

■ ❤❛✈❡ ❘✶✷ ✵✵✵ t♦ ✐♥✈❡st✳ ■ ♥❡❡❞ t❤❡ ♠♦♥❡② t♦ ❣r♦✇ t♦ ❛t ❧❡❛st ❘✸✵ ✵✵✵✳ ■❢ ✐t ✐s ✐♥✈❡st❡❞ ❛t ❛

❝♦♠♣♦✉♥❞ ✐♥t❡r❡st r❛t❡ ♦❢ ✶✸✪ ♣❡r ❛♥♥✉♠✱ ❢♦r ❤♦✇ ❧♦♥❣ ✭✐♥ ❢✉❧❧ ②❡❛rs✮ ❞♦❡s ♠② ✐♥✈❡st♠❡♥t ♥❡❡❞ t♦

❣r♦✇ ❄

❊①❡r❝✐s❡s

✶✳ ❚❤❡ ♣♦♣✉❧❛t✐♦♥ ♦❢ ❛ ❝❡rt❛✐♥ ❜❛❝t❡r✐❛ ✐s ❡①♣❡❝t❡❞ t♦ ❣r♦✇ ❡①♣♦♥❡♥t✐❛❧❧② ❛t ❛ r❛t❡ ♦❢ ✶✺ ✪ ❡✈❡r② ❤♦✉r✳

■❢ t❤❡ ✐♥✐t✐❛❧ ♣♦♣✉❧❛t✐♦♥ ✐s ✺ ✵✵✵✱ ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ❢♦r t❤❡ ♣♦♣✉❧❛t✐♦♥ t♦ r❡❛❝❤ ✶✵✵ ✵✵✵ ❄

✷✳ P❧✉s ❇❛♥❦ ✐s ♦✛❡r✐♥❣ ❛ s❛✈✐♥❣s ❛❝❝♦✉♥t ✇✐t❤ ❛♥ ✐♥t❡r❡st r❛t❡ ✐❢ ✶✵ ✪ ♣❡r ❛♥♥✉♠ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳

❨♦✉ ❝❛♥ ❛✛♦r❞ t♦ s❛✈❡ ❘ ✸✵✵ ♣❡r ♠♦♥t❤✳ ❍♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ ②♦✉ t♦ s❛✈❡ ❘ ✷✵ ✵✵✵ ❄ ✭●✐✈❡ ②♦✉r

❛♥s✇❡r ✐♥ ②❡❛rs ❛♥❞ ♠♦♥t❤s✮

❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ ❙❤♦✇ t❤❛t

x

log

= log (x) − log (y)

✭✸✵✮

a

y

a

a

✷✳ ❙❤♦✇ t❤❛t

√

log (x)

log

b x =

a

✭✸✶✮

a

b

✸✳ ❲✐t❤♦✉t ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r s❤♦✇ t❤❛t✿

75

5

32

log

− 2log

+ log

= log2

✭✸✷✮

16

9

243

✹✳ ●✐✈❡♥ t❤❛t 5n = x ❛♥❞ n = log y

2

❛✳ ❲r✐t❡ y ✐♥ t❡r♠s ♦❢ n

❜✳ ❊①♣r❡ss log 4y ✐♥ t❡r♠s ♦❢ n

8

❝✳ ❊①♣r❡ss 50n+1 ✐♥ t❡r♠s ♦❢ x ❛♥❞ y

✶✵

✺✳ ❙✐♠♣❧✐❢②✱ ✇✐t❤♦✉t t❤❡ ✉s❡ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✿

❛✳ 2

8 3 + log 32

2

√

❜✳ log 9 − log

5

3

5

1

❝✳

5

2 + log 92,12

4−1−9−1

3

✻✳ ❙✐♠♣❧✐❢② t♦ ❛ s✐♥❣❧❡ ♥✉♠❜❡r✱ ✇✐t❤♦✉t ✉s❡ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✿

❛✳ log 125 + log32−log8

5

log8

❜✳ log3 − log0, 3

✼✳ ●✐✈❡♥✿

log 6 = a ❛♥❞ log 5 = b

3

6

❛✳ ❊①♣r❡ss log 2 ✐♥ t❡r♠s ♦❢ a✳

3

❜✳ ❍❡♥❝❡✱ ♦r ♦t❤❡r✇✐s❡✱ ✜♥❞ log 10 ✐♥ t❡r♠s ♦❢ a ❛♥❞ b✳

3

✽✳ ●✐✈❡♥✿

pqk = qp−1 Pr♦✈❡✿

k = 1 − 2log p

q

✾✳ ❊✈❛❧✉❛t❡ ✇✐t❤♦✉t ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r✿ (log 49)5 + log

1

− 13 log 1

7

5

125

9

✶✵✳ ■❢ log5 = 0, 7✱ ❞❡t❡r♠✐♥❡✱ ✇✐t❤♦✉t ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r✿

❛✳ log 5

2

❜✳ 10−1,4

✶✶✳ ●✐✈❡♥✿

M = log (x + 3) + log (x − 3)

2

2

❛✳ ❉❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡s ♦❢ x ❢♦r ✇❤✐❝❤ M ✐s ❞❡✜♥❡❞✳

❜✳ ❙♦❧✈❡ ❢♦r x ✐❢ M = 4✳

✶✷✳ ❙♦❧✈❡✿

x3 logx = 10x2 ✭❆♥s✇❡r✭s✮ ♠❛② ❜❡ ❧❡❢t ✐♥ s✉r❞ ❢♦r♠✱ ✐❢ ♥❡❝❡ss❛r②✳✮

✶✸✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ (log 3)3 ✇✐t❤♦✉t t❤❡ ✉s❡ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r✳

27

√

✶✹✳ ❙✐♠♣❧✐❢② ❇② ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r✿ log 8 + 2log

27

4

3

✶✺✳ ❲r✐t❡ log4500 ✐♥ t❡r♠s ♦❢ a ❛♥❞ b ✐❢ 2 = 10a ❛♥❞ 9 = 10b✳

✶✻✳ ❈❛❧❝✉❧❛t❡✿

52006−52004+24

52004+1

√

✶✼✳ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❢♦r x ✇✐t❤♦✉t t❤❡ ✉s❡ ♦❢ ❛ ❝❛❧❝✉❧❛t♦r ❛♥❞ ✉s✐♥❣ t❤❡ ❢❛❝t t❤❛t 10 ≈ 3, 16 :

6

2log (x + 1) =

− 1

✭✸✸✮

log (x + 1)

✶✽✳ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❢♦r x✿ 66x = 66

✭●✐✈❡ ❛♥s✇❡r ❝♦rr❡❝t t♦ ✷ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳✮

❈❤❛♣t❡r ✶

❙❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s

✶✳✶ ❆r✐t❤♠❡t✐❝ ✫ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s✱ ❘❡❝✉rs✐✈❡ ❋♦r♠✉❧❛❡✶

✶✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

■♥ t❤✐s ❝❤❛♣t❡r ✇❡ ❡①t❡♥❞ t❤❡ ❛r✐t❤♠❡t✐❝ ❛♥❞ q✉❛❞r❛t✐❝ s❡q✉❡♥❝❡s st✉❞✐❡❞ ✐♥ ❡❛r❧✐❡r ❣r❛❞❡s✱ t♦ ❣❡♦♠❡tr✐❝

s❡q✉❡♥❝❡s✳ ❲❡ ❛❧s♦ ❧♦♦❦ ❛t s❡r✐❡s✱ ✇❤✐❝❤ ✐s t❤❡ s✉♠♠✐♥❣ ♦❢ t❤❡ t❡r♠s ✐♥ ❛ s❡q✉❡♥❝❡✳

✶✳✶✳✷ ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡s

❚❤❡ s✐♠♣❧❡st t②♣❡ ♦❢ ♥✉♠❡r✐❝❛❧ s❡q✉❡♥❝❡ ✐s ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✳

❉❡✜♥✐t✐♦♥ ✶✳✶✿ ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡

❆♥ ❛r✐t❤♠❡t✐❝ ✭♦r ❧✐♥❡❛r✮ s❡q✉❡♥❝❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ ✇❤✐❝❤ ❡❛❝❤ ♥❡✇ t❡r♠ ✐s ❝❛❧❝✉❧❛t❡❞

❜② ❛❞❞✐♥❣ ❛ ❝♦♥st❛♥t ✈❛❧✉❡ t♦ t❤❡ ♣r❡✈✐♦✉s t❡r♠

❋♦r ❡①❛♠♣❧❡✱ 1, 2, 3, 4, 5, 6, ... ✐s ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ❜❡❝❛✉s❡ ②♦✉ ❛❞❞ ✶ t♦ t❤❡ ❝✉rr❡♥t t❡r♠ t♦ ❣❡t t❤❡

♥❡①t t❡r♠✿

✜rst t❡r♠✿

✶

s❡❝♦♥❞ t❡r♠✿ ✷❂✶✰✶

t❤✐r❞ t❡r♠✿

✸❂✷✰✶

✳✳✳

nth t❡r♠✿

n = (n − 1) + 1

❚❛❜❧❡ ✶✳✶

✶✳✶✳✷✳✶ ❈♦♠♠♦♥ ❉✐✛❡r❡♥❝❡ ✿

❋✐♥❞ t❤❡ ❝♦♥st❛♥t ✈❛❧✉❡ t❤❛t ✐s ❛❞❞❡❞ t♦ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡s ❛♥❞ ✇r✐t❡ ♦✉t t❤❡ ♥❡①t ✺ t❡r♠s✳

✶✳ 2, 6, 10, 14, 18, 22, ...

✷✳ −5, −3, −1, 1, 3, ...

✸✳ 1, 4, 7, 10, 13, 16, ...

✹✳ −1, 10, 21, 32, 43, 54, ...

✺✳ 3, 0, −3, −6, −9, −12, ...

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✵✷✴✶✳✶✴❃✳

✶✶

✶✷

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

✶✳✶✳✷✳✷ ●❡♥❡r❛❧ ❊q✉❛t✐♦♥ ❢♦r t❤❡ nth✲t❡r♠ ♦❢ ❛♥ ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡

▼♦r❡ ❢♦r♠❛❧❧②✱ t❤❡ ♥✉♠❜❡r ✇❡ st❛rt ♦✉t ✇✐t❤ ✐s ❝❛❧❧❡❞ a1 ✭t❤❡ ✜rst t❡r♠✮✱ ❛♥❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❡❛❝❤

s✉❝❝❡ss✐✈❡ t❡r♠ ✐s ❞❡♥♦t❡❞ ❞✱ ❝❛❧❧❡❞ t❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡✳

❚❤❡ ❣❡♥❡r❛❧ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ❧♦♦❦s ❧✐❦❡✿

a1

=

a1

a2

=

a1 + d

a3

=

a2 + d = (a1 + d) + d = a1 + 2d

✭✶✳✶✮

a4

=

a3 + d = (a1 + 2d) + d = a1 + 3d

...

an

=

a1 + d · (n − 1)

❚❤✉s✱ t❤❡ ❡q✉❛t✐♦♥ ❢♦r t❤❡ nth✲t❡r♠ ✇✐❧❧ ❜❡✿

an = a1 + d · (n − 1)

✭✶✳✷✮

●✐✈❡♥ a1 ❛♥❞ t❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡✱ d✱ t❤❡ ❡♥t✐r❡ s❡t ♦❢ ♥✉♠❜❡rs ❜❡❧♦♥❣✐♥❣ t♦ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ❝❛♥

❜❡ ❣❡♥❡r❛t❡❞✳

❉❡✜♥✐t✐♦♥ ✶✳✷✿ ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡

❆♥ ❛r✐t❤♠❡t✐❝ ✭♦r ❧✐♥❡❛r✮ s❡q✉❡♥❝❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ ✇❤✐❝❤ ❡❛❝❤ ♥❡✇ t❡r♠ ✐s ❝❛❧❝✉❧❛t❡❞

❜② ❛❞❞✐♥❣ ❛ ❝♦♥st❛♥t ✈❛❧✉❡ t♦ t❤❡ ♣r❡✈✐♦✉s t❡r♠✿

an = an−1 + d

✭✶✳✸✮

✇❤❡r❡

• an r❡♣r❡s❡♥ts t❤❡ ♥❡✇ t❡r♠✱ t❤❡ nth✲t❡r♠✱ t❤❛t ✐s ❝❛❧❝✉❧❛t❡❞❀

• an−1 r❡♣r❡s❡♥ts t❤❡ ♣r❡✈✐♦✉s t❡r♠✱ t❤❡ (n − 1)th✲t❡r♠❀

• d r❡♣r❡s❡♥ts s♦♠❡ ❝♦♥st❛♥t✳

t✐♣✿ ❚❡st ❢♦r ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡s

❆ s✐♠♣❧❡ t❡st ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ✐s t♦ ❝❤❡❝❦ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s ✐s ❝♦♥st❛♥t✿

a2 − a1 = a3 − a2 = an − an−1 = d

✭✶✳✹✮

❚❤✐s ✐s q✉✐t❡ ❛♥ ✐♠♣♦rt❛♥t ❡q✉❛t✐♦♥✱ ❛♥❞ ✐s t❤❡ ❞❡✜♥✐t✐✈❡ t❡st ❢♦r ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✳ ■❢ t❤✐s ❝♦♥❞✐t✐♦♥

❞♦❡s ♥♦t ❤♦❧❞✱ t❤❡ s❡q✉❡♥❝❡ ✐s ♥♦t ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✳

✶✳✶✳✷✳✷✳✶ P❧♦tt✐♥❣ ❛ ❣r❛♣❤ ♦❢ t❡r♠s ✐♥ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡

P❧♦tt✐♥❣ ❛ ❣r❛♣❤ ♦❢ t❤❡ t❡r♠s ♦❢ s❡q✉❡♥❝❡ s♦♠❡t✐♠❡s ❤❡❧♣s ✐♥ ❞❡t❡r♠✐♥✐♥❣ t❤❡ t②♣❡ ♦❢ s❡q✉❡♥❝❡ ✐♥✈♦❧✈❡❞✳ ❋♦r

❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✱ ♣❧♦tt✐♥❣ an ✈s✳ n r❡s✉❧ts ✐♥✿

❋✐❣✉r❡ ✶✳✶

✶✸

✶✳✶✳✸ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s

❉❡✜♥✐t✐♦♥ ✶✳✸✿ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s

❆ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐s ❛ s❡q✉❡♥❝❡ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♥✉♠❜❡r ✐♥ t❤❡ s❡q✉❡♥❝❡ ✐s ❡q✉❛❧ t♦ t❤❡ ♣r❡✈✐♦✉s

♥✉♠❜❡r ✐♥ t❤❡ s❡q✉❡♥❝❡✱ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ❝♦♥st❛♥t ♥✉♠❜❡r✳

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ r❛t✐♦ ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ ♥✉♠❜❡rs ✐♥ t❤❡ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐s ❛ ❝♦♥st❛♥t✳ ❲❡ ✇✐❧❧

❡①♣❧❛✐♥ ✇❤❛t ✇❡ ♠❡❛♥ ❜② r❛t✐♦ ❛❢t❡r ❧♦♦❦✐♥❣ ❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✳

✶✳✶✳✸✳✶ ❊①❛♠♣❧❡ ✲ ❆ ❋❧✉ ❊♣✐❞❡♠✐❝

✶✳✶✳✸✳✶✳✶ ❲❤❛t ✐s ✐♥✢✉❡♥③❛❄

■♥✢✉❡♥③❛ ✭❝♦♠♠♦♥❧② ❝❛❧❧❡❞ ✏t❤❡ ✢✉✑✮ ✐s ❝❛✉s❡❞ ❜② t❤❡ ✐♥✢✉❡♥③❛ ✈✐r✉s✱ ✇❤✐❝❤ ✐♥❢❡❝ts t❤❡ r❡s♣✐r❛t♦r② tr❛❝t

✭♥♦s❡✱ t❤r♦❛t✱ ❧✉♥❣s✮✳ ■t ❝❛♥ ❝❛✉s❡ ♠✐❧❞ t♦ s❡✈❡r❡ ✐❧❧♥❡ss t❤❛t ♠♦st ♦❢ ✉s ❣❡t ❞✉r✐♥❣ ✇✐♥t❡r t✐♠❡✳ ❚❤❡ ♠❛✐♥

✇❛② t❤❛t t❤❡ ✐♥✢✉❡♥③❛ ✈✐r✉s ✐s s♣r❡❛❞ ✐s ❢r♦♠ ♣❡rs♦♥ t♦ ♣❡rs♦♥ ✐♥ r❡s♣✐r❛t♦r② ❞r♦♣❧❡ts ♦❢ ❝♦✉❣❤s ❛♥❞ s♥❡❡③❡s✳

✭❚❤✐s ✐s ❝❛❧❧❡❞ ✏❞r♦♣❧❡t s♣r❡❛❞✑✳✮ ❚❤✐s ❝❛♥ ❤❛♣♣❡♥ ✇❤❡♥ ❞r♦♣❧❡ts ❢r♦♠ ❛ ❝♦✉❣❤ ♦r s♥❡❡③❡ ♦❢ ❛♥ ✐♥❢❡❝t❡❞

♣❡rs♦♥ ❛r❡ ♣r♦♣❡❧❧❡❞ ✭❣❡♥❡r❛❧❧②✱ ✉♣ t♦ ❛ ♠❡tr❡✮ t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❞❡♣♦s✐t❡❞ ♦♥ t❤❡ ♠♦✉t❤ ♦r ♥♦s❡ ♦❢ ♣❡♦♣❧❡

♥❡❛r❜②✳ ■t ✐s ❣♦♦❞ ♣r❛❝t✐s❡ t♦ ❝♦✈❡r ②♦✉r ♠♦✉t❤ ✇❤❡♥ ②♦✉ ❝♦✉❣❤ ♦r s♥❡❡③❡ s♦ ❛s ♥♦t t♦ ✐♥❢❡❝t ♦t❤❡rs ❛r♦✉♥❞

②♦✉ ✇❤❡♥ ②♦✉ ❤❛✈❡ t❤❡ ✢✉✳

❆ss✉♠❡ t❤❛t ②♦✉ ❤❛✈❡ t❤❡ ✢✉ ✈✐r✉s✱ ❛♥❞ ②♦✉ ❢♦r❣♦t t♦ ❝♦✈❡r ②♦✉r ♠♦✉t❤ ✇❤❡♥ t✇♦ ❢r✐❡♥❞s ❝❛♠❡ t♦ ✈✐s✐t

✇❤✐❧❡ ②♦✉ ✇❡r❡ s✐❝❦ ✐♥ ❜❡❞✳ ❚❤❡② ❧❡❛✈❡✱ ❛♥❞ t❤❡ ♥❡①t ❞❛② t❤❡② ❛❧s♦ ❤❛✈❡ t❤❡ ✢✉✳ ▲❡t✬s ❛ss✉♠❡ t❤❛t t❤❡② ✐♥

t✉r♥ s♣r❡❛❞ t❤❡ ✈✐r✉s t♦ t✇♦ ♦❢ t❤❡✐r ❢r✐❡♥❞s ❜② t❤❡ s❛♠❡ ❞r♦♣❧❡t s♣r❡❛❞ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛②✳ ❆ss✉♠✐♥❣ t❤✐s

♣❛tt❡r♥ ❝♦♥t✐♥✉❡s ❛♥❞ ❡❛❝❤ s✐❝❦ ♣❡rs♦♥ ✐♥❢❡❝ts ✷ ♦t❤❡r ❢r✐❡♥❞s✱ ✇❡ ❝❛♥ r❡♣r❡s❡♥t t❤❡s❡ ❡✈❡♥ts ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣

♠❛♥♥❡r✿

❋✐❣✉r❡ ✶✳✷✿ ❊❛❝❤ ♣❡rs♦♥ ✐♥❢❡❝ts t✇♦ ♠♦r❡ ♣❡♦♣❧❡ ✇✐t❤ t❤❡ ✢✉ ✈✐r✉s✳

❆❣❛✐♥ ✇❡ ❝❛♥ t❛❜✉❧❛t❡ t❤❡ ❡✈❡♥ts ❛♥❞ ❢♦r♠✉❧❛t❡ ❛♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❣❡♥❡r❛❧ ❝❛s❡✿

❉❛②✱ n ◆✉♠❜❡r ♦❢ ♥❡✇❧②✲✐♥❢❡❝t❡❞ ♣❡♦♣❧❡

✶

2 = 2

✷

4 = 2 × 2 = 2 × 21

✸

8 = 2 × 4 = 2 × 2 × 2 = 2 × 22

✹

16 = 2 × 8 = 2 × 2 × 2 × 2 = 2 × 23

✺

32 = 2 × 16 = 2 × 2 × 2 × 2 × 2 = 2 × 24

✳✳

✳

✳

✳✳

n

= 2 × 2 × 2 × 2 × ... × 2 = 2 × 2n−1

❚❛❜❧❡ ✶✳✷

❚❤❡ ❛❜♦✈❡ t❛❜❧❡ r❡♣r❡s❡♥ts t❤❡ ♥✉♠❜❡r ♦❢ ♥❡✇❧②✲✐♥❢❡❝t❡❞ ♣❡♦♣❧❡ ❛❢t❡r n ❞❛②s s✐♥❝❡ ②♦✉ ✜rst ✐♥❢❡❝t❡❞

②♦✉r ✷ ❢r✐❡♥❞s✳

✶✹

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

❨♦✉ s♥❡❡③❡ ❛♥❞ t❤❡ ✈✐r✉s ✐s ❝❛rr✐❡❞ ♦✈❡r t♦ ✷ ♣❡♦♣❧❡ ✇❤♦ st❛rt t❤❡ ❝❤❛✐♥ ✭a1 = 2✮✳ ❚❤❡ ♥❡①t ❞❛②✱ ❡❛❝❤

♦♥❡ t❤❡♥ ✐♥❢❡❝ts ✷ ♦❢ t❤❡✐r ❢r✐❡♥❞s✳ ◆♦✇ ✹ ♣❡♦♣❧❡ ❛r❡ ♥❡✇❧②✲✐♥❢❡❝t❡❞✳ ❊❛❝❤ ♦❢ t❤❡♠ ✐♥❢❡❝ts ✷ ♣❡♦♣❧❡ t❤❡ t❤✐r❞

❞❛②✱ ❛♥❞ ✽ ♣❡♦♣❧❡ ❛r❡ ✐♥❢❡❝t❡❞✱ ❛♥❞ s♦ ♦♥✳ ❚❤❡s❡ ❡✈❡♥ts ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✿

2; 4; 8; 16; 32; ...

✭✶✳✺✮

◆♦t❡ t❤❡ ❝♦♠♠♦♥ ❢❛❝t♦r ✭✷✮ ❜❡t✇❡❡♥ t❤❡ ❡✈❡♥ts✳ ❘❡❝❛❧❧ ❢r♦♠ t❤❡ ❧✐♥❡❛r ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ❤♦✇ t❤❡ ❝♦♠♠♦♥

❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❡r♠s ✇❡r❡ ❡st❛❜❧✐s❤❡❞✳ ■♥ t❤❡ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✇❡ ❝❛♥ ❞❡t❡r♠✐♥❡ t❤❡ ❝♦♠♠♦♥ r❛t✐♦✱

r✱ ❜②

a2

a3

=

= r

✭✶✳✻✮

a1

a2

❖r✱ ♠♦r❡ ❣❡♥❡r❛❧✱

an = r

✭✶✳✼✮

an−1

✶✳✶✳✸✳✶✳✷ ❈♦♠♠♦♥ ❋❛❝t♦r ♦❢ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡ ✿

❉❡t❡r♠✐♥❡ t❤❡ ❝♦♠♠♦♥ ❢❛❝t♦r ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡s✿

✶✳ 5, 10, 20, 40, 80, ...

✷✳ 1, 1, 1, ...

2

4

8

✸✳ 7, 28, 112, 448, ...

✹✳ 2, 6, 18, 54, ...

✺✳ −3, 30, −300, 3000, ...

✶✳✶✳✸✳✷ ●❡♥❡r❛❧ ❊q✉❛t✐♦♥ ❢♦r t❤❡ nth✲t❡r♠ ♦❢ ❛ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡

❋r♦♠ t❤❡ ❛❜♦✈❡ ❡①❛♠♣❧❡ ✇❡ ❦♥♦✇ a1 = 2 ❛♥❞ r = 2✱ ❛♥❞ ✇❡ ❤❛✈❡ s❡❡♥ ❢r♦♠ t❤❡ t❛❜❧❡ t❤❛t t❤❡ nth✲t❡r♠ ✐s

❣✐✈❡♥ ❜② an = 2 × 2n−1✳ ❚❤✉s✱ ✐♥ ❣❡♥❡r❛❧✱

an = a1 · rn−1

✭✶✳✽✮

✇❤❡r❡ a1 ✐s t❤❡ ✜rst t❡r♠ ❛♥❞ r ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♠♠♦♥ r❛t✐♦✳

❙♦✱ ✐❢ ✇❡ ✇❛♥t t♦ ❦♥♦✇ ❤♦✇ ♠❛♥② ♣❡♦♣❧❡ ❛r❡ ♥❡✇❧②✲✐♥❢❡❝t❡❞ ❛❢t❡r ✶✵ ❞❛②s✱ ✇❡ ♥❡❡❞ t♦ ✇♦r❦ ♦✉t a10✿

an

=

a1 · rn−1

a10

=

2 × 210−1

=

2 × 29

✭✶✳✾✮

=

2 × 512

=

1024

❚❤❛t ✐s✱ ❛❢t❡r ✶✵ ❞❛②s✱ t❤❡r❡ ❛r❡ ✶ ✵✷✹ ♥❡✇❧②✲✐♥❢❡❝t❡❞ ♣❡♦♣❧❡✳

✶✺

❖r✱ ❤♦✇ ♠❛♥② ❞❛②s ✇♦✉❧❞ ♣❛ss ❜❡❢♦r❡ ✶✻ ✸✽✹ ♣❡♦♣❧❡ ❜❡❝♦♠❡ ♥❡✇❧② ✐♥❢❡❝t❡❞ ✇✐t❤ t❤❡ ✢✉ ✈✐r✉s❄

an

=

a1 · rn−1

16 384

=

2 × 2n−1

16 384 ÷ 2

=

2n−1

8 192

=

2n−1

✭✶✳✶✵✮

213

=

2n−1

13

=

n − 1

n

=

14

❚❤❛t ✐s✱ ✶✹ ❞❛②s ♣❛ss ❜❡❢♦r❡ ✶✻ ✸✽✹ ♣❡♦♣❧❡ ❛r❡ ♥❡✇❧②✲✐♥❢❡❝t❡❞✳

✶✳✶✳✸✳✷✳✶ ●❡♥❡r❛❧ ❊q✉❛t✐♦♥ ♦❢ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡ ✿

❉❡t❡r♠✐♥❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡s✿

✶✳ 5, 10, 20, 40, 80, ...

✷✳ 1, 1, 1, ...

2

4

8

✸✳ 7, 28, 112, 448, ...

✹✳ 2, 6, 18, 54, ...

✺✳ −3, 30, −300, 3000, ...

✶✳✶✳✸✳✸ ❊①❡r❝✐s❡s

✶✳ ❲❤❛t ✐s t❤❡ ✐♠♣♦rt❛♥t ❝❤❛r❛❝t❡r✐st✐❝ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡❄

✷✳ ❲r✐t❡ ❞♦✇♥ ❤♦✇ ②♦✉ ✇♦✉❧❞ ❣♦ ❛❜♦✉t ✜♥❞✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ nth t❡r♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡❄

✸✳ ❆ s✐♥❣❧❡ sq✉❛r❡ ✐s ♠❛❞❡ ❢r♦♠ ✹ ♠❛t❝❤st✐❝❦s✳ ❚✇♦ sq✉❛r❡s ✐♥ ❛ r♦✇ ♥❡❡❞s ✼ ♠❛t❝❤st✐❝❦s ❛♥❞ ✸ sq✉❛r❡s

✐♥ ❛ r♦✇ ♥❡❡❞s ✶✵ ♠❛t❝❤st✐❝❦s✳ ❉❡t❡r♠✐♥❡✿

❛✳ t❤❡ ✜rst t❡r♠

❜✳ t❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡

❝✳ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❣❡♥❡r❛❧ t❡r♠

❞✳ ❤♦✇ ♠❛♥② ♠❛t❝❤st✐❝❦s ❛r❡ ✐♥ ❛ r♦✇ ♦❢ ✷✺ sq✉❛r❡s

❋✐❣✉r❡ ✶✳✸

✹✳ 5; x; y ✐s ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ❛♥❞ x; y; 81 ✐s ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✳ ❆❧❧ t❡r♠s ✐♥ t❤❡ s❡q✉❡♥❝❡s ❛r❡

✐♥t❡❣❡rs✳ ❈❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ x ❛♥❞ y✳

✶✳✶✳✹ ❘❡❝✉rs✐✈❡ ❋♦r♠✉❧❛❡ ❢♦r ❙❡q✉❡♥❝❡s

❲❤❡♥ ❞✐s❝✉ss✐♥❣ ❛r✐t❤♠❡t✐❝ ❛♥❞ q✉❛❞r❛t✐❝ s❡q✉❡♥❝❡s✱ ✇❡ ♥♦t✐❝❡❞ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✇♦ ❝♦♥s❡❝✉t✐✈❡

t❡r♠s ✐♥ t❤❡ s❡q✉❡♥❝❡ ❝♦✉❧❞ ❜❡ ✇r✐tt❡♥ ✐♥ ❛ ❣❡♥❡r❛❧ ✇❛②✳

✶✻

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

❋♦r ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✱ ✇❤❡r❡ ❛ ♥❡✇ t❡r♠ ✐s ❝❛❧❝✉❧❛t❡❞ ❜② t❛❦✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❡r♠ ❛♥❞ ❛❞❞✐♥❣ ❛

❝♦♥st❛♥t ✈❛❧✉❡✱ d✿

an = an−1 + d

✭✶✳✶✶✮

❚❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥ s✐♥❝❡ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ nth✲t❡r♠ ♦♥❧② ❜②

❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❡r♠ ✐♥ t❤❡ s❡q✉❡♥❝❡✳ ❈♦♠♣❛r❡ t❤✐s ✇✐t❤ ❡q✉❛t✐♦♥ ✭✶✳✷✮✱

an = a1 + d · (n − 1)

✭✶✳✶✷✮

✇❤❡r❡ ♦♥❡ ❝❛♥ ❞✐r❡❝t❧② ❝❛❧❝✉❧❛t❡ t❤❡ nth✲t❡r♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ✇✐t❤♦✉t ❦♥♦✇✐♥❣ ♣r❡✈✐♦✉s t❡r♠s✳

❋♦r q✉❛❞r❛t✐❝ s❡q✉❡♥❝❡s✱ ✇❡ ♥♦t✐❝❡❞ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s ✐s ❣✐✈❡♥ ❜② ✭✶✳✶✸✮✿

an − an−1 = D · (n − 2) + d

✭✶✳✶✸✮

❚❤❡r❡❢♦r❡✱ ✇❡ r❡✲✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥ ❛s

an = an−1 + D · (n − 2) + d

✭✶✳✶✹✮

✇❤✐❝❤ ✐s t❤❡♥ ❛ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥ ❢♦r ❛ q✉❛❞r❛t✐❝ s❡q✉❡♥❝❡ ✇✐t❤ ❝♦♠♠♦♥ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡✱ D✳

❯s✐♥❣ ✭✶✳✼✮✱ t❤❡ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥ ❢♦r ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐s✿

an = r · an−1

✭✶✳✶✺✮

❘❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥s ❛r❡ ❡①tr❡♠❡❧② ♣♦✇❡r❢✉❧✿ ②♦✉ ❝❛♥ ✇♦r❦ ♦✉t ❡✈❡r② t❡r♠ ✐♥ t❤❡ s❡r✐❡s ❥✉st ❜② ❦♥♦✇✐♥❣

♣r❡✈✐♦✉s t❡r♠s✳ ❆s ②♦✉ ❝❛♥ s❡❡ ❢r♦♠ t❤❡ ❡①❛♠♣❧❡s ❛❜♦✈❡✱ ✇♦r❦✐♥❣ ♦✉t an ✉s✐♥❣ t❤❡ ♣r❡✈✐♦✉s t❡r♠ an−1 ❝❛♥

❜❡ ❛ ♠✉❝❤ s✐♠♣❧❡r ❝♦♠♣✉t❛t✐♦♥ t❤❛♥ ✇♦r❦✐♥❣ ♦✉t an ❢r♦♠ s❝r❛t❝❤ ✉s✐♥❣ ❛ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛✳ ❚❤✐s ♠❡❛♥s t❤❛t

✉s✐♥❣ ❛ r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛ ✇❤❡♥ ✉s✐♥❣ ❛ ❝♦♠♣✉t❡r t♦ ✇♦r❦ ♦✉t ❛ s❡q✉❡♥❝❡ ✇♦✉❧❞ ♠❡❛♥ t❤❡ ❝♦♠♣✉t❡r ✇♦✉❧❞

✜♥✐s❤ ✐ts ❝❛❧❝✉❧❛t✐♦♥s s✐❣♥✐✜❝❛♥t❧② q✉✐❝❦❡r✳

✶✳✶✳✹✳✶ ❘❡❝✉rs✐✈❡ ❋♦r♠✉❧❛ ✿

❲r✐t❡ t❤❡ ✜rst ✺ t❡r♠s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡s✱ ❣✐✈❡♥ t❤❡✐r r❡❝✉rs✐✈❡ ❢♦r♠✉❧❛❡✿

✶✳ an = 2an−1 + 3, a1 = 1

✷✳ an = an−1, a1 = 11

✸✳ an = 2a2 , a

n−1

1 = 2

✶✳✶✳✹✳✷ ❚❤❡ ❋✐❜♦♥❛❝❝✐ ❙❡q✉❡♥❝❡

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡✿

0; 1; 1; 2; 3; 5; 8; 13; 21; 34; ...

✭✶✳✶✻✮

❚❤❡ ❛❜♦✈❡ s❡q✉❡♥❝❡ ✐s ❝❛❧❧❡❞ t❤❡ ❋✐❜♦♥❛❝❝✐ s❡q✉❡♥❝❡✳ ❊❛❝❤ ♥❡✇ t❡r♠ ✐s ❝❛❧❝✉❧❛t❡❞ ❜② ❛❞❞✐♥❣ t❤❡ ♣r❡✈✐♦✉s

t✇♦ t❡r♠s✳ ❍❡♥❝❡✱ ✇❡ ❝❛♥ ✇r✐t❡ ❞♦✇♥ t❤❡ r❡❝✉rs✐✈❡ ❡q✉❛t✐♦♥✿

a

✭✶✳✶✼✮

n = an−1 + an−2

✶✼

✶✳✷ ❙✐❣♠❛ ♥♦t❛t✐♦♥✱ ❋✐♥✐t❡ ✫ ■♥✜♥✐t❡ ❙❡r✐❡s✷

✶✳✷✳✶ ❙❡r✐❡s

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ s✐♠♣❧② ✇♦r❦ ♦♥ t❤❡ ❝♦♥❝❡♣t ♦❢ ❛❞❞✐♥❣ ✉♣ t❤❡ ♥✉♠❜❡rs ❜❡❧♦♥❣✐♥❣ t♦ ❛r✐t❤♠❡t✐❝ ❛♥❞

❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡s✳ ❲❡ ❝❛❧❧ t❤❡ s✉♠ ♦❢ ❛♥② s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ❛ s❡r✐❡s✳

✶✳✷✳✶✳✶ ❙♦♠❡ ❇❛s✐❝s

■❢ ✇❡ ❛❞❞ ✉♣ t❤❡ t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡✱ ✇❡ ♦❜t❛✐♥ ✇❤❛t ✐s ❝❛❧❧❡❞ ❛ s❡r✐❡s✳ ■❢ ✇❡ ♦♥❧② s✉♠ ❛ ✜♥✐t❡ ❛♠♦✉♥t

♦❢ t❡r♠s✱ ✇❡ ❣❡t ❛ ✜♥✐t❡ s❡r✐❡s✳ ❲❡ ✉s❡ t❤❡ s②♠❜♦❧ Sn t♦ ♠❡❛♥ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡

{a1; a2; a3; ...; an}✿

S

✭✶✳✶✽✮

n = a1 + a2 + a3 + ... + an

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs

1; 4; 9; 25; 36; 49; ...

✭✶✳✶✾✮

❛♥❞ ✇❡ ✇✐s❤ t♦ ✜♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✹ t❡r♠s✱ t❤❡♥ ✇❡ ✇r✐t❡

S4 = 1 + 4 + 9 + 25 = 39

✭✶✳✷✵✮

❚❤❡ ❛❜♦✈❡ ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ✜♥✐t❡ s❡r✐❡s s✐♥❝❡ ✇❡ ❛r❡ ♦♥❧② s✉♠♠✐♥❣ ✹ t❡r♠s✳

■❢ ✇❡ s✉♠ ✐♥✜♥✐t❡❧② ♠❛♥② t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡✱ ✇❡ ❣❡t ❛♥ ✐♥✜♥✐t❡ s❡r✐❡s✿

S∞ = a1 + a2 + a3 + ...

✭✶✳✷✶✮

✶✳✷✳✶✳✷ ❙✐❣♠❛ ◆♦t❛t✐♦♥

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ✐♥tr♦❞✉❝❡ ❛ ♥♦t❛t✐♦♥ t❤❛t ✇✐❧❧ ♠❛❦❡ ♦✉r ❧✐✈❡s ❛ ❧✐tt❧❡ ❡❛s✐❡r✳

❆ s✉♠ ♠❛② ❜❡ ✇r✐tt❡♥ ♦✉t ✉s✐♥❣ t❤❡ s✉♠♠❛t✐♦♥ s②♠❜♦❧

✳ ❚❤✐s s②♠❜♦❧ ✐s s✐❣♠❛✱ ✇❤✐❝❤ ✐s t❤❡ ❝❛♣✐t❛❧

❧❡tt❡r ✏❙✑ ✐♥ t❤❡ ●r❡❡❦ ❛❧♣❤❛❜❡t✳ ■t ✐♥❞✐❝❛t❡s t❤❛t ②♦✉ ♠✉st s✉♠ t❤❡ ❡①♣r❡ss✐♦♥ t♦ t❤❡ r✐❣❤t ♦❢ ✐t✿

n

ai = am + am+1 + ... + an−1 + an

✭✶✳✷✷✮

i=m

✇❤❡r❡

• i ✐s t❤❡ ✐♥❞❡① ♦❢ t❤❡ s✉♠❀

• m ✐s t❤❡ ❧♦✇❡r ❜♦✉♥❞ ✭♦r st❛rt ✐♥❞❡①✮✱ s❤♦✇♥ ❜❡❧♦✇ t❤❡ s✉♠♠❛t✐♦♥ s②♠❜♦❧❀

• n ✐s t❤❡ ✉♣♣❡r ❜♦✉♥❞ ✭♦r ❡♥❞ ✐♥❞❡①✮✱ s❤♦✇♥ ❛❜♦✈❡ t❤❡ s✉♠♠❛t✐♦♥ s②♠❜♦❧❀

• ai ❛r❡ t❤❡ t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡✳

❚❤❡ ✐♥❞❡① i ✐s ✐♥❝r❡❛s❡❞ ❢r♦♠ m t♦ n ✐♥ st❡♣s ♦❢ ✶✳

■❢ ✇❡ ❛r❡ s✉♠♠✐♥❣ ❢r♦♠ n = 1 ✭✇❤✐❝❤ ✐♠♣❧✐❡s s✉♠♠✐♥❣ ❢r♦♠ t❤❡ ✜rst t❡r♠ ✐♥ ❛ s❡q✉❡♥❝❡✮✱ t❤❡♥ ✇❡ ❝❛♥

✉s❡ ❡✐t❤❡r Sn✲ ♦r

✲♥♦t❛t✐♦♥ s✐♥❝❡ t❤❡② ♠❡❛♥ t❤❡ s❛♠❡ t❤✐♥❣✿

n

Sn =

ai = a1 + a2 + ... + an

✭✶✳✷✸✮

i=1

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✵✶✴✶✳✶✴❃✳

✶✽

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

❋♦r ❡①❛♠♣❧❡✱ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠✱

5

i

✭✶✳✷✹✮

i=1

✇❡ ❤❛✈❡ t♦ ❛❞❞ t♦❣❡t❤❡r ❛❧❧ t❤❡ t❡r♠s ✐♥ t❤❡ s❡q✉❡♥❝❡ ai = i ❢r♦♠ i = 1 ✉♣ ✉♥t✐❧ i = 5✿

5

i = 1 + 2 + 3 + 4 + 5 = 15

✭✶✳✷✺✮

i=1

✶✳✷✳✶✳✷✳✶ ❊①❛♠♣❧❡s

✶✳

6

2i

=

21 + 22 + 23 + 24 + 25 + 26

i=1

=

2 + 4 + 8 + 16 + 32 + 64

✭✶✳✷✻✮

=

126

✷✳

10

3xi = 3x3 + 3x4 + ... + 3x9 + 3x10

✭✶✳✷✼✮

i=3

❢♦r ❛♥② ✈❛❧✉❡ x✳

t✐♣✿ ◆♦t✐❝❡ t❤❛t ✐♥ t❤❡ s❡❝♦♥❞ ❡①❛♠♣❧❡ ✇❡ ✉s❡❞ t❤r❡❡ ❞♦ts ✭✳✳✳✮ t♦ ✐♥❞✐❝❛t❡ t❤❛t ✇❡ ❤❛❞ ❧❡❢t ♦✉t

♣❛rt ♦❢ t❤❡ s✉♠✳ ❲❡ ❞♦ t❤✐s t♦ ❛✈♦✐❞ ✇r✐t✐♥❣ ♦✉t ❡✈❡r② t❡r♠ ♦❢ ❛ s✉♠✳

✶✳✷✳✶✳✷✳✷ ❙♦♠❡ ❇❛s✐❝ ❘✉❧❡s ❢♦r ❙✐❣♠❛ ◆♦t❛t✐♦♥

✶✳ ●✐✈❡♥ t✇♦ s❡q✉❡♥❝❡s✱ ai ❛♥❞ bi✱

n

(a

a

b

✭✶✳✷✽✮

i=1

i + bi)

=

n

i=1

i +

n

i=1

i

✷✳ ❋♦r ❛♥② ❝♦♥st❛♥t c t❤❛t ✐s ♥♦t ❞❡♣❡♥❞❡♥t ♦♥ t❤❡ ✐♥❞❡① i✱

n

c · a

i=1

i

=

c · a1 + c · a2 + c · a3 + ... + c · an

=

c (a

✭✶✳✷✾✮

1 + a2 + a3 + ... + an)

=

c

n

a

i=1

i

✶✳✷✳✶✳✷✳✸ ❊①❡r❝✐s❡s

✶✳ ❲❤❛t ✐s

4

2❄

k=1

✷✳ ❉❡t❡r♠✐♥❡

3

i✳

i=−1

✸✳ ❊①♣❛♥❞

5

i✳

k=0

✹✳ ❈❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ a ✐❢✿

3

a · 2k−1 = 28

✭✶✳✸✵✮

k=1

✶✾

✶✳✷✳✷ ❋✐♥✐t❡ ❆r✐t❤♠❡t✐❝ ❙❡r✐❡s

❘❡♠❡♠❜❡r t❤❛t ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ✐s ❛ s❡t ♦❢ ♥✉♠❜❡rs✱ s✉❝❤ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ ❛♥② t❡r♠ ❛♥❞

t❤❡ ♣r❡✈✐♦✉s t❡r♠ ✐s ❛ ❝♦♥st❛♥t ♥✉♠❜❡r✱ d✱ ❝❛❧❧❡❞ t❤❡ ❝♦♥st❛♥t ❞✐✛❡r❡♥❝❡✿

an = a1 + d (n − 1)

✭✶✳✸✶✮

✇❤❡r❡

• n ✐s t❤❡ ✐♥❞❡① ♦❢ t❤❡ s❡q✉❡♥❝❡❀

• an ✐s t❤❡ nth✲t❡r♠ ♦❢ t❤❡ s❡q✉❡♥❝❡❀

• a1 ✐s t❤❡ ✜rst t❡r♠❀

• d ✐s t❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡✳

❲❤❡♥ ✇❡ s✉♠ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t❡r♠s ✐♥ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✱ ✇❡ ❣❡t ❛ ✜♥✐t❡ ❛r✐t❤♠❡t✐❝ s❡r✐❡s✳

❚❤❡ s✐♠♣❧❡st ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ✐s ✇❤❡♥ a1 = 1 ❛♥❞ d = 0 ✐♥ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠ ✭✶✳✸✶✮❀ ✐♥ ♦t❤❡r ✇♦r❞s ❛❧❧

t❤❡ t❡r♠s ✐♥ t❤❡ s❡q✉❡♥❝❡ ❛r❡ ✶✿

ai

=

a1 + d (i − 1)

=

1 + 0 · (i − 1)

✭✶✳✸✷✮

=

1

{ai} = {1; 1; 1; 1; 1; ...}

■❢ ✇❡ ✇✐s❤ t♦ s✉♠ t❤✐s s❡q✉❡♥❝❡ ❢r♦♠ i = 1 t♦ ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ✇❡ ✇♦✉❧❞ ✇r✐t❡

n

n

ai =

1 = 1 + 1 + 1 + ... + 1

(ntimes)

✭✶✳✸✸✮

i=1

i=1

❙✐♥❝❡ ❛❧❧ t❤❡ t❡r♠s ❛r❡ ❡q✉❛❧ t♦ ✶✱ ✐t ♠❡❛♥s t❤❛t ✐❢ ✇❡ s✉♠ t♦ n ✇❡ ✇✐❧❧ ❜❡ ❛❞❞✐♥❣ n✲♥✉♠❜❡r ♦❢ ✶✬s t♦❣❡t❤❡r✱

✇❤✐❝❤ ✐s s✐♠♣❧② ❡q✉❛❧ t♦ n✿

n

1 = n

✭✶✳✸✹✮

i=1

❆♥♦t❤❡r s✐♠♣❧❡ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ✐s ✇❤❡♥ a1 = 1 ❛♥❞ d = 1✱ ✇❤✐❝❤ ✐s t❤❡ s❡q✉❡♥❝❡ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✿

ai

=

a1 + d (i − 1)

=

1 + 1 · (i − 1)

✭✶✳✸✺✮

=

i

{ai} = {1; 2; 3; 4; 5; ...}

■❢ ✇❡ ✇✐s❤ t♦ s✉♠ t❤✐s s❡q✉❡♥❝❡ ❢r♦♠ i = 1 t♦ ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✱ ✇❡ ✇♦✉❧❞ ✇r✐t❡

n

i = 1 + 2 + 3 + ... + n

✭✶✳✸✻✮

i=1

❚❤✐s ✐s ❛♥ ❡q✉❛t✐♦♥ ✇✐t❤ ❛ ✈❡r② ✐♠♣♦rt❛♥t s♦❧✉t✐♦♥ ❛s ✐t ❣✐✈❡s t❤❡ ❛♥s✇❡r t♦ t❤❡ s✉♠ ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳

♥♦t❡✿ ▼❛t❤❡♠❛t✐❝✐❛♥✱ ❑❛r❧ ❋r✐❡❞r✐❝❤ ●❛✉ss✱ ❞✐s❝♦✈❡r❡❞ t❤✐s ♣r♦♦❢ ✇❤❡♥ ❤❡ ✇❛s ♦♥❧② ✽ ②❡❛rs ♦❧❞✳

❍✐s t❡❛❝❤❡r ❤❛❞ ❞❡❝✐❞❡❞ t♦ ❣✐✈❡ ❤✐s ❝❧❛ss ❛ ♣r♦❜❧❡♠ ✇❤✐❝❤ ✇♦✉❧❞ ❞✐str❛❝t t❤❡♠ ❢♦r t❤❡ ❡♥t✐r❡ ❞❛②

❜② ❛s❦✐♥❣ t❤❡♠ t♦ ❛❞❞ ❛❧❧ t❤❡ ♥✉♠❜❡rs ❢r♦♠ ✶ t♦ ✶✵✵✳ ❨♦✉♥❣ ❑❛r❧ r❡❛❧✐s❡❞ ❤♦✇ t♦ ❞♦ t❤✐s ❛❧♠♦st

✐♥st❛♥t❛♥❡♦✉s❧② ❛♥❞ s❤♦❝❦❡❞ t❤❡ t❡❛❝❤❡r ✇✐t❤ t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ✺✵✺✵✳

✷✵

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

❲❡ ✜rst ✇r✐t❡ Sn ❛s ❛ s✉♠ ♦❢ t❡r♠s ✐♥ ❛s❝❡♥❞✐♥❣ ♦r❞❡r✿

Sn = 1 + 2 + ... + (n − 1) + n

✭✶✳✸✼✮

❲❡ t❤❡♥ ✇r✐t❡ t❤❡ s❛♠❡ s✉♠ ❜✉t ✇✐t❤ t❤❡ t❡r♠s ✐♥ ❞❡s❝❡♥❞✐♥❣ ♦r❞❡r✿

Sn = n + (n − 1) + ... + 2 + 1

✭✶✳✸✽✮

❲❡ t❤❡♥ ❛❞❞ ❝♦rr❡s♣♦♥❞✐♥❣ ♣❛✐rs ♦❢ t❡r♠s ❢r♦♠ ❡q✉❛t✐♦♥s ✭✶✳✸✼✮ ❛♥❞ ✭✶✳✸✽✮✱ ❛♥❞ ✇❡ ✜♥❞ t❤❛t t❤❡ s✉♠ ❢♦r

❡❛❝❤ ♣❛✐r ✐s t❤❡ s❛♠❡✱ (n + 1)✿

2Sn = (n + 1) + (n + 1) + ... + (n + 1) + (n + 1)

✭✶✳✸✾✮

❲❡ t❤❡♥ ❤❛✈❡ n✲♥✉♠❜❡r ♦❢ (n + 1)✲t❡r♠s✱ ❛♥❞ ❜② s✐♠♣❧✐❢②✐♥❣ ✇❡ ❛rr✐✈❡ ❛t t❤❡ ✜♥❛❧ r❡s✉❧t✿

2Sn

=

n (n + 1)

✭✶✳✹✵✮

Sn

=

n (n + 1)

2

n

n

Sn =

i =

(n + 1)

✭✶✳✹✶✮

2

i=1

◆♦t❡ t❤❛t t❤✐s ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ q✉❛❞r❛t✐❝ s❡q✉❡♥❝❡✳

✶✳✷✳✷✳✶ ●❡♥❡r❛❧ ❋♦r♠✉❧❛ ❢♦r ❛ ❋✐♥✐t❡ ❆r✐t❤♠❡t✐❝ ❙❡r✐❡s

■❢ ✇❡ ✇✐s❤ t♦ s✉♠ ❛♥② ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✱ t❤❡r❡ ✐s ♥♦ ♥❡❡❞ t♦ ✇♦r❦ ✐t ♦✉t t❡r♠✲❢♦r✲t❡r♠✳ ❲❡ ✇✐❧❧ ♥♦✇

❞❡t❡r♠✐♥❡ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ t♦ ❡✈❛❧✉❛t❡ ❛ ✜♥✐t❡ ❛r✐t❤♠❡t✐❝ s❡r✐❡s✳ ❲❡ st❛rt ✇✐t❤ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦r

❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ❛♥❞ s✉♠ ✐t ❢r♦♠ i = 1 t♦ ❛♥② ♣♦s✐t✐✈❡ ✐♥t❡❣❡r n✿

n

a

[a

i=1

i

=

n

i=1

1 + d (i − 1)]

=

n

(a

i=1

1 + di − d)

=

n

[(a

i=1

1 − d) + di]

=

n

(a

(di)

i=1

1 − d) +

n

i=1

=

n

(a

i

✭✶✳✹✷✮

i=1

1 − d) + d

n

i=1

=

(a1 − d) n + dn (n + 1)

2

=

n (2a

2

1 − 2d + dn + d)

=

n (2a

2

1 + dn − d)

=

n [ 2a

2

1 + d (n − 1)]

❙♦✱ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦r ❞❡t❡r♠✐♥✐♥❣ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s ❣✐✈❡♥ ❜②

n

n

Sn =

[a1 + d (i − 1)] =

[2a1 + d (n − 1)]

✭✶✳✹✸✮

2

i=1

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ✇✐s❤ t♦ ❦♥♦✇ t❤❡ s❡r✐❡s S20 ❢♦r t❤❡ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ai = 3 + 7 (i − 1)✱ ✇❡ ❝♦✉❧❞ ❡✐t❤❡r

✷✶

❝❛❧❝✉❧❛t❡ ❡❛❝❤ t❡r♠ ✐♥❞✐✈✐❞✉❛❧❧② ❛♥❞ s✉♠ t❤❡♠✿

S20

=

20

[3 + 7 (i − 1)]

i=1

=

3 + 10 + 17 + 24 + 31 + 38 + 45 + 52+

59 + 66 + 73 + 80 + 87 + 94 + 101+

✭✶✳✹✹✮

108 + 115 + 122 + 129 + 136

=

1390

♦r✱ ♠♦r❡ s❡♥s✐❜❧②✱ ✇❡ ❝♦✉❧❞ ✉s❡ ❡q✉❛t✐♦♥ ✭✶✳✹✸✮ ♥♦t✐♥❣ t❤❛t a1 = 3✱ d = 7 ❛♥❞ n = 20 s♦ t❤❛t

S20

=

20

[3 + 7 (i − 1)]

i=1

=

20 [2 · 3 + 7 (20 − 1)]

✭✶✳✹✺✮

2

=

1390

❚❤✐s ❡①❛♠♣❧❡ ❞❡♠♦♥str❛t❡s ❤♦✇ ✉s❡❢✉❧ ❡q✉❛t✐♦♥ ✭✶✳✹✸✮ ✐s✳

✶✳✷✳✷✳✷ ❊①❡r❝✐s❡s

✶✳ ❚❤❡ s✉♠ t♦ n t❡r♠s ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s Sn = n (7n + 15)✳

2

❛✳ ❍♦✇ ♠❛♥② t❡r♠s ♦❢ t❤❡ s❡r✐❡s ♠✉st ❜❡ ❛❞❞❡❞ t♦ ❣✐✈❡ ❛ s✉♠ ♦❢ ✹✷✺❄

❜✳ ❉❡t❡r♠✐♥❡ t❤❡ ✻t❤ t❡r♠ ♦❢ t❤❡ s❡r✐❡s✳

✷✳ ❚❤❡ s✉♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s ✶✵✵ t✐♠❡s ✐ts ✜rst t❡r♠✱ ✇❤✐❧❡ t❤❡ ❧❛st t❡r♠ ✐s ✾ t✐♠❡s t❤❡ ✜rst t❡r♠✳

❈❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ t❡r♠s ✐♥ t❤❡ s❡r✐❡s ✐❢ t❤❡ ✜rst t❡r♠ ✐s ♥♦t ❡q✉❛❧ t♦ ③❡r♦✳

✸✳ ❚❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s ✸✳ ❈❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ n ❢♦r ✇❤✐❝❤ t❤❡ nth t❡r♠

♦❢ t❤❡ s❡r✐❡s ✐s ✾✸✱ ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s ✐s ✾✼✺✳

✹✳ ❚❤❡ s✉♠ ♦❢ n t❡r♠s ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s 5n2 − 11n ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢ n✳ ❉❡t❡r♠✐♥❡ t❤❡ ❝♦♠♠♦♥

❞✐✛❡r❡♥❝❡✳

✺✳ ❚❤❡ s✉♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s ✶✵✵ t✐♠❡s t❤❡ ✈❛❧✉❡ ♦❢ ✐ts ✜rst t❡r♠✱ ✇❤✐❧❡ t❤❡ ❧❛st t❡r♠ ✐s ✾ t✐♠❡s

t❤❡ ✜rst t❡r♠✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ t❡r♠s ✐♥ t❤❡ s❡r✐❡s ✐❢ t❤❡ ✜rst t❡r♠ ✐s ♥♦t ❡q✉❛❧ t♦ ③❡r♦✳

✻✳ ❚❤❡ t❤✐r❞ t❡r♠ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡ ✐s ✲✼ ❛♥❞ t❤❡ ✼t❤ t❡r♠ ✐s ✾✳ ❉❡t❡r♠✐♥❡ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst

✺✶ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✳

✼✳ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ ♦❢ t❤❡ ❛r✐t❤♠❡t✐❝ s❡r✐❡s 4 + 7 + 10 + · · · + 901✳

✽✳ ❚❤❡ ❝♦♠♠♦♥ ❞✐✛❡r❡♥❝❡ ♦❢ ❛♥ ❛r✐t❤♠❡t✐❝ s❡r✐❡s ✐s ✸✳ ❈❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡s ♦❢ n ❢♦r ✇❤✐❝❤ t❤❡ nth t❡r♠

♦❢ t❤❡ s❡r✐❡s ✐s ✾✸ ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s ✐s ✾✼✺✳

✶✳✷✳✸ ❋✐♥✐t❡ ❙q✉❛r❡❞ ❙❡r✐❡s

❲❤❡♥ ✇❡ s✉♠ ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ t❡r♠s ✐♥ ❛ q✉❛❞r❛t✐❝ s❡q✉❡♥❝❡✱ ✇❡ ❣❡t ❛ ✜♥✐t❡ q✉❛❞r❛t✐❝ s❡r✐❡s✳ ❚❤❡ ❣❡♥❡r❛❧

❢♦r♠ ♦❢ ❛ q✉❛❞r❛t✐❝ s❡r✐❡s ✐s q✉✐t❡ ❝♦♠♣❧✐❝❛t❡❞✱ s♦ ✇❡ ✇✐❧❧ ♦♥❧② ❧♦♦❦ ❛t t❤❡ s✐♠♣❧❡ ❝❛s❡ ✇❤❡♥ D = 2 ❛♥❞

d = (a2 − a1) = 3✱ ✇❤❡r❡ D ✐s t❤❡ ❝♦♠♠♦♥ s❡❝♦♥❞ ❞✐✛❡r❡♥❝❡ ❛♥❞ d ✐s t❤❡ ✜♥✐t❡ ❞✐✛❡r❡♥❝❡✳ ❚❤✐s ✐s t❤❡

s❡q✉❡♥❝❡ ♦❢ sq✉❛r❡s ♦❢ t❤❡ ✐♥t❡❣❡rs✿

ai

=

i2

a

✭✶✳✹✻✮

i

=

12; 22; 32; 42; 52; 62; ...

=

1; 4; 9; 16; 25; 36; ...

✷✷

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

■❢ ✇❡ ✇✐s❤ t♦ s✉♠ t❤✐s s❡q✉❡♥❝❡ ❛♥❞ ❝r❡❛t❡ ❛ s❡r✐❡s✱ t❤❡♥ ✇❡ ✇r✐t❡

n

Sn =

i 2 = 1 + 4 + 9 + ... + n2

✭✶✳✹✼✮

i=1

✇❤✐❝❤ ❝❛♥ ❜❡ ✇r✐tt❡♥✱ ✐♥ ❣❡♥❡r❛❧✱ ❛s

Sn =

n

i2 = n (2n + 1) (n + 1)

✭✶✳✹✽✮

i=1

6

❚❤❡ ♣r♦♦❢ ❢♦r ❡q✉❛t✐♦♥ ✭✶✳✹✽✮ ❝❛♥ ❜❡ ❢♦✉♥❞ ✉♥❞❡r t❤❡ ❆❞✈❛♥❝❡❞ ❜❧♦❝❦ t❤❛t ❢♦❧❧♦✇s✿

✶✳✷✳✸✳✶ ❉❡r✐✈❛t✐♦♥ ♦❢ t❤❡ ❋✐♥✐t❡ ❙q✉❛r❡❞ ❙❡r✐❡s

❲❡ ✇✐❧❧ ♥♦✇ ♣r♦✈❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ✜♥✐t❡ sq✉❛r❡❞ s❡r✐❡s✿

n

Sn =

i 2 = 1 + 4 + 9 + ... + n2

✭✶✳✹✾✮

i=1

❲❡ st❛rt ♦✛ ✇✐t❤ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ (k + 1)3✳

(k + 1)3

=

k3 + 3k2 + 3k + 1

✭✶✳✺✵✮

(k + 1)3 − k3

=

3k2 + 3k + 1

k = 1

:

23 − 13 = 3(1)2 + 3 (1) + 1

k = 2

:

33 − 23 = 3(2)2 + 3 (2) + 1

✭✶✳✺✶✮

k = 3

:

43 − 33 = 3(3)2 + 3 (3) + 1

✳✳✳

k = n

:

(n + 1)3 − n3 = 3n2 + 3n + 1

■❢ ✇❡ ❛❞❞ ❛❧❧ t❤❡ t❡r♠s ♦♥ t❤❡ r✐❣❤t ❛♥❞ ❧❡❢t✱ ✇❡ ❛rr✐✈❡ ❛t

(n + 1)3 − 1

=

n

3i2 + 3i + 1

i=1

n3 + 3n2 + 3n + 1 − 1

=

3

n

i2 + 3

n

i +

n

1

i=1

i=1

i=1

n3 + 3n2 + 3n

=

3

n

i2 + 3n (n + 1) + n

i=1

2

n

i2

=

1

n3 + 3n2 + 3n − 3n (n + 1) − n

✭✶✳✺✷✮

i=1

3

2

=

1

n3 + 3n2 + 3n − 3 n2 − 3 n − n

3

2

2

=

1

n3 + 3 n2 + 1 n

3

2

2

=

n

2n2 + 3n + 1

6

✷✸

❚❤❡r❡❢♦r❡✱

n

i2 = n (2n + 1) (n + 1)

✭✶✳✺✸✮

i=1

6

✶✳✷✳✹ ❋✐♥✐t❡ ●❡♦♠❡tr✐❝ ❙❡r✐❡s

❲❤❡♥ ✇❡ s✉♠ ❛ ❦♥♦✇♥ ♥✉♠❜❡r ♦❢ t❡r♠s ✐♥ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✱ ✇❡ ❣❡t ❛ ✜♥✐t❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✳ ❲❡ ❝❛♥

✇r✐t❡ ♦✉t ❡❛❝❤ t❡r♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐♥ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠✿

an = a1 · rn−1

✭✶✳✺✹✮

✇❤❡r❡

• n ✐s t❤❡ ✐♥❞❡① ♦❢ t❤❡ s❡q✉❡♥❝❡❀

• an ✐s t❤❡ nth✲t❡r♠ ♦❢ t❤❡ s❡q✉❡♥❝❡❀

• a1 ✐s t❤❡ ✜rst t❡r♠❀

• r ✐s t❤❡ ❝♦♠♠♦♥ r❛t✐♦ ✭t❤❡ r❛t✐♦ ♦❢ ❛♥② t❡r♠ t♦ t❤❡ ♣r❡✈✐♦✉s t❡r♠✮✳

❇② s✐♠♣❧② ❛❞❞✐♥❣ t♦❣❡t❤❡r t❤❡ ✜rst n t❡r♠s✱ ✇❡ ❛r❡ ❛❝t✉❛❧❧② ✇r✐t✐♥❣ ♦✉t t❤❡ s❡r✐❡s

Sn = a1 + a1r + a1r2 + ... + a1rn−2 + a1rn−1

✭✶✳✺✺✮

❲❡ ♠❛② ♠✉❧t✐♣❧② t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❜② r ♦♥ ❜♦t❤ s✐❞❡s✱ ❣✐✈✐♥❣ ✉s

rSn = a1r + a1r2 + a1r3 + ... + a1rn−1 + a1rn

✭✶✳✺✻✮

❨♦✉ ♠❛② ♥♦t✐❝❡ t❤❛t ❛❧❧ t❤❡ t❡r♠s ♦♥ t❤❡ r✐❣❤t s✐❞❡ ♦❢ ✭✶✳✺✺✮ ❛♥❞ ✭✶✳✺✻✮ ❛r❡ t❤❡ s❛♠❡✱ ❡①❝❡♣t t❤❡ ✜rst ❛♥❞

❧❛st t❡r♠s✳ ■❢ ✇❡ s✉❜tr❛❝t ✭✶✳✺✺✮ ❢r♦♠ ✭✶✳✺✻✮✱ ✇❡ ❛r❡ ❧❡❢t ✇✐t❤ ❥✉st

rSn − Sn = a1rn − a1

✭✶✳✺✼✮

Sn (r − 1) = a1 (rn − 1)

❉✐✈✐❞✐♥❣ ❜② (r − 1) ♦♥ ❜♦t❤ s✐❞❡s✱ ✇❡ ❛rr✐✈❡ ❛t t❤❡ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿

n

a1 (rn − 1)

Sn =

a1 · ri−1 =

✭✶✳✺✽✮

r − 1

i=1

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✈✐❞❡♦ s✉♠♠❛r✐s❡s ✇❤❛t ②♦✉ ❤❛✈❡ ❧❡❛r♥t s♦ ❢❛r ❛❜♦✉t s❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s✿

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ s❡r✐❡s ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❱❣❱❏r❙❏①❦❉❦✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✶✳✹

✷✹

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

✶✳✷✳✹✳✶ ❊①❡r❝✐s❡s

✶✳ Pr♦✈❡ t❤❛t

a (1 − rn)

a + ar + ar2 + ... + arn−1 =

✭✶✳✺✾✮

(1 − r)

✷✳ ❋✐♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✶✶ t❡r♠s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s 6 + 3 + 3 + 3 + ...

2

4

✸✳ ❙❤♦✇ t❤❛t t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s

n−1

1

54 + 18 + 6 + ... + 5

✭✶✳✻✵✮

3

✐s ❣✐✈❡♥ ❜② 81 − 34−n✳

✹✳ ❚❤❡ ❡✐❣❤t❤ t❡r♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐s ✻✹✵✳ ❚❤❡ t❤✐r❞ t❡r♠ ✐s ✷✵✳ ❋✐♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✼

t❡r♠s✳

✺✳ ❙♦❧✈❡ ❢♦r

t

n✿

n

8 1

= 15 3 ✳

t=1

2

4

✻✳ ❚❤❡ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst t❤r❡❡ t❡r♠s ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✹t❤✲✱ ✺t❤✲

❛♥❞ ✻t❤✲t❡r♠s ♦❢ t❤❡ s❛♠❡ s❡r✐❡s ✐s 8 : 27✳ ❉❡t❡r♠✐♥❡ t❤❡ ❝♦♠♠♦♥ r❛t✐♦ ❛♥❞ t❤❡ ✜rst ✷ t❡r♠s ✐❢ t❤❡

t❤✐r❞ t❡r♠ ✐s ✽✳

✼✳ ●✐✈❡♥ t❤❡ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ 1; − 3; 9; · · · ❞❡t❡r♠✐♥❡✿

❛✳ ❚❤❡ ✽th t❡r♠ ♦❢ t❤❡ s❡q✉❡♥❝❡

❜✳ ❚❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✽ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✳

✽✳ ❉❡t❡r♠✐♥❡✿

4

3 · 2n−1

✭✶✳✻✶✮

n=1

✶✳✷✳✺ ■♥✜♥✐t❡ ❙❡r✐❡s

❚❤✉s ❢❛r ✇❡ ❤❛✈❡ ❜❡❡♥ ✇♦r❦✐♥❣ ♦♥❧② ✇✐t❤ ✜♥✐t❡ s✉♠s✱ ♠❡❛♥✐♥❣ t❤❛t ✇❤❡♥❡✈❡r ✇❡ ❞❡t❡r♠✐♥❡❞ t❤❡ s✉♠ ♦❢ ❛

s❡r✐❡s✱ ✇❡ ♦♥❧② ❝♦♥s✐❞❡r❡❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ✇❡

❛❞❞ ✐♥✜♥✐t❡❧② ♠❛♥② t❡r♠s t♦❣❡t❤❡r✳ ❨♦✉ ♠✐❣❤t t❤✐♥❦ t❤❛t t❤✐s ✐s ❛ s✐❧❧② q✉❡st✐♦♥ ✲ s✉r❡❧② t❤❡ ❛♥s✇❡r ✇✐❧❧ ❜❡

∞ ✇❤❡♥ ♦♥❡ s✉♠s ✐♥✜♥✐t❡❧② ♠❛♥② ♥✉♠❜❡rs✱ ♥♦ ♠❛tt❡r ❤♦✇ s♠❛❧❧ t❤❡② ❛r❡❄ ❚❤❡ s✉r♣r✐s✐♥❣ ❛♥s✇❡r ✐s t❤❛t

✇❤✐❧❡ ✐♥ s♦♠❡ ❝❛s❡s ♦♥❡ ✇✐❧❧ r❡❛❝❤ ∞ ✭❧✐❦❡ ✇❤❡♥ ②♦✉ tr② t♦ ❛❞❞ ❛❧❧ t❤❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs t♦❣❡t❤❡r✮✱ t❤❡r❡ ❛r❡

s♦♠❡ ❝❛s❡s ♦♥❡ ✇✐❧❧ ❣❡t ❛ ✜♥✐t❡ ❛♥s✇❡r✳ ■❢ ②♦✉ ❞♦♥✬t ❜❡❧✐❡✈❡ t❤✐s✱ tr② ❞♦✐♥❣ t❤❡ ❢♦❧❧♦✇✐♥❣ s✉♠✱ ❛ ❣❡♦♠❡tr✐❝

s❡r✐❡s✱ ♦♥ ②♦✉r ❝❛❧❝✉❧❛t♦r ♦r ❝♦♠♣✉t❡r✿

1

1

1

1

1

+

+

+

+

+ ...

✭✶✳✻✷✮

2

4

8

16

32

❨♦✉ ♠✐❣❤t t❤✐♥❦ t❤❛t ✐❢ ②♦✉ ❦❡❡♣ ❛❞❞✐♥❣ ♠♦r❡ ❛♥❞ ♠♦r❡ t❡r♠s ②♦✉ ✇✐❧❧ ❡✈❡♥t✉❛❧❧② ❣❡t ❧❛r❣❡r ❛♥❞ ❧❛r❣❡r

♥✉♠❜❡rs✱ ❜✉t ✐♥ ❢❛❝t ②♦✉ ✇♦♥✬t ❡✈❡♥ ❣❡t ♣❛st ✶ ✲ tr② ✐t ❛♥❞ s❡❡ ❢♦r ②♦✉rs❡❧❢✦

❲❡ ❞❡♥♦t❡ t❤❡ s✉♠ ♦❢ ❛♥ ✐♥✜♥✐t❡ ♥✉♠❜❡r ♦❢ t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡ ❜②

∞

S

✭✶✳✻✸✮

∞ =

a

i=1

i

❲❤❡♥ ✇❡ s✉♠ t❤❡ t❡r♠s ♦❢ ❛ s❡r✐❡s✱ ❛♥❞ t❤❡ ❛♥s✇❡r ✇❡ ❣❡t ❛❢t❡r ❡❛❝❤ s✉♠♠❛t✐♦♥ ❣❡ts ❝❧♦s❡r ❛♥❞ ❝❧♦s❡r t♦

s♦♠❡ ♥✉♠❜❡r✱ ✇❡ s❛② t❤❛t t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳ ■❢ ❛ s❡r✐❡s ❞♦❡s ♥♦t ❝♦♥✈❡r❣❡✱ t❤❡♥ ✇❡ s❛② t❤❛t ✐t ❞✐✈❡r❣❡s✳

✶✳✷✳✺✳✶ ■♥✜♥✐t❡ ●❡♦♠❡tr✐❝ ❙❡r✐❡s

❚❤❡r❡ ✐s ❛ s✐♠♣❧❡ t❡st ❢♦r ❦♥♦✇✐♥❣ ✐♥st❛♥t❧② ✇❤✐❝❤ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❝♦♥✈❡r❣❡s ❛♥❞ ✇❤✐❝❤ ❞✐✈❡r❣❡s✳ ❲❤❡♥ r✱

t❤❡ ❝♦♠♠♦♥ r❛t✐♦✱ ✐s str✐❝t❧② ❜❡t✇❡❡♥ ✲✶ ❛♥❞ ✶✱ ✐✳❡✳ −1 < r < 1✱ t❤❡ ✐♥✜♥✐t❡ s❡r✐❡s ✇✐❧❧ ❝♦♥✈❡r❣❡✱ ♦t❤❡r✇✐s❡

✐t ✇✐❧❧ ❞✐✈❡r❣❡✳ ❚❤❡r❡ ✐s ❛❧s♦ ❛ ❢♦r♠✉❧❛ ❢♦r ✇♦r❦✐♥❣ ♦✉t t❤❡ ✈❛❧✉❡ t♦ ✇❤✐❝❤ t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳

✷✺

▲❡t✬s st❛rt ♦✛ ✇✐t❤ ❢♦r♠✉❧❛ ✭✶✳✺✽✮ ❢♦r t❤❡ ✜♥✐t❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿

n

a1 (rn − 1)

Sn =

a1 · ri−1 =

✭✶✳✻✹✮

r − 1

i=1

◆♦✇ ✇❡ ✇✐❧❧ ✐♥✈❡st✐❣❛t❡ t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ rn ❢♦r −1 < r < 1 ❛s ♥ ❜❡❝♦♠❡s ❧❛r❣❡r✳

❚❛❦❡ r = 1✿

2

1

n = 1

:

rn = r1 = 1

= 1

2

2

2

n = 2

:

rn = r2 = 1

= 1 · 1 = 1 < 1

✭✶✳✻✺✮

2

2

2

4

2

3

n = 3

:

rn = r3 = 1

= 1 · 1 · 1 = 1 < 1

2

2

2

2

8

4

❙✐♥❝❡ r ✐s ✐♥ t❤❡ r❛♥❣❡ −1 < r < 1✱ ✇❡ s❡❡ t❤❛t rn ❣❡ts ❝❧♦s❡r t♦ ✵ ❛s ♥ ❣❡ts ❧❛r❣❡r✳

❚❤❡r❡❢♦r❡✱

Sn

=

a1 (rn−1)

r−1

S∞

=

a1 (0−1)

for − 1 < r < 1

r−1

✭✶✳✻✻✮

=

−a1

r−1

=

a1

1−r

❚❤❡ s✉♠ ♦❢ ❛♥ ✐♥✜♥✐t❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✐s ❣✐✈❡♥ ❜② t❤❡ ❢♦r♠✉❧❛

∞

S∞ =

a

for

− 1 < r < 1

✭✶✳✻✼✮

i=1

1ri−1 =

a1

1−r

✇❤❡r❡ a1 ✐s t❤❡ ✜rst t❡r♠ ♦❢ t❤❡ s❡r✐❡s ❛♥❞ r ✐s t❤❡ ❝♦♠♠♦♥ r❛t✐♦✳

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ s❡r✐❡s ✲ ✷

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❯❴✽●❘▲❏♣❧❩❣✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✶✳✺

✶✳✷✳✺✳✷ ❊①❡r❝✐s❡s

✶✳ ❲❤❛t ❞♦❡s 2 n ❛♣♣r♦❛❝❤ ❛s n t❡♥❞s t♦✇❛r❞s ∞❄

5

✷✳ ●✐✈❡♥ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿

2 · (5)5 + 2 · (5)4 + 2 · (5)3 + ...

✭✶✳✻✽✮

❛✳ ❙❤♦✇ t❤❛t t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s

❜✳ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ t♦ ✐♥✜♥✐t② ♦❢ t❤❡ s❡r✐❡s

❝✳ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✽ t❡r♠s ♦❢ t❤❡ s❡r✐❡s✱ ❝♦rr❡❝t t♦ t✇♦ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳

✷✻

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

❞✳ ❉❡t❡r♠✐♥❡

∞

2 · 56−n ❝♦rr❡❝t t♦ t✇♦ ❞❡❝✐♠❛❧ ♣❧❛❝❡s ✉s✐♥❣ ♣r❡✈✐♦✉s❧② ❝❛❧❝✉❧❛t❡❞ r❡s✉❧ts✳

n=9

✸✳ ❋✐♥❞ t❤❡ s✉♠ t♦ ✐♥✜♥✐t② ♦❢ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s 3 + 1 + 1 + 1 + ...

3

9

✹✳ ❉❡t❡r♠✐♥❡ ❢♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ x✱ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s

2

2

2 +

(x + 1) +

(x + 1)2 + ...

✭✶✳✻✾✮

3

9

✇✐❧❧ ❝♦♥✈❡r❣❡✳

✺✳ ❚❤❡ s✉♠ t♦ ✐♥✜♥✐t② ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✇✐t❤ ♣♦s✐t✐✈❡ t❡r♠s ✐s 41 ❛♥❞ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst t✇♦ t❡r♠s

6

✐s 22✳ ❋✐♥❞ a✱ t❤❡ ✜rst t❡r♠✱ ❛♥❞ r✱ t❤❡ ❝♦♠♠♦♥ r❛t✐♦ ❜❡t✇❡❡♥ ❝♦♥s❡❝✉t✐✈❡ t❡r♠s✳

3

✶✳✷✳✻ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ ■s 1 + 2 + 3 + 4 + ... ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ✜♥✐t❡ s❡r✐❡s ♦r ❛♥ ✐♥✜♥✐t❡ s❡r✐❡s❄

✷✳ ❈❛❧❝✉❧❛t❡

6

k+2

1

3

✭✶✳✼✵✮

3

k=2

✸✳ ■❢ x + 1❀ x − 1❀ 2x − 5 ❛r❡ t❤❡ ✜rst ✸ t❡r♠s ♦❢ ❛ ❝♦♥✈❡r❣❡♥t ❣❡♦♠❡tr✐❝ s❡r✐❡s✱ ❝❛❧❝✉❧❛t❡ t❤❡✿

❛✳ ❱❛❧✉❡ ♦❢ x✳

❜✳ ❙✉♠ t♦ ✐♥✜♥✐t② ♦❢ t❤❡ s❡r✐❡s✳

✹✳ ❲r✐t❡ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✷✵ t❡r♠s ♦❢ t❤❡ s❡r✐❡s 6 + 3 + 3 + 3 + ... ✐♥

✱✲♥♦t❛t✐♦♥✳

2

4

✺✳ ●✐✈❡♥ t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿ 2 · 55 + 2 · 54 + 2 · 53 + ...

❛✳ ❙❤♦✇ t❤❛t t❤❡ s❡r✐❡s ❝♦♥✈❡r❣❡s✳

❜✳ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ✽ t❡r♠s ♦❢ t❤❡ s❡r✐❡s✱ ❝♦rr❡❝t t♦ ❚❲❖ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳

❝✳ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ t♦ ✐♥✜♥✐t② ♦❢ t❤❡ s❡r✐❡s✳

❞✳ ❯s❡ ②♦✉r ❛♥s✇❡r t♦ ❧✐st✱ ♣✳ ✷✻ ❛❜♦✈❡ t♦ ❞❡t❡r♠✐♥❡

∞

2 · 5(6−n)

✭✶✳✼✶✮

n=9

❝♦rr❡❝t t♦ ❚❲❖ ❞❡❝✐♠❛❧ ♣❧❛❝❡s✳

✻✳ ❋♦r t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✱

n−1

1

54 + 18 + 6 + ... + 5

✭✶✳✼✷✮

3

❝❛❧❝✉❧❛t❡ t❤❡ s♠❛❧❧❡st ✈❛❧✉❡ ♦❢ n ❢♦r ✇❤✐❝❤ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s ✐s ❣r❡❛t❡r t❤❛♥ 80.99✳

✼✳ ❉❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡ ♦❢

∞

k−1

12 1

✳

k=1

5

✽✳ ❆ ♥❡✇ s♦❝❝❡r ❝♦♠♣❡t✐t✐♦♥ r❡q✉✐r❡s ❡❛❝❤ ♦❢ ✽ t❡❛♠s t♦ ♣❧❛② ❡✈❡r② ♦t❤❡r t❡❛♠ ♦♥❝❡✳

❛✳ ❈❛❧❝✉❧❛t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠❛t❝❤❡s t♦ ❜❡ ♣❧❛②❡❞ ✐♥ t❤❡ ❝♦♠♣❡t✐t✐♦♥✳

❜✳ ■❢ ❡❛❝❤ ♦❢ n t❡❛♠s ♣❧❛②❡❞ ❡❛❝❤ ♦t❤❡r ♦♥❝❡✱ ❞❡t❡r♠✐♥❡ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♠❛t❝❤❡s

✐♥ t❡r♠s ♦❢ n✳

✾✳ ❚❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡s ♦❢ sq✉❛r❡ ♦❢ ❧❡♥❣t❤ ✹ ✉♥✐ts ❛r❡ ❥♦✐♥❡❞ t♦ ❢♦r♠ ✹ ♥❡✇ s♠❛❧❧❡r

sq✉❛r❡s✳ ❚❤✐s ♠✐❞♣♦✐♥ts ♦❢ t❤❡ ♥❡✇ s♠❛❧❧❡r sq✉❛r❡s ❛r❡ t❤❡♥ ❥♦✐♥❡❞ ✐♥ t❤❡ s❛♠❡ ✇❛② t♦ ♠❛❦❡ ❡✈❡♥

s♠❛❧❧❡r sq✉❛r❡s✳ ❚❤✐s ♣r♦❝❡ss ✐s r❡♣❡❛t❡❞ ✐♥❞❡✜♥✐t❡❧②✳ ❈❛❧❝✉❧❛t❡ t❤❡ s✉♠ ♦❢ t❤❡ ❛r❡❛s ♦❢ ❛❧❧ t❤❡ sq✉❛r❡s

s♦ ❢♦r♠❡❞✳

✶✵✳ ❚❤❡♠❜✐ ✇♦r❦❡❞ ♣❛rt✲t✐♠❡ t♦ ❜✉② ❛ ▼❛t❤❡♠❛t✐❝s ❜♦♦❦ ✇❤✐❝❤ ❝♦st ❘✷✾✱✺✵✳ ❖♥ ✶ ❋❡❜r✉❛r② s❤❡ s❛✈❡❞

❘✶✱✻✵✱ ❛♥❞ s❛✈❡s ❡✈❡r②❞❛② ✸✵ ❝❡♥ts ♠♦r❡ t❤❛♥ s❤❡ s❛✈❡❞ t❤❡ ♣r❡✈✐♦✉s ❞❛②✳ ✭❙♦✱ ♦♥ t❤❡ s❡❝♦♥❞ ❞❛②✱ s❤❡

s❛✈❡❞ ❘✶✱✾✵✱ ❛♥❞ s♦ ♦♥✳✮ ❆❢t❡r ❤♦✇ ♠❛♥② ❞❛②s ❞✐❞ s❤❡ ❤❛✈❡ ❡♥♦✉❣❤ ♠♦♥❡② t♦ ❜✉② t❤❡ ❜♦♦❦❄

✷✼

✶✶✳ ❈♦♥s✐❞❡r t❤❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s✿

1

1

5 + 2

+ 1

+ ...

✭✶✳✼✸✮

2

4

❛✳ ■❢ A ✐s t❤❡ s✉♠ t♦ ✐♥✜♥✐t② ❛♥❞ B ✐s t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s✱ ✇r✐t❡ ❞♦✇♥ t❤❡ ✈❛❧✉❡ ♦❢✿

✐✳ A

✐✐✳ B ✐♥ t❡r♠s ♦❢ n✳

❜✳ ❋♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ n ✐s (A − B) < 1 ❄

24

✶✷✳ ❆ ❝❡rt❛✐♥ ♣❧❛♥t r❡❛❝❤❡s ❛ ❤❡✐❣❤t ♦❢ ✶✶✽ ♠♠ ❛❢t❡r ♦♥❡ ②❡❛r ✉♥❞❡r ✐❞❡❛❧ ❝♦♥❞✐t✐♦♥s ✐♥ ❛ ❣r❡❡♥❤♦✉s❡✳

❉✉r✐♥❣ t❤❡ ♥❡①t ②❡❛r✱ t❤❡ ❤❡✐❣❤t ✐♥❝r❡❛s❡s ❜② ✶✷ ♠♠✳ ■♥ ❡❛❝❤ s✉❝❝❡ss✐✈❡ ②❡❛r✱ t❤❡ ❤❡✐❣❤t ✐♥❝r❡❛s❡s ❜②

5 ♦❢ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✬s ❣r♦✇t❤✳ ❙❤♦✇ t❤❛t t❤❡ ♣❧❛♥t ✇✐❧❧ ♥❡✈❡r r❡❛❝❤ ❛ ❤❡✐❣❤t ♦❢ ♠♦r❡ t❤❛♥ ✶✺✵ ♠♠✳

8

✶✸✳ ❈❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ n ✐❢

n

(20 − 4a) = −20✳

a=1

✶✹✳ ▼✐❝❤❛❡❧ s❛✈❡❞ ❘✹✵✵ ❞✉r✐♥❣ t❤❡ ✜rst ♠♦♥t❤ ♦❢ ❤✐s ✇♦r❦✐♥❣ ❧✐❢❡✳ ■♥ ❡❛❝❤ s✉❜s❡q✉❡♥t ♠♦♥t❤✱ ❤❡ s❛✈❡❞

✶✵✪ ♠♦r❡ t❤❛♥ ✇❤❛t ❤❡ ❤❛❞ s❛✈❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s ♠♦♥t❤✳

❛✳ ❍♦✇ ♠✉❝❤ ❞✐❞ ❤❡ s❛✈❡ ✐♥ t❤❡ ✼t❤ ✇♦r❦✐♥❣ ♠♦♥t❤❄

❜✳ ❍♦✇ ♠✉❝❤ ❞✐❞ ❤❡ s❛✈❡ ❛❧❧ t♦❣❡t❤❡r ✐♥ ❤✐s ✜rst ✶✷ ✇♦r❦✐♥❣ ♠♦♥t❤s❄

❝✳ ■♥ ✇❤✐❝❤ ♠♦♥t❤ ♦❢ ❤✐s ✇♦r❦✐♥❣ ❧✐❢❡ ❞✐❞ ❤❡ s❛✈❡ ♠♦r❡ t❤❛♥ ❘✶✱✺✵✵ ❢♦r t❤❡ ✜rst t✐♠❡❄

✶✺✳ ❆ ♠❛♥ ✇❛s ✐♥❥✉r❡❞ ✐♥ ❛♥ ❛❝❝✐❞❡♥t ❛t ✇♦r❦✳ ❍❡ r❡❝❡✐✈❡s ❛ ❞✐s❛❜✐❧✐t② ❣r❛♥t ♦❢ ❘✹✱✽✵✵ ✐♥ t❤❡ ✜rst ②❡❛r✳

❚❤✐s ❣r❛♥t ✐♥❝r❡❛s❡s ✇✐t❤ ❛ ✜①❡❞ ❛♠♦✉♥t ❡❛❝❤ ②❡❛r✳

❛✳ ❲❤❛t ✐s t❤❡ ❛♥♥✉❛❧ ✐♥❝r❡❛s❡ ✐❢✱ ♦✈❡r ✷✵ ②❡❛rs✱ ❤❡ ✇♦✉❧❞ ❤❛✈❡ r❡❝❡✐✈❡❞ ❛ t♦t❛❧ ♦❢ ❘✶✹✸✱✺✵✵❄

❜✳ ❍✐s ✐♥✐t✐❛❧ ❛♥♥✉❛❧ ❡①♣❡♥❞✐t✉r❡ ✐s ❘✷✱✻✵✵ ❛♥❞ ✐♥❝r❡❛s❡s ❛t ❛ r❛t❡ ♦❢ ❘✹✵✵ ♣❡r ②❡❛r✳ ❆❢t❡r ❤♦✇ ♠❛♥②

②❡❛rs ❞♦❡s ❤✐s ❡①♣❡♥s❡s ❡①❝❡❡❞ ❤✐s ✐♥❝♦♠❡❄

✶✻✳ ❚❤❡ ❈❛♣❡ ❚♦✇♥ ❍✐❣❤ ❙❝❤♦♦❧ ✇❛♥ts t♦ ❜✉✐❧❞ ❛ s❝❤♦♦❧ ❤❛❧❧ ❛♥❞ ✐s ❜✉s② ✇✐t❤ ❢✉♥❞r❛✐s✐♥❣✳ ▼r✳ ▼❛♥✉❡❧✱

❛♥ ❡①✲❧❡❛r♥❡r ♦❢ t❤❡ s❝❤♦♦❧ ❛♥❞ ❛ s✉❝❝❡ss❢✉❧ ♣♦❧✐t✐❝✐❛♥✱ ♦✛❡rs t♦ ❞♦♥❛t❡ ♠♦♥❡② t♦ t❤❡ s❝❤♦♦❧✳ ❍❛✈✐♥❣

❡♥❥♦②❡❞ ♠❛t❤❡♠❛t✐❝s ❛t s❝❤♦♦❧✱ ❤❡ ❞❡❝✐❞❡s t♦ ❞♦♥❛t❡ ❛♥ ❛♠♦✉♥t ♦❢ ♠♦♥❡② ♦♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❜❛s✐s✳ ❍❡

s❡ts ❛ ♠❛t❤❡♠❛t✐❝❛❧ q✉✐③ ✇✐t❤ ✷✵ q✉❡st✐♦♥s✳ ❋♦r t❤❡ ❝♦rr❡❝t ❛♥s✇❡r t♦ t❤❡ ✜rst q✉❡st✐♦♥ ✭❛♥② ❧❡❛r♥❡r

♠❛② ❛♥s✇❡r✮✱ t❤❡ s❝❤♦♦❧ ✇✐❧❧ r❡❝❡✐✈❡ ✶ ❝❡♥t✱ ❢♦r ❛ ❝♦rr❡❝t ❛♥s✇❡r t♦ t❤❡ s❡❝♦♥❞ q✉❡st✐♦♥✱ t❤❡ s❝❤♦♦❧ ✇✐❧❧

r❡❝❡✐✈❡ ✷ ❝❡♥ts✱ ❛♥❞ s♦ ♦♥✳ ❚❤❡ ❞♦♥❛t✐♦♥s ✶✱ ✷✱ ✹✱ ✳✳✳ ❢♦r♠ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✳ ❈❛❧❝✉❧❛t❡ ✭●✐✈❡ ②♦✉r

❛♥s✇❡r t♦ t❤❡ ♥❡❛r❡st ❘❛♥❞✮

❛✳ ❚❤❡ ❛♠♦✉♥t ♦❢ ♠♦♥❡② t❤❛t t❤❡ s❝❤♦♦❧ ✇✐❧❧ r❡❝❡✐✈❡ ❢♦r t❤❡ ❝♦rr❡❝t ❛♥s✇❡r t♦ t❤❡ ✷✵th q✉❡st✐♦♥✳

❜✳ ❚❤❡ t♦t❛❧ ❛♠♦✉♥t ♦❢ ♠♦♥❡② t❤❛t t❤❡ s❝❤♦♦❧ ✇✐❧❧ r❡❝❡✐✈❡ ✐❢ ❛❧❧ ✷✵ q✉❡st✐♦♥s ❛r❡ ❛♥s✇❡r❡❞ ❝♦rr❡❝t❧②✳

✶✼✳ ❚❤❡ ✜rst t❡r♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐s ✾✱ ❛♥❞ t❤❡ r❛t✐♦ ♦❢ t❤❡ s✉♠ ♦❢ t❤❡ ✜rst ❡✐❣❤t t❡r♠s t♦ t❤❡ s✉♠

♦❢ t❤❡ ✜rst ❢♦✉r t❡r♠s ✐s 97 : 81✳ ❋✐♥❞ t❤❡ ✜rst t❤r❡❡ t❡r♠s ♦❢ t❤❡ s❡q✉❡♥❝❡✱ ✐❢ ✐t ✐s ❣✐✈❡♥ t❤❛t ❛❧❧ t❤❡

t❡r♠s ❛r❡ ♣♦s✐t✐✈❡✳

✶✽✳ (k − 4) ; (k + 1) ; m; 5k ✐s ❛ s❡t ♦❢ ♥✉♠❜❡rs✱ t❤❡ ✜rst t❤r❡❡ ♦❢ ✇❤✐❝❤ ❢♦r♠ ❛♥ ❛r✐t❤♠❡t✐❝ s❡q✉❡♥❝❡✱ ❛♥❞

t❤❡ ❧❛st t❤r❡❡ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✳ ❋✐♥❞ k ❛♥❞ m ✐❢ ❜♦t❤ ❛r❡ ♣♦s✐t✐✈❡✳

✶✾✳ ●✐✈❡♥✿ ❚❤❡ s❡q✉❡♥❝❡ 6 + p; 10 + p; 15 + p ✐s ❣❡♦♠❡tr✐❝✳

❛✳ ❉❡t❡r♠✐♥❡ p✳

❜✳ ❙❤♦✇ t❤❛t t❤❡ ❝♦♠♠♦♥ r❛t✐♦ ✐s 5✳

4

❝✳ ❉❡t❡r♠✐♥❡ t❤❡ ✶✵t❤ t❡r♠ ♦❢ t❤✐s s❡q✉❡♥❝❡ ❝♦rr❡❝t t♦ ♦♥❡ ❞❡❝✐♠❛❧ ♣❧❛❝❡✳

✷✵✳ ❚❤❡ s❡❝♦♥❞ ❛♥❞ ❢♦✉rt❤ t❡r♠s ♦❢ ❛ ❝♦♥✈❡r❣❡♥t ❣❡♦♠❡tr✐❝ s❡r✐❡s ❛r❡ ✸✻ ❛♥❞ ✶✻✱ r❡s♣❡❝t✐✈❡❧②✳ ❋✐♥❞ t❤❡

s✉♠ t♦ ✐♥✜♥✐t② ♦❢ t❤✐s s❡r✐❡s✱ ✐❢ ❛❧❧ ✐ts t❡r♠s ❛r❡ ♣♦s✐t✐✈❡✳

✷✶✳ ❊✈❛❧✉❛t❡✿

5

k(k+1)

k=2

2

✷✷✳ Sn = 4n2 + 1 r❡♣r❡s❡♥ts t❤❡ s✉♠ ♦❢ t❤❡ ✜rst n t❡r♠s ♦❢ ❛ ♣❛rt✐❝✉❧❛r s❡r✐❡s✳ ❋✐♥❞ t❤❡ s❡❝♦♥❞ t❡r♠✳

✷✸✳ ❋✐♥❞ p ✐❢✿

∞

27pk =

12

(24 − 3t)

k=1

t=1

✷✹✳ ❋✐♥❞ t❤❡ ✐♥t❡❣❡r t❤❛t ✐s t❤❡ ❝❧♦s❡st ❛♣♣r♦①✐♠❛t✐♦♥ t♦✿

102001 + 102003

✭✶✳✼✹✮

102002 + 102002

✷✽

❈❍❆P❚❊❘ ✶✳ ❙❊◗❯❊◆❈❊❙ ❆◆❉ ❙❊❘■❊❙

✷✺✳ ❋✐♥❞ t❤❡ ♣❛tt❡r♥ ❛♥❞ ❤❡♥❝❡ ❝❛❧❝✉❧❛t❡✿

1 − 2 + 3 − 4 + 5 − 6... + 677 − 678 + ... − 1000

✭✶✳✼✺✮

✷✻✳ ❉❡t❡r♠✐♥❡ ✐❢

∞

(x + 2)p ❝♦♥✈❡r❣❡s✳ ■❢ ✐t ❞♦❡s✱ t❤❡♥ ✇♦r❦ ♦✉t ✇❤❛t ✐t ❝♦♥✈❡r❣❡s t♦ ✐❢✿

p=1

❛✳ x = −52

❜✳ x = −5

✷✼✳ ❈❛❧❝✉❧❛t❡✿

∞

5 · 4−i

i=1

✷✽✳ ❚❤❡ s✉♠ ♦❢ t❤❡ ✜rst p t❡r♠s ♦❢ ❛ s❡q✉❡♥❝❡ ✐s p (p + 1)✳ ❋✐♥❞ t❤❡ ✶✵t❤ t❡r♠✳

✷✾✳ ❚❤❡ ♣♦✇❡rs ♦❢ ✷ ❛r❡ r❡♠♦✈❡❞ ❢r♦♠ t❤❡ s❡t ♦❢ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs

1; 2; 3; 4; 5; 6; ...; 1998; 1999; 2000

✭✶✳✼✻✮

❋✐♥❞ t❤❡ s✉♠ ♦❢ r❡♠❛✐♥✐♥❣ ✐♥t❡❣❡rs✳

✸✵✳ ❖❜s❡r✈❡ t❤❡ ♣❛tt❡r♥ ❜❡❧♦✇✿

❋✐❣✉r❡ ✶✳✻

❛✳ ■❢ t❤❡ ♣❛tt❡r♥ ❝♦♥t✐♥✉❡s✱ ✜♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❧❡tt❡rs ✐♥ t❤❡ ❝♦❧✉♠♥ ❝♦♥t❛✐♥✐♥❣ ▼✬s✳

❜✳ ■❢ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❧❡tt❡rs ✐♥ t❤❡ ♣❛tt❡r♥ ✐s ✸✻✶✱ ✇❤✐❝❤ ❧❡tt❡r ✇✐❧❧ ❜❡ ✐♥ t❤❡ ❧❛st ❝♦❧✉♠♥✳

✸✶✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥ ✇❛s ❛s❦❡❞ ✐♥ ❛ t❡st✿ ❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ 22005 + 22005✳ ❍❡r❡ ❛r❡ s♦♠❡ ♦❢ t❤❡

st✉❞❡♥ts✬ ❛♥s✇❡rs✿

❛✳ ▼❡❣❛♥ s❛✐❞ t❤❡ ❛♥s✇❡r ✐s 42005✳

❜✳ ❙t❡❢❛♥ ✇r♦t❡ ❞♦✇♥ 24010✳

❝✳ ◆✐♥❛ t❤✐♥❦s ✐t ✐s 22006✳

❞✳ ❆♥♥❛tt❡ ❣❛✈❡ t❤❡ ❛♥s✇❡r 22005×2005✳

❲❤♦ ✐s ❝♦rr❡❝t❄ ✭✏◆♦♥❡ ♦❢ t❤❡♠✧ ✐s ❛❧s♦ ❛ ♣♦ss✐❜✐❧✐t②✳✮

✸✷✳ ❆ s❤r✉❜ ♦❢ ❤❡✐❣❤t ✶✶✵ ❝♠ ✐s ♣❧❛♥t❡❞✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ✜rst ②❡❛r✱ t❤❡ s❤r✉❜ ✐s ✶✷✵ ❝♠ t❛❧❧✳ ❚❤❡r❡❛❢t❡r✱

t❤❡ ❣r♦✇t❤ ♦❢ t❤❡ s❤r✉❜ ❡❛❝❤ ②❡❛r ✐s ❤❛❧❢ ♦❢ ✐ts ❣r♦✇t❤ ✐♥ t❤❡ ♣r❡✈✐♦✉s ②❡❛r✳ ❙❤♦✇ t❤❛t t❤❡ ❤❡✐❣❤t ♦❢

t❤❡ s❤r✉❜ ✇✐❧❧ ♥❡✈❡r ❡①❝❡❡❞ ✶✸✵ ❝♠✳

❈❤❛♣t❡r ✷

❋✐♥❛♥❝❡

✷✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❙❡q✉❡♥❝❡s ✫ ❙❡r✐❡s✱ ❋✉t✉r❡ ❱❛❧✉❡ ♦❢ P❛②♠❡♥ts✶

✷✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

■♥ ❡❛r❧✐❡r ❣r❛❞❡s s✐♠♣❧❡ ✐♥t❡r❡st ❛♥❞ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st ✇❡r❡ st✉❞✐❡❞✱ t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❝♦♥❝❡♣t ♦❢ ❞❡♣r❡❝✐✲

❛t✐♦♥✳ ◆♦♠✐♥❛❧ ❛♥❞ ❡✛❡❝t✐✈❡ ✐♥t❡r❡st r❛t❡s ✇❡r❡ ❛❧s♦ ❞❡s❝r✐❜❡❞✳ ❙✐♥❝❡ t❤✐s ❝❤❛♣t❡r ❡①♣❛♥❞s ♦♥ ❡❛r❧✐❡r ✇♦r❦✱

✐t ✇♦✉❧❞ ❜❡ ❜❡st ✐❢ ②♦✉ r❡✈✐s❡❞ t❤❡ ✇♦r❦ ❞♦♥❡ ✐♥ ●r❛❞❡s ✶✵ ❛♥❞ ✶✶✳

■❢ ②♦✉ ♠❛st❡r t❤❡ t❡❝❤♥✐q✉❡s ✐♥ t❤✐s ❝❤❛♣t❡r✱ ✇❤❡♥ ②♦✉ st❛rt ✇♦r❦✐♥❣ ❛♥❞ ❡❛r♥✐♥❣ ②♦✉ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦

❛♣♣❧② t❤❡ t❡❝❤♥✐q✉❡s ✐♥ t❤✐s ❝❤❛♣t❡r t♦ ❝r✐t✐❝❛❧❧② ❛ss❡ss ❤♦✇ t♦ ✐♥✈❡st ②♦✉r ♠♦♥❡②✳ ❆♥❞ ✇❤❡♥ ②♦✉ ❛r❡ ❧♦♦❦✐♥❣

❛t ❛♣♣❧②✐♥❣ ❢♦r ❛ ❜♦♥❞ ❢r♦♠ ❛ ❜❛♥❦ t♦ ❜✉② ❛ ❤♦♠❡✱ ②♦✉ ✇✐❧❧ ❝♦♥✜❞❡♥t❧② ❜❡ ❛❜❧❡ t♦ ❣❡t ♦✉t t❤❡ ❝❛❧❝✉❧❛t♦r ❛♥❞

✇♦r❦ ♦✉t ✇✐t❤ ❛♠❛③❡♠❡♥t ❤♦✇ ♠✉❝❤ ②♦✉ ❝♦✉❧❞ ❛❝t✉❛❧❧② s❛✈❡ ❜② ♠❛❦✐♥❣ ❛❞❞✐t✐♦♥❛❧ r❡♣❛②♠❡♥ts✳ ■♥❞❡❡❞✱ t❤✐s

❝❤❛♣t❡r ✇✐❧❧ ♣r♦✈✐❞❡ ②♦✉ ✇✐t❤ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦♥❝❡♣ts ②♦✉ ✇✐❧❧ ♥❡❡❞ t♦ ❝♦♥✜❞❡♥t❧② ♠❛♥❛❣❡ ②♦✉r ✜♥❛♥❝❡s

❛♥❞ ✇✐t❤ s♦♠❡ s✉❝❝❡ss❢✉❧ ✐♥✈❡st✐♥❣✱ s✐t ❜❛❝❦ ♦♥ ②♦✉r ②❛❝❤t ❛♥❞ ❡♥❥♦② t❤❡ ♠✐❧❧✐♦♥❛✐r❡ ❧✐❢❡st②❧❡✳

✷✳✶✳✷ ❋✐♥❞✐♥❣ t❤❡ ▲❡♥❣t❤ ♦❢ t❤❡ ■♥✈❡st♠❡♥t ♦r ▲♦❛♥

■♥ ●r❛❞❡ ✶✶✱ ✇❡ ✉s❡❞ t❤❡ ❈♦♠♣♦✉♥❞ ■♥t❡r❡st ❢♦r♠✉❧❛ A = P (1 + i)n t♦ ❞❡t❡r♠✐♥❡ t❤❡ t❡r♠ ♦❢ t❤❡ ✐♥✈❡st♠❡♥t

♦r ❧♦❛♥✱ ❜② tr✐❛❧ ❛♥❞ ❡rr♦r✳ ❘❡♠❡♠❜❡r t❤❛t P ✐s t❤❡ ✐♥✐t✐❛❧ ❛♠♦✉♥t✱ A ✐s t❤❡ ❝✉rr❡♥t ❛♠♦✉♥t✱ i ✐s t❤❡ ✐♥t❡r❡st

r❛t❡ ❛♥❞ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ t✐♠❡ ✉♥✐ts ✭♥✉♠❜❡r ♦❢ ♠♦♥t❤s ♦r ②❡❛rs✮✳ ❙♦ ✐❢ ✇❡ ✐♥✈❡st ❛♥ ❛♠♦✉♥t ❛♥❞ ❦♥♦✇

✇❤❛t t❤❡ ✐♥t❡r❡st r❛t❡ ✐s✱ t❤❡♥ ✇❡ ❝❛♥ ✇♦r❦ ♦✉t ❤♦✇ ❧♦♥❣ ✐t ✇✐❧❧ t❛❦❡ ❢♦r t❤❡ ♠♦♥❡② t♦ ❣r♦✇ t♦ t❤❡ r❡q✉✐r❡❞

❛♠♦✉♥t✳

◆♦✇ t❤❛t ②♦✉ ❤❛✈❡ ❧❡❛r♥t ❛❜♦✉t ❧♦❣❛r✐t❤♠s✱ ②♦✉ ❛r❡ r❡❛❞② t♦ ✇♦r❦ ♦✉t t❤❡ ♣r♦♣❡r ❛❧❣❡❜r❛✐❝ s♦❧✉t✐♦♥✳ ■❢

②♦✉ ♥❡❡❞ t♦ r❡♠✐♥❞ ②♦✉rs❡❧❢ ❤♦✇ ❧♦❣❛r✐t❤♠s ✇♦r❦✱ ❣♦ t♦ ❈❤❛♣t❡r ✭♦♥ ♣❛❣❡ ✮✳

❚❤❡ ❜❛s✐❝ ✜♥❛♥❝❡ ❡q✉❛t✐♦♥ ✐s✿

A = P · (1 + i)n

✭✷✳✶✮

■❢ ②♦✉ ❞♦♥✬t ❦♥♦✇ ✇❤❛t A✱ P ✱ i ❛♥❞ n r❡♣r❡s❡♥t✱ t❤❡♥ ②♦✉ s❤♦✉❧❞ ❞❡✜♥✐t❡❧② r❡✈✐s❡ t❤❡ ✇♦r❦ ❢r♦♠ ●r❛❞❡ ✶✵

❛♥❞ ✶✶✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✾✻✴✶✳✶✴❃✳

✷✾

✸✵

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❙♦❧✈✐♥❣ ❢♦r n✿

A

=

P (1 + i)n

(1 + i)n

=

(A/P )

log ((1 + i)n)

=

log (A/P )

✭✷✳✷✮

nlog (1 + i)

=

log (A/P )

n

=

log (A/P ) /log (1 + i)

❘❡♠❡♠❜❡r✱ ②♦✉ ❞♦ ♥♦t ❤❛✈❡ t♦ ♠❡♠♦r✐s❡ t❤✐s ❢♦r♠✉❧❛✳ ■t ✐s ✈❡r② ❡❛s② t♦ ❞❡r✐✈❡ ❛♥② t✐♠❡ ②♦✉ ♥❡❡❞ ✐t✳ ■t ✐s

s✐♠♣❧② ❛ ♠❛tt❡r ♦❢ ✇r✐t✐♥❣ ❞♦✇♥ ✇❤❛t ②♦✉ ❤❛✈❡✱ ❞❡❝✐❞✐♥❣ ✇❤❛t ②♦✉ ♥❡❡❞✱ ❛♥❞ s♦❧✈✐♥❣ ❢♦r t❤❛t ✈❛r✐❛❜❧❡✳

❊①❡r❝✐s❡ ✷✳✶✿ ❚❡r♠ ♦❢ ■♥✈❡st♠❡♥t ✲ ▲♦❣❛r✐t❤♠s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✶✳✮

❙✉♣♣♦s❡ ✇❡ ✐♥✈❡st❡❞ ❘✸ ✺✵✵ ✐♥t♦ ❛ s❛✈✐♥❣s ❛❝❝♦✉♥t ✇❤✐❝❤ ♣❛②s ✼✱✺✪ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st✳ ❆❢t❡r ❛♥

✉♥❦♥♦✇♥ ♣❡r✐♦❞ ♦❢ t✐♠❡ ♦✉r ❛❝❝♦✉♥t ✐s ✇♦rt❤ ❘✹ ✵✹✹✱✻✾✳ ❋♦r ❤♦✇ ❧♦♥❣ ❞✐❞ ✇❡ ✐♥✈❡st t❤❡ ♠♦♥❡②❄

❍♦✇ ❞♦❡s t❤✐s ❝♦♠♣❛r❡ ✇✐t❤ t❤❡ tr✐❛❧ ❛♥❞ ❡rr♦r ❛♥s✇❡r ❢r♦♠ ❈❤❛♣t❡rs ✳

✷✳✶✳✸ ❆ ❙❡r✐❡s ♦❢ P❛②♠❡♥ts

✐ ✐❇②t❤✐sst❛❣❡✱②♦✉❦♥♦✇❤♦✇t♦❞♦❝❛❧❝✉❧❛t✐♦♥ss✉❝❤❛s❵■❢■✇❛♥t❘✶✵✵✵✐♥✸②❡❛rs✬t✐♠❡✱❤♦✇♠✉❝❤❞♦

■ ♥❡❡❞ t♦ ✐♥✈❡st ♥♦✇ ❛t ✶✵✪ ❄✬

❲❤❛t ✐❢ ✇❡ ❡①t❡♥❞ t❤✐s ❛s ❢♦❧❧♦✇s✿ ❵■❢ ■ ✇❛♥t t♦ ❞r❛✇ ❘✶ ✵✵✵ ♥❡①t ②❡❛r✱ ❘✶ ✵✵✵ t❤❡ ♥❡①t ②❡❛r ❛♥❞ ❘✶ ✵✵✵

❛❢t❡r t❤r❡❡ ②❡❛rs ✳✳✳ ❤♦✇ ♠✉❝❤ ❞♦ ■ ♥❡❡❞ t♦ ✐♥✐t✐❛❧❧② ♣✉t ✐♥t♦ ❛ ❜❛♥❦ ❛❝❝♦✉♥t ❡❛r♥✐♥❣ ✶✵✪ ♣✳❛✳ t♦ ❜❡ ❛❜❧❡ t♦

❛✛♦r❞ t♦ ❜❡ ❛❜❧❡ t♦ ❞♦ t❤✐s❄✬

❚❤❡ ♦❜✈✐♦✉s ✇❛② ♦❢ ✇♦r❦✐♥❣ t❤❛t ♦✉t ✐s t♦ ✇♦r❦ ♦✉t ❤♦✇ ♠✉❝❤ ②♦✉ ♥❡❡❞ ♥♦✇ t♦ ❛✛♦r❞ t❤❡ ♣❛②✲

♠❡♥ts ✐♥❞✐✈✐❞✉❛❧❧② ❛♥❞ s✉♠ t❤❡♠✳ ❲❡✬❧❧ ✇♦r❦ ♦✉t ❤♦✇ ♠✉❝❤ ✐s ♥❡❡❞❡❞ ♥♦✇ t♦ ❛✛♦r❞ t❤❡ ♣❛②♠❡♥t ♦❢

❘✶ ✵✵✵ ✐♥ ❛ ②❡❛r

−

= R1 000 × (1, 10) 1 = R909, 09 ✱ t❤❡ ❛♠♦✉♥t ♥❡❡❞❡❞ ♥♦✇ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ②❡❛r✬s

❘✶ ✵✵✵

−

= R1 000 × (1, 10) 2 = R826, 45

❛♥❞ t❤❡ ❛♠♦✉♥t ♥❡❡❞❡❞ ♥♦✇ ❢♦r t❤❡ ❘✶ ✵✵✵ ❛❢t❡r ✸ ②❡❛rs

−

= R1 000 × (1, 10) 3 = R751, 31 ✳ ❆❞❞ t❤❡s❡ t♦❣❡t❤❡r ❣✐✈❡s ②♦✉ t❤❡ ❛♠♦✉♥t ♥❡❡❞❡❞ t♦ ❛✛♦r❞ ❛❧❧ t❤r❡❡

♣❛②♠❡♥ts ❛♥❞ ②♦✉ ❣❡t ❘✷ ✹✽✻✱✽✺✳

❙♦✱ ✐❢ ②♦✉ ♣✉t ❘✷ ✹✽✻✱✽✺ ✐♥t♦ ❛ ✶✵✪ ❜❛♥❦ ❛❝❝♦✉♥t ♥♦✇✱ ②♦✉ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ❞r❛✇ ♦✉t ❘✶ ✵✵✵ ✐♥ ❛ ②❡❛r✱

❘✶ ✵✵✵ ❛ ②❡❛r ❛❢t❡r t❤❛t✱ ❛♥❞ ❘✶ ✵✵✵ ❛ ②❡❛r ❛❢t❡r t❤❛t ✲ ❛♥❞ ②♦✉r ❜❛♥❦ ❛❝❝♦✉♥t ✇✐❧❧ ❝♦♠❡ ❞♦✇♥ t♦ ❘✵✳ ❨♦✉

✇♦✉❧❞ ❤❛✈❡ ❤❛❞ ❡①❛❝t❧② t❤❡ r✐❣❤t ❛♠♦✉♥t ♦❢ ♠♦♥❡② t♦ ❞♦ t❤❛t ✭♦❜✈✐♦✉s❧②✦✮✳

❨♦✉ ❝❛♥ ❝❤❡❝❦ t❤✐s ❛s ❢♦❧❧♦✇s✿

❆♠♦✉♥t ❛t ❚✐♠❡ ✵ ✭✐✳❡✳ ◆♦✇✮

❂ ❘✷ ✹✽✻✱✽✺

❆♠♦✉♥t ❛t ❚✐♠❡ ✶ ✭✐✳❡✳ ❛ ②❡❛r ❧❛t❡r✮ ❂ ✷ ✹✽✻✱✽✺✭✶✰✶✵✪✮

❂ ❘✷ ✼✸✺✱✺✹

❆♠♦✉♥t ❛❢t❡r ✇✐t❤❞r❛✇✐♥❣ ❘✶ ✵✵✵

❂ ✷ ✼✸✺✱✺✹ ✲ ✶ ✵✵✵

❂ ❘✶ ✼✸✺✱✺✹

❆♠♦✉♥t ❛t ❚✐♠❡ ✷ ✭✐✳❡✳ ❛ ②❡❛r ❧❛t❡r✮ ❂ ✶ ✼✸✺✱✺✹✭✶✰✶✵✪✮

❂ ❘✶ ✾✵✾✱✵✾

❆♠♦✉♥t ❛❢t❡r ✇✐t❤❞r❛✇✐♥❣ ❘✶ ✵✵✵

❂ ❘✶ ✾✵✾✱✵✾ ✲ ✶ ✵✵✵ ❂ ❘✾✵✾✱✵✾

❆♠♦✉♥t ❛t ❚✐♠❡ ✸ ✭✐✳❡✳ ❛ ②❡❛r ❧❛t❡r✮ ❂ ✾✵✾✱✵✾✭✶✰✶✵✪✮

❂ ❘✶ ✵✵✵

❆♠♦✉♥t ❛❢t❡r ✇✐t❤❞r❛✇✐♥❣ ❘✶ ✵✵✵

❂ ✶ ✵✵✵ ✲ ✶ ✵✵✵

❂ ❘✵

❚❛❜❧❡ ✷✳✶

✸✶

P❡r❢❡❝t✦ ❖❢ ❝♦✉rs❡✱ ❢♦r ♦♥❧② t❤r❡❡ ②❡❛rs✱ t❤❛t ✇❛s ♥♦t t♦♦ ❜❛❞✳ ❇✉t ✇❤❛t ✐❢ ■ ❛s❦❡❞ ②♦✉ ❤♦✇ ♠✉❝❤ ②♦✉

♥❡❡❞❡❞ t♦ ♣✉t ✐♥t♦ ❛ ❜❛♥❦ ❛❝❝♦✉♥t ♥♦✇✱ t♦ ❜❡ ❛❜❧❡ t♦ ❛✛♦r❞ ❘✶✵✵ ❛ ♠♦♥t❤ ❢♦r t❤❡ ♥❡①t ✶✺ ②❡❛rs✳ ■❢ ②♦✉ ✉s❡❞

t❤❡ ❛❜♦✈❡ ❛♣♣r♦❛❝❤ ②♦✉ ✇♦✉❧❞ st✐❧❧ ❣❡t t❤❡ r✐❣❤t ❛♥s✇❡r✱ ❜✉t ✐t ✇♦✉❧❞ t❛❦❡ ②♦✉ ✇❡❡❦s✦

❚❤❡r❡ ✐s ✲ ■✬♠ s✉r❡ ②♦✉ ❣✉❡ss❡❞ ✲ ❛♥ ❡❛s✐❡r ✇❛②✦ ❚❤✐s s❡❝t✐♦♥ ✇✐❧❧ ❢♦❝✉s ♦♥ ❞❡s❝r✐❜✐♥❣ ❤♦✇ t♦ ✇♦r❦ ✇✐t❤✿

• ❛♥♥✉✐t✐❡s ✲ ❛ ✜①❡❞ s✉♠ ♣❛②❛❜❧❡ ❡❛❝❤ ②❡❛r ♦r ❡❛❝❤ ♠♦♥t❤✱ ❡✐t❤❡r t♦ ♣r♦✈✐❞❡ ❛ ♣r❡✲❞❡t❡r♠✐♥❡❞ s✉♠ ❛t

t❤❡ ❡♥❞ ♦❢ ❛ ♥✉♠❜❡r ♦❢ ②❡❛rs ♦r ♠♦♥t❤s ✭r❡❢❡rr❡❞ t♦ ❛s ❛ ❢✉t✉r❡ ✈❛❧✉❡ ❛♥♥✉✐t②✮ ♦r ❛ ✜①❡❞ ❛♠♦✉♥t ♣❛✐❞

❡❛❝❤ ②❡❛r ♦r ❡❛❝❤ ♠♦♥t❤ t♦ r❡♣❛② ✭❛♠♦rt✐s❡✮ ❛ ❧♦❛♥ ✭r❡❢❡rr❡❞ t♦ ❛s ❛ ♣r❡s❡♥t ✈❛❧✉❡ ❛♥♥✉✐t②✮✳

• ❜♦♥❞ r❡♣❛②♠❡♥ts ✲ ❛ ✜①❡❞ s✉♠ ♣❛②❛❜❧❡ ❛t r❡❣✉❧❛r ✐♥t❡r✈❛❧s t♦ ♣❛② ♦✛ ❛ ❧♦❛♥✳ ❚❤✐s ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢

❛ ♣r❡s❡♥t ✈❛❧✉❡ ❛♥♥✉✐t②✳

• s✐♥❦✐♥❣ ❢✉♥❞s ✲ ❛♥ ❛❝❝♦✉♥t✐♥❣ t❡r♠ ❢♦r ❝❛s❤ s❡t ❛s✐❞❡ ❢♦r ❛ ♣❛rt✐❝✉❧❛r ♣✉r♣♦s❡ ❛♥❞ ✐♥✈❡st❡❞ s♦ t❤❛t

t❤❡ ❝♦rr❡❝t ❛♠♦✉♥t ♦❢ ♠♦♥❡② ✇✐❧❧ ❜❡ ❛✈❛✐❧❛❜❧❡ ✇❤❡♥ ✐t ✐s ♥❡❡❞❡❞✳ ❚❤✐s ✐s ❛♥ ❡①❛♠♣❧❡ ♦❢ ❛ ❢✉t✉r❡ ✈❛❧✉❡

❛♥♥✉✐t②✳

✷✳✶✳✸✳✶ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s

❇❡❢♦r❡ ✇❡ ♣r♦❣r❡ss✱ ②♦✉ ♥❡❡❞ t♦ ❣♦ ❜❛❝❦ ❛♥❞ r❡❛❞ ❈❤❛♣t❡r ✭❢r♦♠ ♣❛❣❡ ✮ t♦ r❡✈✐s❡ s❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s✳

■♥ s✉♠♠❛r②✱ ✐❢ ②♦✉ ❤❛✈❡ ❛ s❡r✐❡s ♦❢ n t❡r♠s ✐♥ t♦t❛❧ ✇❤✐❝❤ ❧♦♦❦s ❧✐❦❡ t❤✐s✿

a + ar + ar2 + ... + arn−1 = a 1 + r + r2 + ...rn−1

✭✷✳✸✮

t❤✐s ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ ❛s✿

a(rn−1)

usefulwhenr > 1

r−1

✭✷✳✹✮

a(1−rn)

usefulwhen0 ≤ r < 1

1−r

✷✳✶✳✸✳✷ Pr❡s❡♥t ❱❛❧✉❡s ♦❢ ❛ s❡r✐❡s ♦❢ P❛②♠❡♥ts

❙♦ ❤❛✈✐♥❣ r❡✈✐❡✇❡❞ t❤❡ ♠❛t❤❡♠❛t✐❝s ♦❢ ❙❡q✉❡♥❝❡s ❛♥❞ ❙❡r✐❡s✱ ②♦✉ ♠✐❣❤t ❜❡ ✇♦♥❞❡r✐♥❣ ❤♦✇ t❤✐s ✐s ♠❡❛♥t t♦

❤❛✈❡ ❛♥② ♣r❛❝t✐❝❛❧ ♣✉r♣♦s❡✦ ●✐✈❡♥ t❤❛t ✇❡ ❛r❡ ✐♥ t❤❡ ✜♥❛♥❝❡ s❡❝t✐♦♥✱ ②♦✉ ✇♦✉❧❞ ❜❡ r✐❣❤t t♦ ❣✉❡ss t❤❛t t❤❡r❡

♠✉st ❜❡ s♦♠❡ ✜♥❛♥❝✐❛❧ ✉s❡ t♦ ❛❧❧ t❤✐s✳ ❍❡r❡ ✐s ❛♥ ❡①❛♠♣❧❡ ✇❤✐❝❤ ❤❛♣♣❡♥s ✐♥ ♠❛♥② ♣❡♦♣❧❡✬s ❧✐✈❡s ✲ s♦ ②♦✉

❦♥♦✇ ②♦✉ ❛r❡ ❧❡❛r♥✐♥❣ s♦♠❡t❤✐♥❣ ♣r❛❝t✐❝❛❧✳

▲❡t ✉s s❛② ②♦✉ ✇♦✉❧❞ ❧✐❦❡ t♦ ❜✉② ❛ ♣r♦♣❡rt② ❢♦r ❘✸✵✵ ✵✵✵✱ s♦ ②♦✉ ❣♦ t♦ t❤❡ ❜❛♥❦ t♦ ❛♣♣❧② ❢♦r ❛ ♠♦rt❣❛❣❡

❜♦♥❞✳ ❚❤❡ ❜❛♥❦ ✇❛♥ts ✐t t♦ ❜❡ r❡♣❛✐❞ ❜② ❛♥♥✉❛❧❧② ♣❛②♠❡♥ts ❢♦r t❤❡ ♥❡①t ✷✵ ②❡❛rs✱ st❛rt✐♥❣ ❛t ❡♥❞ ♦❢ t❤✐s

②❡❛r✳ ❚❤❡② ✇✐❧❧ ❝❤❛r❣❡ ②♦✉ ✶✺✪ ✐♥t❡r❡st ♣❡r ❛♥♥✉♠✳ ❆t t❤❡ ❡♥❞ ♦❢ t❤❡ ✷✵ ②❡❛rs t❤❡ ❜❛♥❦ ✇♦✉❧❞ ❤❛✈❡ r❡❝❡✐✈❡❞

❜❛❝❦ t❤❡ t♦t❛❧ ❛♠♦✉♥t ②♦✉ ❜♦rr♦✇❡❞ t♦❣❡t❤❡r ✇✐t❤ ❛❧❧ t❤❡ ✐♥t❡r❡st t❤❡② ❤❛✈❡ ❡❛r♥❡❞ ❢r♦♠ ❧❡♥❞✐♥❣ ②♦✉ t❤❡

♠♦♥❡②✳ ❨♦✉ ✇♦✉❧❞ ♦❜✈✐♦✉s❧② ✇❛♥t t♦ ✇♦r❦ ♦✉t ✇❤❛t t❤❡ ❛♥♥✉❛❧ r❡♣❛②♠❡♥t ✐s ❣♦✐♥❣ t♦ ❜❡✦

▲❡t X ❜❡ t❤❡ ❛♥♥✉❛❧ r❡♣❛②♠❡♥t✱ i ✐s t❤❡ ✐♥t❡r❡st r❛t❡✱ ❛♥❞ M ✐s t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ♠♦rt❣❛❣❡ ❜♦♥❞ ②♦✉

✇✐❧❧ ❜❡ t❛❦✐♥❣ ♦✉t✳

❚✐♠❡ ❧✐♥❡s ❛r❡ ♣❛rt✐❝✉❧❛r❧② ✉s❡❢✉❧ t♦♦❧s ❢♦r ✈✐s✉❛❧✐③✐♥❣ t❤❡ s❡r✐❡s ♦❢ ♣❛②♠❡♥ts ❢♦r ❝❛❧❝✉❧❛t✐♦♥s✱ ❛♥❞ ✇❡ ❝❛♥

r❡♣r❡s❡♥t t❤❡s❡ ♣❛②♠❡♥ts ♦♥ ❛ t✐♠❡ ❧✐♥❡ ❛s✿

❋✐❣✉r❡ ✷✳✶✿ ❚✐♠❡ ▲✐♥❡ ❢♦r ❛♥ ❛♥♥✉✐t② ✭✐♥ ❛rr❡❛rs✮ ♦❢ X ❢♦r n ♣❡r✐♦❞s✳

✸✷

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❚❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ ❛❧❧ t❤❡ ♣❛②♠❡♥ts ✭✇❤✐❝❤ ✐♥❝❧✉❞❡s ✐♥t❡r❡st✮ ♠✉st ❡q✉❛t❡ t♦ t❤❡ ✭♣r❡s❡♥t✮ ✈❛❧✉❡ ♦❢ t❤❡

♠♦rt❣❛❣❡ ❧♦❛♥ ❛♠♦✉♥t✳

▼❛t❤❡♠❛t✐❝❛❧❧②✱ ②♦✉ ❝❛♥ ✇r✐t❡ t❤✐s ❛s✿

−

−

−

−

M = X(1 + i) 1 + X(1 + i) 2 + X(1 + i) 3 + ... + X(1 + i) 20

✭✷✳✺✮

❚❤❡ ♣❛✐♥❢✉❧ ✇❛② ♦❢ s♦❧✈✐♥❣ t❤✐s ♣r♦❜❧❡♠ ✇♦✉❧❞ ❜❡ t♦ ❞♦ t❤❡ ❝❛❧❝✉❧❛t✐♦♥ ❢♦r ❡❛❝❤ ♦❢ t❤❡ t❡r♠s ❛❜♦✈❡ ✲ ✇❤✐❝❤

✐s ✷✵ ❞✐✛❡r❡♥t ❝❛❧❝✉❧❛t✐♦♥s✳ ◆♦t ♦♥❧② ✇♦✉❧❞ ②♦✉ ♣r♦❜❛❜❧② ❣❡t ❜♦r❡❞ ❛❧♦♥❣ t❤❡ ✇❛②✱ ❜✉t ②♦✉ ❛r❡ ❛❧s♦ ❧✐❦❡❧② t♦

♠❛❦❡ ❛ ♠✐st❛❦❡✳

◆❛t✉r❛❧❧②✱ t❤❡r❡ ✐s ❛ s✐♠♣❧❡r ✇❛② ♦❢ ❞♦✐♥❣ t❤✐s✦ ❨♦✉ ❝❛♥ r❡✇r✐t❡ t❤❡ ❛❜♦✈❡ ❡q✉❛t✐♦♥ ❛s ❢♦❧❧♦✇s✿

M = X v1 + v2 + v3 + ... + v20

✭✷✳✻✮

−

wherevi = (1 + i) 1 = 1/ (1 + i)

❖❢ ❝♦✉rs❡✱ ②♦✉ ❞♦ ♥♦t ❤❛✈❡ t♦ ✉s❡ t❤❡ ♠❡t❤♦❞ ♦❢ s✉❜st✐t✉t✐♦♥ t♦ s♦❧✈❡ t❤✐s✳ ❲❡ ❥✉st ✜♥❞ t❤✐s ❛ ✉s❡❢✉❧ ♠❡t❤♦❞

❜❡❝❛✉s❡ ②♦✉ ❝❛♥ ❣❡t r✐❞ ♦❢ t❤❡ ♥❡❣❛t✐✈❡ ❡①♣♦♥❡♥ts ✲ ✇❤✐❝❤ ❝❛♥ ❜❡ q✉✐t❡ ❝♦♥❢✉s✐♥❣✦ ❆s ❛♥ ❡①❡r❝✐s❡ ✲ t♦ s❤♦✇

②♦✉ ❛r❡ ❛ r❡❛❧ ✜♥❛♥❝✐❛❧ ✇❤✐③③ ✲ tr② t♦ s♦❧✈❡ t❤✐s ✇✐t❤♦✉t s✉❜st✐t✉t✐♦♥✳ ■t ✐s ❛❝t✉❛❧❧② q✉✐t❡ ❡❛s②✳

◆♦✇✱ t❤❡ ✐t❡♠ ✐♥ sq✉❛r❡ ❜r❛❝❦❡ts ✐s t❤❡ s✉♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✱ ❛s ❞✐s❝✉ss✐♦♥ ✐♥ ✳ ❚❤✐s ❝❛♥ ❜❡

r❡✲✇r✐tt❡♥ ❛s ❢♦❧❧♦✇s✱ ✉s✐♥❣ ✇❤❛t ✇❡ ❦♥♦✇ ❢r♦♠ ❈❤❛♣t❡r ♦❢ t❤✐s t❡①t ❜♦♦❦✿

v1 + v2 + v3 + ... + vn

=

v 1 + v + v2 + ... + vn−1

=

v

1−vn

1−v

✭✷✳✼✮

=

1−vn

1/v−1

=

1−(1+i)−n

i

◆♦t❡ t❤❛t ✇❡ t♦♦❦ ♦✉t ❛ ❝♦♠♠♦♥ ❢❛❝t♦r ♦❢ v ❜❡❢♦r❡ ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✳

❙♦ ✇❡ ❝❛♥ ✇r✐t❡✿



−



1 − (1 + i) n

M = X

✭✷✳✽✮





i

❚❤✐s ❝❛♥ ❜❡ r❡✲✇r✐tt❡♥✿

M

M i

X =

=

✭✷✳✾✮

(

−n

1−(1+i)−n)

1 − (1 + i)

i

❙♦✱ t❤✐s ❢♦r♠✉❧❛ ✐s ✉s❡❢✉❧ ✐❢ ②♦✉ ❦♥♦✇ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ♠♦rt❣❛❣❡ ❜♦♥❞ ②♦✉ ♥❡❡❞ ❛♥❞ ✇❛♥t t♦ ✇♦r❦ ♦✉t t❤❡

r❡♣❛②♠❡♥t✱ ♦r ✐❢ ②♦✉ ❦♥♦✇ ❤♦✇ ❜✐❣ ❛ r❡♣❛②♠❡♥t ②♦✉ ❝❛♥ ❛✛♦r❞ ❛♥❞ ✇❛♥t t♦ s❡❡ ✇❤❛t ♣r♦♣❡rt② ②♦✉ ❝❛♥ ❜✉②✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ■ ✇❛♥t t♦ ❜✉② ❛ ❤♦✉s❡ ❢♦r ❘✸✵✵ ✵✵✵ ♦✈❡r ✷✵ ②❡❛rs✱ ❛♥❞ t❤❡ ❜❛♥❦ ✐s ❣♦✐♥❣ t♦ ❝❤❛r❣❡ ♠❡

✶✺✪ ♣❡r ❛♥♥✉♠ ♦♥ t❤❡ ♦✉tst❛♥❞✐♥❣ ❜❛❧❛♥❝❡✱ t❤❡♥ t❤❡ ❛♥♥✉❛❧ r❡♣❛②♠❡♥t ✐s✿

X

=

M i

1−(1+i)−n

=

R300 000×0,15

✭✷✳✶✵✮

1−(1+0,15)−20

=

R4 792 844

❚❤✐s ♠❡❛♥s✱ ❡❛❝❤ ②❡❛r ❢♦r t❤❡ ♥❡①t ✷✵ ②❡❛rs✱ ■ ♥❡❡❞ t♦ ♣❛② t❤❡ ❜❛♥❦ ❘✹✼ ✾✷✽✱✹✹ ♣❡r ②❡❛r ❜❡❢♦r❡ ■ ❤❛✈❡ ♣❛✐❞

♦✛ t❤❡ ♠♦rt❣❛❣❡ ❜♦♥❞✳

✸✸

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✐❢ ■ ❦♥♦✇ ■ ✇✐❧❧ ♦♥❧② ❤❛✈❡ ❘✸✵ ✵✵✵ ❛ ②❡❛r t♦ r❡♣❛② ♠② ❜♦♥❞✱ t❤❡♥ ❤♦✇ ❜✐❣ ❛ ❤♦✉s❡

❝❛♥ ■ ❜✉②❄ ❚❤❛t ✐s ❡❛s② ✳✳✳✳

(1−(1+i)−n)

M

=

X

i

(1−(1,15)−20)

=

R30 000

✭✷✳✶✶✮

0,15

=

R187 779, 90

❙♦✱ ❢♦r ❘✸✵ ✵✵✵ ❛ ②❡❛r ❢♦r ✷✵ ②❡❛rs✱ ■ ❝❛♥ ❛✛♦r❞ t♦ ❜✉② ❛ ❤♦✉s❡ ♦❢ ❘✶✽✼ ✽✵✵ ✭r♦✉♥❞❡❞ t♦ t❤❡ ♥❡❛r❡st ❤✉♥❞r❡❞✮✳

❚❤❡ ❜❛❞ ♥❡✇s ✐s t❤❛t ❘✶✽✼ ✽✵✵ ❞♦❡s ♥♦t ❝♦♠❡ ❝❧♦s❡ t♦ t❤❡ ❘✸✵✵ ✵✵✵ ②♦✉ ✇❛♥t❡❞ t♦ ♣❛②✦ ❚❤❡ ❣♦♦❞ ♥❡✇s

✐s t❤❛t ②♦✉ ❞♦ ♥♦t ❤❛✈❡ t♦ ♠❡♠♦r✐s❡ t❤✐s ❢♦r♠✉❧❛✳ ■♥ ❢❛❝t ✱ ✇❤❡♥ ②♦✉ ❛♥s✇❡r q✉❡st✐♦♥s ❧✐❦❡ t❤✐s ✐♥ ❛♥ ❡①❛♠✱

②♦✉ ✇✐❧❧ ❜❡ ❡①♣❡❝t❡❞ t♦ st❛rt ❢r♦♠ t❤❡ ❜❡❣✐♥♥✐♥❣ ✲ ✇r✐t✐♥❣ ♦✉t t❤❡ ♦♣❡♥✐♥❣ ❡q✉❛t✐♦♥ ✐♥ ❢✉❧❧✱ s❤♦✇✐♥❣ t❤❛t

✐t ✐s t❤❡ s✉♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✱ ❞❡r✐✈✐♥❣ t❤❡ ❛♥s✇❡r✱ ❛♥❞ t❤❡♥ ❝♦♠✐♥❣ ✉♣ ✇✐t❤ t❤❡ ❝♦rr❡❝t ♥✉♠❡r✐❝❛❧

❛♥s✇❡r✳

❊①❡r❝✐s❡ ✷✳✷✿ ▼♦♥t❤❧② ♠♦rt❣❛❣❡ r❡♣❛②♠❡♥ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✶✳✮

❙❛♠ ✐s ❧♦♦❦✐♥❣ t♦ ❜✉② ❤✐s ✜rst ✢❛t✱ ❛♥❞ ❤❛s ❘✶✺ ✵✵✵ ✐♥ ❝❛s❤ s❛✈✐♥❣s ✇❤✐❝❤ ❤❡ ✇✐❧❧ ✉s❡ ❛s ❛ ❞❡♣♦s✐t✳

❍❡ ❤❛s ✈✐❡✇❡❞ ❛ ✢❛t ✇❤✐❝❤ ✐s ♦♥ t❤❡ ♠❛r❦❡t ❢♦r ❘✷✺✵ ✵✵✵✱ ❛♥❞ ❤❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✇♦r❦ ♦✉t ❤♦✇

♠✉❝❤ t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥ts ✇♦✉❧❞ ❜❡✳ ❍❡ ✇✐❧❧ ❜❡ t❛❦✐♥❣ ♦✉t ❛ ✸✵ ②❡❛r ♠♦rt❣❛❣❡ ✇✐t❤ ♠♦♥t❤❧②

r❡♣❛②♠❡♥ts✳ ❚❤❡ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡ ✐s ✶✶✪✳

❊①❡r❝✐s❡ ✷✳✸✿ ▼♦♥t❤❧② ♠♦rt❣❛❣❡ r❡♣❛②♠❡♥ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✷✳✮

❨♦✉ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ♣✉r❝❤❛s✐♥❣ ❛ ✢❛t ❢♦r ❘✷✵✵ ✵✵✵ ❛♥❞ t❤❡ ❜❛♥❦✬s ♠♦rt❣❛❣❡ r❛t❡ ✐s ❝✉rr❡♥t❧② ✾✪

♣❡r ❛♥♥✉♠ ♣❛②❛❜❧❡ ♠♦♥t❤❧②✳ ❨♦✉ ❤❛✈❡ s❛✈✐♥❣s ♦❢ ❘✶✵ ✵✵✵ ✇❤✐❝❤ ②♦✉ ✐♥t❡♥❞ t♦ ✉s❡ ❢♦r ❛ ❞❡♣♦s✐t✳

❍♦✇ ♠✉❝❤ ✇♦✉❧❞ ②♦✉r ♠♦♥t❤❧② ♠♦rt❣❛❣❡ ♣❛②♠❡♥t ❜❡ ✐❢ ②♦✉ ✇❡r❡ ❝♦♥s✐❞❡r✐♥❣ ❛ ♠♦rt❣❛❣❡ ♦✈❡r ✷✵

②❡❛rs✳

✷✳✶✳✸✳✷✳✶ ❙❤♦✇ ♠❡ t❤❡ ♠♦♥❡② Â[U+0085]

◆♦✇ t❤❛t ②♦✉✬✈❡ ❞♦♥❡ t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ❢♦r t❤❡ ✇♦r❦❡❞ ❡①❛♠♣❧❡ ❛♥❞ ❦♥♦✇ ✇❤❛t t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥ts ❛r❡✱

②♦✉ ❝❛♥ ✇♦r❦ ♦✉t s♦♠❡ s✉r♣r✐s✐♥❣ ✜❣✉r❡s✳ ❋♦r ❡①❛♠♣❧❡✱ ❘✶ ✼✵✾✱✹✽ ♣❡r ♠♦♥t❤ ❢♦r ✷✹✵ ♠♦♥t❤ ♠❛❦❡s ❢♦r ❛

t♦t❛❧ ♦❢ ❘✹✶✵ ✷✼✺✱✷✵ ✭❂❘✶ ✼✵✾✱✹✽ × ✷✹✵✮✳ ❚❤❛t ✐s ♠♦r❡ t❤❛♥ ❞♦✉❜❧❡ t❤❡ ❛♠♦✉♥t t❤❛t ②♦✉ ❜♦rr♦✇❡❞✦ ❚❤✐s

s❡❡♠s ❧✐❦❡ ❛ ❧♦t✳ ❍♦✇❡✈❡r✱ ♥♦✇ t❤❛t ②♦✉✬✈❡ st✉❞✐❡❞ t❤❡ ❡✛❡❝ts ♦❢ t✐♠❡ ✭❛♥❞ ✐♥t❡r❡st✮ ♦♥ ♠♦♥❡②✱ ②♦✉ s❤♦✉❧❞

❦♥♦✇ t❤❛t t❤✐s ❛♠♦✉♥t ✐s s♦♠❡✇❤❛t ♠❡❛♥✐♥❣❧❡ss✳ ❚❤❡ ✈❛❧✉❡ ♦❢ ♠♦♥❡② ✐s ❞❡♣❡♥❞❛♥t ♦♥ ✐ts t✐♠✐♥❣✳

◆♦♥❡t❤❡❧❡ss✱ ②♦✉ ♠✐❣❤t ♥♦t ❜❡ ♣❛rt✐❝✉❧❛r❧② ❤❛♣♣② t♦ s✐t ❜❛❝❦ ❢♦r ✷✵ ②❡❛rs ♠❛❦✐♥❣ ②♦✉r ❘✶ ✼✵✾✱✹✽ ♠♦rt❣❛❣❡

♣❛②♠❡♥t ❡✈❡r② ♠♦♥t❤ ❦♥♦✇✐♥❣ t❤❛t ❤❛❧❢ t❤❡ ♠♦♥❡② ②♦✉ ❛r❡ ♣❛②✐♥❣ ❛r❡ ❣♦✐♥❣ t♦✇❛r❞ ✐♥t❡r❡st✳ ❇✉t t❤❡r❡ ✐s

❛ ✇❛② t♦ ❛✈♦✐❞ t❤♦s❡ ❤❡❛✈② ✐♥t❡r❡st ❝❤❛r❣❡s✳ ■t ❝❛♥ ❜❡ ❞♦♥❡ ❢♦r ❧❡ss t❤❛♥ ❘✸✵✵ ❡①tr❛ ❡✈❡r② ♠♦♥t❤✳✳✳

❙♦ ♦✉r ♣❛②♠❡♥t ✐s ♥♦✇ ❘✷ ✵✵✵✳ ❚❤❡ ✐♥t❡r❡st r❛t❡ ✐s st✐❧❧ ✾✪ ♣❡r ❛♥♥✉♠ ♣❛②❛❜❧❡ ♠♦♥t❤❧② ✭✵✱✼✺✪ ♣❡r

♠♦♥t❤✮✱ ❛♥❞ ♦✉r ♣r✐♥❝✐♣❛❧ ❛♠♦✉♥t ❜♦rr♦✇❡❞ ✐s ❘✶✾✵ ✵✵✵✳ ▼❛❦✐♥❣ t❤✐s ❤✐❣❤❡r r❡♣❛②♠❡♥t ❛♠♦✉♥t ❡✈❡r②

♠♦♥t❤✱ ❤♦✇ ❧♦♥❣ ✇✐❧❧ ✐t t❛❦❡ t♦ ♣❛② ♦✛ t❤❡ ♠♦rt❣❛❣❡❄

❚❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ t❤❡ str❡❛♠ ♦❢ ♣❛②♠❡♥ts ♠✉st ❜❡ ❡q✉❛❧ t♦ ❘✶✾✵ ✵✵✵ ✭t❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ t❤❡ ❜♦rr♦✇❡❞

✸✹

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❛♠♦✉♥t✮✳ ❙♦ ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ ❢♦r n ✐♥✿

−

R2 000 × 1 − (1 + 0, 75%) n /0, 75%

=

R190 000

−

1 − (1 + 0, 75%) n

=

190 000×0,75%

2 000

−

log(1 + 0, 75%) n

=

log

1 − 190 000×0,0075

2 000

−n × log (1 + 0, 75%)

=

log

1 − 190 000×0,0075

✭✷✳✶✷✮

2 000

−n × 0, 007472

=

−1, 2465

n

=

166, 8months

=

13, 9years

❙♦ t❤❡ ♠♦rt❣❛❣❡ ✇✐❧❧ ❜❡ ❝♦♠♣❧❡t❡❧② r❡♣❛✐❞ ✐♥ ❧❡ss t❤❛♥ ✶✹ ②❡❛rs✱ ❛♥❞ ②♦✉ ✇♦✉❧❞ ❤❛✈❡ ♠❛❞❡ ❛ t♦t❛❧ ♣❛②♠❡♥t

♦❢ ✶✻✻✱✽× ❘✷ ✵✵✵ ❂ ❘✸✸✸ ✻✵✵✳

❈❛♥ ②♦✉ s❡❡ ✇❤❛t ✐s ❤❛♣♣❡♥❡❞❄ ▼❛❦✐♥❣ r❡❣✉❧❛r ♣❛②♠❡♥ts ♦❢ ❘✷ ✵✵✵ ✐♥st❡❛❞ ♦❢ t❤❡ r❡q✉✐r❡❞ ❘✶✱✼✵✾✱✹✽✱

②♦✉ ✇✐❧❧ ❤❛✈❡ s❛✈❡❞ ❘✼✻ ✻✼✺✱✷✵ ✭❂ ❘✹✶✵ ✷✼✺✱✷✵ ✲ ❘✸✸✸ ✻✵✵✮ ✐♥ ✐♥t❡r❡st✱ ❛♥❞ ②❡t ②♦✉ ❤❛✈❡ ♦♥❧② ♣❛✐❞ ❛♥

❛❞❞✐t✐♦♥❛❧ ❛♠♦✉♥t ♦❢ ❘✷✾✵✱✺✷ ❢♦r ✶✻✻✱✽ ♠♦♥t❤s✱ ♦r ❘✹✽ ✹✺✽✱✼✹✳ ❨♦✉ s✉r❡❧② ❦♥♦✇ ❜② ♥♦✇ t❤❛t t❤❡ ❞✐✛❡r❡♥❝❡

❜❡t✇❡❡♥ t❤❡ ❛❞❞✐t✐♦♥❛❧ ❘✹✽ ✹✺✽✱✼✹ t❤❛t ②♦✉ ❤❛✈❡ ♣❛✐❞ ❛♥❞ t❤❡ ❘✼✻ ✻✼✺✱✷✵ ✐♥t❡r❡st t❤❛t ②♦✉ ❤❛✈❡ s❛✈❡❞ ✐s

❛ttr✐❜✉t❛❜❧❡ t♦✱ ②❡s✱ ②♦✉ ❤❛✈❡ ❣♦t ✐t✱ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st✦

✷✳✶✳✸✳✸ ❋✉t✉r❡ ❱❛❧✉❡ ♦❢ ❛ s❡r✐❡s ♦❢ P❛②♠❡♥ts

■♥ t❤❡ s❛♠❡ ✇❛② t❤❛t ✇❤❡♥ ✇❡ ❤❛✈❡ ❛ s✐♥❣❧❡ ♣❛②♠❡♥t✱ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ ❛ ♣r❡s❡♥t ✈❛❧✉❡ ♦r ❛ ❢✉t✉r❡ ✈❛❧✉❡ ✲

✇❡ ❝❛♥ ❛❧s♦ ❞♦ t❤❛t ✇❤❡♥ ✇❡ ❤❛✈❡ ❛ s❡r✐❡s ♦❢ ♣❛②♠❡♥ts✳

■♥ t❤❡ ❛❜♦✈❡ s❡❝t✐♦♥✱ ✇❡ ❤❛❞ ❛ ❢❡✇ ♣❛②♠❡♥ts✱ ❛♥❞ ✇❡ ✇❛♥t❡❞ t♦ ❦♥♦✇ ✇❤❛t t❤❡② ❛r❡ ✇♦rt❤ ♥♦✇ ✲ s♦ ✇❡

❝❛❧❝✉❧❛t❡❞ ♣r❡s❡♥t ✈❛❧✉❡s✳ ❇✉t t❤❡ ♦t❤❡r ♣♦ss✐❜❧❡ s✐t✉❛t✐♦♥ ✐s t❤❛t ✇❡ ✇❛♥t t♦ ❧♦♦❦ ❛t t❤❡ ❢✉t✉r❡ ✈❛❧✉❡ ♦❢ ❛

s❡r✐❡s ♦❢ ♣❛②♠❡♥ts✳

▼❛②❜❡ ②♦✉ ✇❛♥t t♦ s❛✈❡ ✉♣ ❢♦r ❛ ❝❛r✱ ✇❤✐❝❤ ✇✐❧❧ ❝♦st ❘✹✺ ✵✵✵ ✲ ❛♥❞ ②♦✉ ✇♦✉❧❞ ❧✐❦❡ t♦ ❜✉② ✐t ✐♥ ✷ ②❡❛rs

t✐♠❡✳ ❨♦✉ ❤❛✈❡ ❛ s❛✈✐♥❣s ❛❝❝♦✉♥t ✇❤✐❝❤ ♣❛②s ✐♥t❡r❡st ♦❢ ✶✷✪ ♣❡r ❛♥♥✉♠✳ ❨♦✉ ♥❡❡❞ t♦ ✇♦r❦ ♦✉t ❤♦✇ ♠✉❝❤

t♦ ♣✉t ✐♥t♦ ②♦✉r ❜❛♥❦ ❛❝❝♦✉♥t ♥♦✇✱ ❛♥❞ t❤❡♥ ❛❣❛✐♥ ❡❛❝❤ ♠♦♥t❤ ❢♦r ✷ ②❡❛rs✱ ✉♥t✐❧ ②♦✉ ❛r❡ r❡❛❞② t♦ ❜✉② t❤❡

❝❛r✳❈❛♥ ②♦✉ s❡❡ t❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤✐s ❡①❛♠♣❧❡ ❛♥❞ t❤❡ ♦♥❡s ❛t t❤❡ st❛rt ♦❢ t❤❡ ❝❤❛♣t❡r ✇❤❡r❡ ✇❡ ✇❡r❡

♦♥❧② ♠❛❦✐♥❣ ❛ s✐♥❣❧❡ ♣❛②♠❡♥t ✐♥t♦ t❤❡ ❜❛♥❦ ❛❝❝♦✉♥t ✲ ✇❤❡r❡❛s ♥♦✇ ✇❡ ❛r❡ ♠❛❦✐♥❣ ❛ s❡r✐❡s ♦❢ ♣❛②♠❡♥ts ✐♥t♦

t❤❡ s❛♠❡ ❛❝❝♦✉♥t❄ ❚❤✐s ✐s ❛ s✐♥❦✐♥❣ ❢✉♥❞✳

❙♦✱ ✉s✐♥❣ ♦✉r ✉s✉❛❧ ♥♦t❛t✐♦♥✱ ❧❡t ✉s ✇r✐t❡ ♦✉t t❤❡ ❛♥s✇❡r✳ ▼❛❦❡ s✉r❡ ②♦✉ ❛❣r❡❡ ❤♦✇ ✇❡ ❝♦♠❡ ✉♣ ✇✐t❤

t❤✐s✳ ❇❡❝❛✉s❡ ✇❡ ❛r❡ ♠❛❦✐♥❣ ♠♦♥t❤❧② ♣❛②♠❡♥ts✱ ❡✈❡r②t❤✐♥❣ ♥❡❡❞s t♦ ❜❡ ✐♥ ♠♦♥t❤s✳ ❙♦ ❧❡t A ❜❡ t❤❡ ❝❧♦s✐♥❣

❜❛❧❛♥❝❡ ②♦✉ ♥❡❡❞ t♦ ❜✉② ❛ ❝❛r✱ P ✐s ❤♦✇ ♠✉❝❤ ②♦✉ ♥❡❡❞ t♦ ♣❛② ✐♥t♦ t❤❡ ❜❛♥❦ ❛❝❝♦✉♥t ❡❛❝❤ ♠♦♥t❤✱ ❛♥❞ i12

✐s t❤❡ ♠♦♥t❤❧② ✐♥t❡r❡st r❛t❡✳ ✭❈❛r❡❢✉❧ ✲ ❜❡❝❛✉s❡ ✶✷✪ ✐s t❤❡ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡✱ s♦ ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ✇♦r❦ ♦✉t

❧❛t❡r ✇❤❛t t❤❡ ♠♦♥t❤❧② ✐♥t❡r❡st r❛t❡ ✐s✦✮

A = P (1 + i12)24 + P (1 + i12)23 + ... + P (1 + i12)1

✭✷✳✶✸✮

❍❡r❡ ❛r❡ s♦♠❡ ✐♠♣♦rt❛♥t ♣♦✐♥ts t♦ r❡♠❡♠❜❡r ✇❤❡♥ ❞❡r✐✈✐♥❣ t❤✐s ❢♦r♠✉❧❛✿

✶✳ ❲❡ ❛r❡ ❝❛❧❝✉❧❛t✐♥❣ ❢✉t✉r❡ ✈❛❧✉❡s✱ s♦ ✐♥ t❤✐s ❡①❛♠♣❧❡ ✇❡ ✉s❡

−

(1 + i12)n ❛♥❞ ♥♦t (1 + i12) n✳ ❈❤❡❝❦ ❜❛❝❦

t♦ t❤❡ st❛rt ♦❢ t❤❡ ❝❤❛♣t❡r ✐❢ t❤✐s ✐s ♥♦t ♦❜✈✐♦✉s t♦ ②♦✉ ❜② ♥♦✇✳

✷✳ ■❢ ②♦✉ ❞r❛✇ ❛ t✐♠❡❧✐♥❡ ②♦✉ ✇✐❧❧ s❡❡ t❤❛t t❤❡ t✐♠❡ ❜❡t✇❡❡♥ t❤❡ ✜rst ♣❛②♠❡♥t ❛♥❞ ✇❤❡♥ ②♦✉ ❜✉② t❤❡ ❝❛r

✐s ✷✹ ♠♦♥t❤s✱ ✇❤✐❝❤ ✐s ✇❤② ✇❡ ✉s❡ ✷✹ ✐♥ t❤❡ ✜rst ❡①♣♦♥❡♥t✳

✸✳ ❆❣❛✐♥✱ ❧♦♦❦✐♥❣ ❛t t❤❡ t✐♠❡❧✐♥❡✱ ②♦✉ ❝❛♥ s❡❡ t❤❛t t❤❡ ✷✹t❤ ♣❛②♠❡♥t ✐s ❜❡✐♥❣ ♠❛❞❡ ♦♥❡ ♠♦♥t❤ ❜❡❢♦r❡ ②♦✉

❜✉② t❤❡ ❝❛r ✲ ✇❤✐❝❤ ✐s ✇❤② t❤❡ ❧❛st ❡①♣♦♥❡♥t ✐s ❛ ✶✳

✹✳ ❆❧✇❛②s ❝❤❡❝❦ t❤❛t ②♦✉ ❤❛✈❡ ❣♦t t❤❡ r✐❣❤t ♥✉♠❜❡r ♦❢ ♣❛②♠❡♥ts ✐♥ t❤❡ ❡q✉❛t✐♦♥✳ ❈❤❡❝❦ r✐❣❤t ♥♦✇ t❤❛t

②♦✉ ❛❣r❡❡ t❤❛t t❤❡r❡ ❛r❡ ✷✹ t❡r♠s ✐♥ t❤❡ ❢♦r♠✉❧❛ ❛❜♦✈❡✳

✸✺

❙♦✱ ♥♦✇ t❤❛t ✇❡ ❤❛✈❡ t❤❡ r✐❣❤t st❛rt✐♥❣ ♣♦✐♥t✱ ❧❡t ✉s s✐♠♣❧✐❢② t❤✐s ❡q✉❛t✐♦♥✿

A

=

P (1 + i12)24 + (1 + i12)23 + ... + (1 + i12)1

✭✷✳✶✹✮

=

P X24 + X23 + ... + X1 usingX = (1 + i12)

◆♦t❡ t❤❛t t❤✐s t✐♠❡ ❳ ❤❛s ❛ ♣♦s✐t✐✈❡ ❡①♣♦♥❡♥t ♥♦t ❛ ♥❡❣❛t✐✈❡ ❡①♣♦♥❡♥t✱ ❜❡❝❛✉s❡ ✇❡ ❛r❡ ❞♦✐♥❣ ❢✉t✉r❡ ✈❛❧✉❡s✳

❚❤✐s ✐s ♥♦t ❛ r✉❧❡ ②♦✉ ❤❛✈❡ t♦ ♠❡♠♦r✐s❡ ✲ ②♦✉ ❝❛♥ s❡❡ ❢r♦♠ t❤❡ ❡q✉❛t✐♦♥ ✇❤❛t t❤❡ ♦❜✈✐♦✉s ❝❤♦✐❝❡ ♦❢ ❳ s❤♦✉❧❞

❜❡✳ ▲❡t ✉s r❡✲♦r❞❡r t❤❡ t❡r♠s✿

A = P X1 + X2 + ... + X24 = P · X 1 + X + X2 + ... + X23

✭✷✳✶✺✮

❚❤✐s ✐s ❥✉st ❛♥♦t❤❡r s✉♠ ♦❢ ❛ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡✱ ✇❤✐❝❤ ❛s ②♦✉ ❦♥♦✇ ❝❛♥ ❜❡ s✐♠♣❧✐✜❡❞ ❛s✿

A

=

P · X [Xn − 1] / ((1 + i12) − 1)

✭✷✳✶✻✮

=

P · X [Xn − 1] /i12

❙♦ ✐❢ ✇❡ ✇❛♥t t♦ ✉s❡ ♦✉r ♥✉♠❜❡rs✱ ✇❡ ❦♥♦✇ t❤❛t A ❂ ❘✹✺ ✵✵✵✱ n❂✷✹ ✭❜❡❝❛✉s❡ ✇❡ ❛r❡ ❧♦♦❦✐♥❣ ❛t ♠♦♥t❤❧②

♣❛②♠❡♥ts✱ s♦ t❤❡r❡ ❛r❡ ✷✹ ♠♦♥t❤s ✐♥✈♦❧✈❡❞✮ ❛♥❞ i = 12% ♣❡r ❛♥♥✉♠✳

❇❯❚ ✭❛♥❞ ✐t ✐s ❛ ❜✐❣ ❜✉t✮ ✇❡ ♥❡❡❞ ❛ ♠♦♥t❤❧② ✐♥t❡r❡st r❛t❡✳ ❉♦ ♥♦t ❢♦r❣❡t t❤❛t t❤❡ tr✐❝❦ ✐s t♦ ❦❡❡♣ t❤❡

t✐♠❡ ♣❡r✐♦❞s ❛♥❞ t❤❡ ✐♥t❡r❡st r❛t❡s ✐♥ t❤❡ s❛♠❡ ✉♥✐ts ✲ s♦ ✐❢ ✇❡ ❤❛✈❡ ♠♦♥t❤❧② ♣❛②♠❡♥ts✱ ♠❛❦❡ s✉r❡ ②♦✉ ✉s❡

❛ ♠♦♥t❤❧② ✐♥t❡r❡st r❛t❡✦ ❯s✐♥❣ t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ ●r❛❞❡ ✶✶✱ ✇❡ ❦♥♦✇ t❤❛t (1 + i) = (1 + i12)12✳ ❙♦ ✇❡ ❝❛♥

s❤♦✇ t❤❛t i12 = 0, 0094888 = 0, 94888%✳

❚❤❡r❡❢♦r❡✱

45 000

=

P (1, 0094888) (1, 0094888)24 − 1 /0, 0094888

✭✷✳✶✼✮

P

=

1662, 67

❚❤✐s ♠❡❛♥s ②♦✉ ♥❡❡❞ t♦ ✐♥✈❡st ❘✶✻✻ ✷✻✼ ❡❛❝❤ ♠♦♥t❤ ✐♥t♦ t❤❛t ❜❛♥❦ ❛❝❝♦✉♥t t♦ ❜❡ ❛❜❧❡ t♦ ♣❛② ❢♦r ②♦✉r ❝❛r

✐♥ ✷ ②❡❛rs t✐♠❡✳

✷✳✶✳✸✳✹ ❊①❡r❝✐s❡s ✲ Pr❡s❡♥t ❛♥❞ ❋✉t✉r❡ ❱❛❧✉❡s

✶✳ ❨♦✉ ❤❛✈❡ t❛❦❡♥ ♦✉t ❛ ♠♦rt❣❛❣❡ ❜♦♥❞ ❢♦r ❘✽✼✺ ✵✵✵ t♦ ❜✉② ❛ ✢❛t✳ ❚❤❡ ❜♦♥❞ ✐s ❢♦r ✸✵ ②❡❛rs ❛♥❞ t❤❡

✐♥t❡r❡st r❛t❡ ✐s ✶✷✪ ♣❡r ❛♥♥✉♠ ♣❛②❛❜❧❡ ♠♦♥t❤❧②✳

❛✳ ❲❤❛t ✐s t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥t ♦♥ t❤❡ ❜♦♥❞❄

❜✳ ❍♦✇ ♠✉❝❤ ✐♥t❡r❡st ✇✐❧❧ ❜❡ ♣❛✐❞ ✐♥ t♦t❛❧ ♦✈❡r t❤❡ ✸✵ ②❡❛rs❄

✷✳ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ♠✉st ❜❡ ✐♥✈❡st❡❞ ♥♦✇ t♦ ♦❜t❛✐♥ r❡❣✉❧❛r ❛♥♥✉✐t② ♣❛②♠❡♥ts ♦❢ ❘ ✺ ✺✵✵ ♣❡r ♠♦♥t❤

❢♦r ✜✈❡ ②❡❛rs ❄ ❚❤❡ ♠♦♥❡② ✐s ✐♥✈❡st❡❞ ❛t ✶✶✱✶✪ ♣✳❛✳✱ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✳ ✭❆♥s✇❡r t♦ t❤❡ ♥❡❛r❡st

❤✉♥❞r❡❞ r❛♥❞✮

✷✳✷ ■♥✈❡st♠❡♥ts ✫ ▲♦❛♥s✱ ▲♦❛♥ ❙❝❤❡❞✉❧❡s✱ ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣✱

❋♦r♠✉❧❛❡✷

✷✳✷✳✶ ■♥✈❡st♠❡♥ts ❛♥❞ ▲♦❛♥s

❇② ♥♦✇✱ ②♦✉ s❤♦✉❧❞ ❜❡ ✇❡❧❧ ❡q✉✐♣♣❡❞ t♦ ♣❡r❢♦r♠ ❝❛❧❝✉❧❛t✐♦♥s ✇✐t❤ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st✳ ❚❤✐s s❡❝t✐♦♥ ❛✐♠s t♦

❛❧❧♦✇ ②♦✉ t♦ ✉s❡ t❤❡s❡ ✈❛❧✉❛❜❧❡ s❦✐❧❧s t♦ ❝r✐t✐❝❛❧❧② ❛♥❛❧②s❡ ✐♥✈❡st♠❡♥t ❛♥❞ ❧♦❛♥ ♦♣t✐♦♥s t❤❛t ②♦✉ ✇✐❧❧ ❝♦♠❡

❛❝r♦ss ✐♥ ②♦✉r ❧❛t❡r ❧✐❢❡✳ ❚❤✐s ✇❛②✱ ②♦✉ ✇✐❧❧ ❜❡ ❛❜❧❡ t♦ ♠❛❦❡ ✐♥❢♦r♠❡❞ ❞❡❝✐s✐♦♥s ♦♥ ♦♣t✐♦♥s ♣r❡s❡♥t❡❞ t♦ ②♦✉✳

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✾✹✴✶✳✶✴❃✳

✸✻

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❆t t❤✐s st❛❣❡✱ ②♦✉ s❤♦✉❧❞ ✉♥❞❡rst❛♥❞ t❤❡ ♠❛t❤❡♠❛t✐❝❛❧ t❤❡♦r② ❜❡❤✐♥❞ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st✳ ❍♦✇❡✈❡r✱ t❤❡

♥✉♠❡r✐❝❛❧ ✐♠♣❧✐❝❛t✐♦♥s ♦❢ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st ❛r❡ ♦❢t❡♥ s✉❜t❧❡ ❛♥❞ ❢❛r ❢r♦♠ ♦❜✈✐♦✉s✳

❘❡❝❛❧❧ t❤❡ ❡①❛♠♣❧❡ ❵❙❤♦✇ ♠❡ t❤❡ ♠♦♥❡②✬ ✐♥ ✧Pr❡s❡♥t ❱❛❧✉❡s ♦❢ ❛ s❡r✐❡s ♦❢ P❛②♠❡♥ts✧✳ ❋♦r ❛♥ ❡①tr❛

♣❛②♠❡♥t ♦❢ ❘✷✾ ✵✺✷ ❛ ♠♦♥t❤✱ ✇❡ ❝♦✉❧❞ ❤❛✈❡ ♣❛✐❞ ♦✛ ♦✉r ❧♦❛♥ ✐♥ ❧❡ss t❤❛♥ ✶✹ ②❡❛rs ✐♥st❡❛❞ ♦❢ ✷✵ ②❡❛rs✳ ❚❤✐s

♣r♦✈✐❞❡s ❛ ❣♦♦❞ ✐❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❧♦♥❣ t❡r♠ ❡✛❡❝t ♦❢ ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st t❤❛t ✐s ♦❢t❡♥ s✉r♣r✐s✐♥❣✳ ■♥ t❤❡

❢♦❧❧♦✇✐♥❣ s❡❝t✐♦♥✱ ✇❡✬❧❧ ❛✐♠ t♦ ❡①♣❧❛✐♥ t❤❡ r❡❛s♦♥ ❢♦r t❤❡ ❞r❛st✐❝ r❡❞✉❝t✐♦♥ ✐♥ t❤❡ t✐♠❡ ✐t t❛❦❡s t♦ r❡♣❛② t❤❡

❧♦❛♥✳

✷✳✷✳✶✳✶ ▲♦❛♥ ❙❝❤❡❞✉❧❡s

❙♦ ❢❛r✱ ✇❡ ❤❛✈❡ ❜❡❡♥ ✇♦r❦✐♥❣ ♦✉t ❧♦❛♥ r❡♣❛②♠❡♥t ❛♠♦✉♥ts ❜② t❛❦✐♥❣ ❛❧❧ t❤❡ ♣❛②♠❡♥ts ❛♥❞ ❞✐s❝♦✉♥t✐♥❣ t❤❡♠

❜❛❝❦ t♦ t❤❡ ♣r❡s❡♥t t✐♠❡✳ ❲❡ ❛r❡ ♥♦t ❝♦♥s✐❞❡r✐♥❣ t❤❡ r❡♣❛②♠❡♥ts ✐♥❞✐✈✐❞✉❛❧❧②✳ ❚❤✐♥❦ ❛❜♦✉t t❤❡ t✐♠❡ ②♦✉

♠❛❦❡ ❛ r❡♣❛②♠❡♥t t♦ t❤❡ ❜❛♥❦✳ ❚❤❡r❡ ❛r❡ ♥✉♠❡r♦✉s q✉❡st✐♦♥s t❤❛t ❝♦✉❧❞ ❜❡ r❛✐s❡❞✿ ❤♦✇ ♠✉❝❤ ❞♦ ②♦✉ st✐❧❧

♦✇❡ t❤❡♠❄ ❙✐♥❝❡ ②♦✉ ❛r❡ ♣❛②✐♥❣ ♦✛ t❤❡ ❧♦❛♥✱ s✉r❡❧② ②♦✉ ♠✉st ♦✇❡ t❤❡♠ ❧❡ss ♠♦♥❡②✱ ❜✉t ❤♦✇ ♠✉❝❤ ❧❡ss❄ ❲❡

❦♥♦✇ t❤❛t ✇❡✬❧❧ ❜❡ ♣❛②✐♥❣ ✐♥t❡r❡st ♦♥ t❤❡ ♠♦♥❡② ✇❡ st✐❧❧ ♦✇❡ t❤❡ ❜❛♥❦✳ ❲❤❡♥ ❡①❛❝t❧② ❞♦ ✇❡ ♣❛② ✐♥t❡r❡st❄

❍♦✇ ♠✉❝❤ ✐♥t❡r❡st ❛r❡ ✇❡ ♣❛②✐♥❣❄

❚❤❡ ❛♥s✇❡r t♦ t❤❡s❡ q✉❡st✐♦♥s ❧✐❡ ✐♥ s♦♠❡t❤✐♥❣ ❝❛❧❧❡❞ t❤❡ ❧♦❛❞ s❝❤❡❞✉❧❡✳

❲❡ ✇✐❧❧ ❝♦♥t✐♥✉❡ t♦ ✉s❡ t❤❡ ❡❛r❧✐❡r ❡①❛♠♣❧❡✳ ❚❤❡r❡ ✐s ❛ ❧♦❛♥ ❛♠♦✉♥t ♦❢ ❘✶✾✵ ✵✵✵✳ ❲❡ ❛r❡ ♣❛②✐♥❣ ✐t ♦✛

♦✈❡r ✷✵ ②❡❛rs ❛t ❛♥ ✐♥t❡r❡st ♦❢ ✾✪ ♣❡r ❛♥♥✉♠ ♣❛②❛❜❧❡ ♠♦♥t❤❧②✳ ❲❡ ✇♦r❦❡❞ ♦✉t t❤❛t t❤❡ r❡♣❛②♠❡♥ts s❤♦✉❧❞

❜❡ ❘✶ ✼✵✾✱✹✽✳

❈♦♥s✐❞❡r t❤❡ ✜rst ♣❛②♠❡♥t ♦❢ ❘✶ ✼✵✾✱✹✽ ♦♥❡ ♠♦♥t❤ ✐♥t♦ t❤❡ ❧♦❛♥✳ ❋✐rst✱ ✇❡ ❝❛♥ ✇♦r❦ ♦✉t ❤♦✇ ♠✉❝❤

✐♥t❡r❡st ✇❡ ♦✇❡ t❤❡ ❜❛♥❦ ❛t t❤✐s ♠♦♠❡♥t✳ ❲❡ ❜♦rr♦✇❡❞ ❘✶✾✵ ✵✵✵ ❛ ♠♦♥t❤ ❛❣♦✱ s♦ ✇❡ s❤♦✉❧❞ ♦✇❡✿

I

=

M × i12

=

R190 000 × 0, 75%

✭✷✳✶✽✮

=

R1 425

❲❡ ❛r❡ ♣❛②✐♥❣ t❤❡♠ ❘✶ ✹✷✺ ✐♥ ✐♥t❡r❡st✳ ❲❡ ❝❛❧❧ t❤✐s t❤❡ ✐♥t❡r❡st ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ r❡♣❛②♠❡♥t✳ ❲❡ ❛r❡

♦♥❧② ♣❛②✐♥❣ ♦✛ ❘✶ ✼✵✾✱✹✽ ✲ ❘✶ ✹✷✺ ❂ ❘✷✽✹✳✹✽ ♦❢ ✇❤❛t ✇❡ ♦✇❡✦ ❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ ❝❛♣✐t❛❧ ❝♦♠♣♦♥❡♥t✳ ❚❤❛t

♠❡❛♥s ✇❡ st✐❧❧ ♦✇❡ ❘✶✾✵ ✵✵✵ ✲ ❘✷✽✹✱✹✽ ❂ ❘✶✽✾ ✼✶✺✱✺✷✳ ❚❤✐s ✐s ❝❛❧❧❡❞ t❤❡ ❝❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣✳ ▲❡t✬s s❡❡

✇❤❛t ❤❛♣♣❡♥s ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❝♦♥❞ ♠♦♥t❤✳ ❚❤❡ ❛♠♦✉♥t ♦❢ ✐♥t❡r❡st ✇❡ ♥❡❡❞ t♦ ♣❛② ✐s t❤❡ ✐♥t❡r❡st ♦♥ t❤❡

❝❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣✳

I

=

M × i12

=

R189 715, 52 × 0, 75%

✭✷✳✶✾✮

=

R1 422, 87

❙✐♥❝❡ ✇❡ ❞♦♥✬t ♦✇❡ t❤❡ ❜❛♥❦ ❛s ♠✉❝❤ ❛s ✇❡ ❞✐❞ ❧❛st t✐♠❡✱ ✇❡ ❛❧s♦ ♦✇❡ ❛ ❧✐tt❧❡ ❧❡ss ✐♥t❡r❡st✳ ❚❤❡ ❝❛♣✐t❛❧

❝♦♠♣♦♥❡♥t ♦❢ t❤❡ r❡♣❛②♠❡♥t ✐s ♥♦✇ ❘✶ ✼✵✾✱✹✽ ✲ ❘✶ ✹✷✷✱✽✼ ❂ ❘✷✽✻✱✻✶✳ ❚❤❡ ❝❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣ ✇✐❧❧ ❜❡

❘✶✽✾ ✼✶✺✱✺✷ ✲ ❘✷✽✻✱✻✶ ❂ ❘✶✽✾ ✹✷✽✱✾✶✳ ❚❤✐s ✇❛②✱ ✇❡ ❝❛♥ ❜r❡❛❦ ❡❛❝❤ ♦❢ ♦✉r r❡♣❛②♠❡♥ts ❞♦✇♥ ✐♥t♦ ❛♥ ✐♥t❡r❡st

♣❛rt ❛♥❞ t❤❡ ♣❛rt t❤❛t ❣♦❡s t♦✇❛r❞s ♣❛②✐♥❣ ♦✛ t❤❡ ❧♦❛♥✳

❚❤✐s ✐s ❛ s✐♠♣❧❡ ❛♥❞ r❡♣❡t✐t✐✈❡ ♣r♦❝❡ss✳ ❚❛❜❧❡ ✷✳✷ ✐s ❛ t❛❜❧❡ s❤♦✇✐♥❣ t❤❡ ❜r❡❛❦❞♦✇♥ ♦❢ t❤❡ ✜rst ✶✷

♣❛②♠❡♥ts✳ ❚❤✐s ✐s ❝❛❧❧❡❞ ❛ ❧♦❛♥ s❝❤❡❞✉❧❡✳

✸✼

❚✐♠❡

❘❡♣❛②♠❡♥t

■♥t❡r❡st ❈♦♠♣♦♥❡♥t ❈❛♣✐t❛❧ ❈♦♠♣♦♥❡♥t ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣

✵

❘ ✶✾✵ ✵✵✵ ✵✵

✶

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✷✺ ✵✵

❘ ✷✽✹ ✹✽

❘ ✶✽✾ ✼✶✺ ✺✷

✷

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✷✷ ✽✼

❘ ✷✽✻ ✻✶

❘ ✶✽✾ ✹✷✽ ✾✶

✸

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✷✵ ✼✷

❘ ✷✽✽ ✼✻

❘ ✶✽✾ ✶✹✵ ✶✹

✹

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✶✽ ✺✺

❘ ✷✾✵ ✾✸

❘ ✶✽✽ ✽✹✾ ✷✶

✺

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✶✻ ✸✼

❘ ✷✾✸ ✶✶

❘ ✶✽✽ ✺✺✻ ✶✵

✻

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✶✹ ✶✼

❘ ✷✾✺ ✸✶

❘ ✶✽✽ ✷✻✵ ✼✾

✼

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✶✶ ✾✻

❘ ✷✾✼ ✺✷

❘ ✶✽✼ ✾✻✸ ✷✼

✽

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✵✾ ✼✷

❘ ✷✾✾ ✼✻

❘ ✶✽✼ ✻✻✸ ✺✶

✾

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✵✼ ✹✽

❘ ✸✵✷ ✵✵

❘ ✶✽✼ ✸✻✶ ✺✶

✶✵

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✵✺ ✷✶

❘ ✸✵✹ ✷✼

❘ ✶✽✼ ✵✺✼ ✷✹

✶✶

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✵✷ ✾✸

❘ ✸✵✻ ✺✺

❘ ✶✽✻ ✼✺✵ ✻✾

✶✷

❘ ✶ ✼✵✾ ✹✽ ❘ ✶ ✹✵✵ ✻✸

❘ ✸✵✽ ✽✺

❘ ✶✽✻ ✹✹✶ ✽✹

❚❛❜❧❡ ✷✳✷✿ ❆ ❧♦❛♥ s❝❤❡❞✉❧❡ ✇✐t❤ r❡♣❛②♠❡♥ts ♦❢ ❘✶ ✼✵✾✱✹✽ ♣❡r ♠♦♥t❤✳

◆♦✇✱ ❧❡t✬s s❡❡ t❤❡ s❛♠❡ t❤✐♥❣ ❛❣❛✐♥✱ ❜✉t ✇✐t❤ ❘✷ ✵✵✵ ❜❡✐♥❣ r❡♣❛✐❞ ❡❛❝❤ ②❡❛r✳ ❲❡ ❡①♣❡❝t t❤❡ ♥✉♠❜❡rs t♦

❝❤❛♥❣❡✳ ❍♦✇❡✈❡r✱ ❤♦✇ ♠✉❝❤ ✇✐❧❧ t❤❡② ❝❤❛♥❣❡ ❜②❄ ❆s ❜❡❢♦r❡✱ ✇❡ ♦✇❡ ❘✶ ✹✷✺ ✐♥ ✐♥t❡r❡st ✐♥ ✐♥t❡r❡st✳ ❆❢t❡r

♦♥❡ ♠♦♥t❤✳ ❍♦✇❡✈❡r✱ ✇❡ ❛r❡ ♣❛②✐♥❣ ❘✷ ✵✵✵ t❤✐s t✐♠❡✳ ❚❤❛t ❧❡❛✈❡s ❘✺✼✺ t❤❛t ❣♦❡s t♦✇❛r❞s ♣❛②✐♥❣ ♦✛ t❤❡

❝❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣✱ r❡❞✉❝✐♥❣ ✐t t♦ ❘✶✽✾ ✹✷✺✳ ❇② t❤❡ ❡♥❞ ♦❢ t❤❡ s❡❝♦♥❞ ♠♦♥t❤✱ t❤❡ ✐♥t❡r❡st ♦✇❡❞ ✐s ❘✶ ✹✷✵✱✻✾

✭❚❤❛t✬s ❘✶✽✾ ✹✷✺×i12✮✳ ❖✉r ❘✷ ✵✵✵ ♣❛②s ❢♦r t❤❛t ✐♥t❡r❡st✱ ❛♥❞ r❡❞✉❝❡s t❤❡ ❝❛♣✐t❛❧ ❛♠♦✉♥t ♦✇❡❞ ❜② ❘✷ ✵✵✵

✲ ❘✶ ✹✷✵✱✻✾ ❂ ❘✺✼✾✱✸✶✳ ❚❤✐s r❡❞✉❝❡s t❤❡ ❛♠♦✉♥t ♦✉tst❛♥❞✐♥❣ t♦ ❘✶✽✽ ✽✹✺✱✻✾✳

❉♦✐♥❣ t❤❡ s❛♠❡ ❝❛❧❝✉❧❛t✐♦♥s ❛s ❜❡❢♦r❡ ②✐❡❧❞s ❛ ♥❡✇ ❧♦❛♥ s❝❤❡❞✉❧❡ s❤♦✇♥ ✐♥ ❚❛❜❧❡ ✷✳✸✳

❚✐♠❡

❘❡♣❛②♠❡♥t

■♥t❡r❡st ❈♦♠♣♦♥❡♥t ❈❛♣✐t❛❧ ❈♦♠♣♦♥❡♥t ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣

✵

❘ ✶✾✵ ✵✵✵ ✵✵

✶

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✹✷✺ ✵✵

❘ ✺✼✺ ✵✵

❘ ✶✽✾ ✹✷✺ ✵✵

✷

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✹✷✵ ✻✾

❘ ✺✼✾ ✸✶

❘ ✶✽✽ ✽✹✺ ✻✾

✸

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✹✶✻ ✸✹

❘ ✺✽✸ ✻✻

❘ ✶✽✽ ✷✻✷ ✵✸

✹

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✹✶✶ ✾✼

❘ ✺✽✽ ✵✸

❘ ✶✽✼ ✻✼✹ ✵✵

✺

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✹✵✼ ✺✺

❘ ✺✾✷ ✹✺

❘ ✶✽✼ ✵✽✶ ✺✺

✻

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✹✵✸ ✶✶

❘ ✺✾✻ ✽✾

❘ ✶✽✻ ✹✽✹ ✻✻

✼

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✸✾✽ ✻✸

❘ ✻✵✶ ✸✼

❘ ✶✽✺ ✽✽✸ ✸✵

✽

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✸✾✹ ✶✷

❘ ✻✵✺ ✽✽

❘ ✶✽✺ ✷✼✼ ✹✷

✾

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✸✽✾ ✺✽

❘ ✻✶✵ ✹✷

❘ ✶✽✹ ✻✻✼ ✵✵

✶✵

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✸✽✺ ✵✵

❘ ✻✶✺ ✵✵

❘ ✶✽✹ ✵✺✷ ✵✵

✶✶

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✸✽✵ ✸✾

❘ ✻✶✾ ✻✶

❘ ✶✽✸ ✹✸✷ ✸✾

✶✷

❘ ✷ ✵✵✵ ✵✵ ❘ ✶ ✸✼✺ ✼✹

❘ ✻✷✹ ✷✻

❘ ✶✽✷ ✽✵✽ ✶✹

✸✽

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❚❛❜❧❡ ✷✳✸✿ ❆ ❧♦❛♥ s❝❤❡❞✉❧❡ ✇✐t❤ r❡♣❛②♠❡♥ts ♦❢ ❘✷ ✵✵✵ ♣❡r ♠♦♥t❤✳

❚❤❡ ✐♠♣♦rt❛♥t ♥✉♠❜❡rs t♦ ♥♦t✐❝❡ ✐s t❤❡ ✏❈❛♣✐t❛❧ ❈♦♠♣♦♥❡♥t✧ ❝♦❧✉♠♥✳ ◆♦t❡ t❤❛t ✇❤❡♥ ✇❡ ❛r❡ ♣❛②✐♥❣

♦✛ ❘✷ ✵✵✵ ❛ ♠♦♥t❤ ❛s ❝♦♠♣❛r❡❞ t♦ ❘✶ ✼✵✾✱✹✽ ❛ ♠♦♥t❤✱ t❤✐s ❝♦❧✉♠♥ ♠♦r❡ t❤❛♥ ❞♦✉❜❧❡✳ ■♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢

♣❛②✐♥❣ ♦✛ ❛ ❧♦❛♥✱ ✈❡r② ❧✐tt❧❡ ♦❢ ♦✉r ♠♦♥❡② ✐s ✉s❡❞ t♦ ♣❛② ♦✛ t❤❡ ❝❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣✳ ❚❤❡r❡❢♦r❡✱ ❡✈❡♥ ❛ s♠❛❧❧

✐♥❝r❡❛s❡ ✐♥ r❡♣❛②♠❡♥t ❛♠♦✉♥ts ❝❛♥ s✐❣♥✐✜❝❛♥t❧② ✐♥❝r❡❛s❡ t❤❡ s♣❡❡❞ ❛t ✇❤✐❝❤ ✇❡ ❛r❡ ♣❛②✐♥❣ ♦✛ t❤❡ ❝❛♣✐t❛❧✳

❲❤❛t✬s ♠♦r❡✱ ❧♦♦❦ ❛t t❤❡ ❛♠♦✉♥t ✇❡ ❛r❡ st✐❧❧ ♦✇✐♥❣ ❛❢t❡r ♦♥❡ ②❡❛r ✭✐✳❡✳ ❛t t✐♠❡ ✶✷✮✳ ❲❤❡♥ ✇❡ ✇❡r❡

♣❛②✐♥❣ ❘✶ ✼✵✾✱✹✽ ❛ ♠♦♥t❤✱ ✇❡ st✐❧❧ ♦✇❡ ❘✶✽✻ ✹✹✶✱✽✹✳ ❍♦✇❡✈❡r✱ ✐❢ ✇❡ ✐♥❝r❡❛s❡ t❤❡ r❡♣❛②♠❡♥ts t♦ ❘✷ ✵✵✵ ❛

♠♦♥t❤✱ t❤❡ ❛♠♦✉♥t ♦✉tst❛♥❞✐♥❣ ❞❡❝r❡❛s❡s ❜② ♦✈❡r ❘✸ ✵✵✵ t♦ ❘✶✽✷ ✽✵✽✱✶✹✳ ❚❤✐s ♠❡❛♥s ✇❡ ✇♦✉❧❞ ❤❛✈❡ ♣❛✐❞

♦✛ ♦✈❡r ❘✼ ✵✵✵ ✐♥ ♦✉r ✜rst ②❡❛r ✐♥st❡❛❞ ♦❢ ❧❡ss t❤❛♥ ❘✹ ✵✵✵✳ ❚❤✐s ✐♥❝r❡❛s❡❞ s♣❡❡❞ ❛t ✇❤✐❝❤ ✇❡ ❛r❡ ♣❛②✐♥❣

♦✛ t❤❡ ❝❛♣✐t❛❧ ♣♦rt✐♦♥ ♦❢ t❤❡ ❧♦❛♥ ✐s ✇❤❛t ❛❧❧♦✇s ✉s t♦ ♣❛② ♦✛ t❤❡ ✇❤♦❧❡ ❧♦❛♥ ✐♥ ❛r♦✉♥❞ ✶✹ ②❡❛rs ✐♥st❡❛❞ ♦❢

t❤❡ ♦r✐❣✐♥❛❧ ✷✵✳ ◆♦t❡ ❤♦✇❡✈❡r✱ t❤❡ ❡✛❡❝t ♦❢ ♣❛②✐♥❣ ❘✷ ✵✵✵ ✐♥st❡❛❞ ♦❢ ❘✶ ✼✵✾✱✹✽ ✐s ♠♦r❡ s✐❣♥✐✜❝❛♥t ✐♥ t❤❡

❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ❧♦❛♥ t❤❛♥ ♥❡❛r t❤❡ ❡♥❞ ♦❢ t❤❡ ❧♦❛♥✳

■t ✐s ♥♦t❡❞ t❤❛t ✐♥ t❤✐s ✐♥st❛♥❝❡✱ ❜② ♣❛②✐♥❣ s❧✐❣❤t❧② ♠♦r❡ t❤❛♥ ✇❤❛t t❤❡ ❜❛♥❦ ✇♦✉❧❞ ❛s❦ ②♦✉ t♦ ♣❛②✱ ②♦✉

❝❛♥ ♣❛② ♦✛ ❛ ❧♦❛♥ ❛ ❧♦t q✉✐❝❦❡r✳ ❚❤❡ ♥❛t✉r❛❧ q✉❡st✐♦♥ t♦ ❛s❦ ❤❡r❡ ✐s✿ ✇❤② ❛r❡ ❜❛♥❦s ❛s❦✐♥❣ ✉s t♦ ♣❛② t❤❡

❧♦✇❡r ❛♠♦✉♥t ❢♦r ♠✉❝❤ ❧♦♥❣❡r t❤❡♥❄ ❆r❡ t❤❡② tr②✐♥❣ t♦ ❝❤❡❛t ✉s ♦✉t ♦❢ ♦✉r ♠♦♥❡②❄

❚❤❡r❡ ✐s ♥♦ s✐♠♣❧❡ ❛♥s✇❡r t♦ t❤✐s✳ ❇❛♥❦s ♣r♦✈✐❞❡ ❛ s❡r✈✐❝❡ t♦ ✉s ✐♥ r❡t✉r♥ ❢♦r ❛ ❢❡❡✱ s♦ t❤❡② ❛r❡ ♦✉t t♦

♠❛❦❡ ❛ ♣r♦✜t✳ ❍♦✇❡✈❡r✱ t❤❡② ♥❡❡❞ t♦ ❜❡ ❝❛r❡❢✉❧ ♥♦t t♦ ❝❤❡❛t t❤❡✐r ❝✉st♦♠❡rs ❢♦r ❢❡❛r t❤❛t t❤❡②✬❧❧ s✐♠♣❧② ✉s❡

❛♥♦t❤❡r ❜❛♥❦✳ ❚❤❡ ❝❡♥tr❛❧ ✐ss✉❡ ❤❡r❡ ✐s ♦♥❡ ♦❢ s❝❛❧❡✳ ❋♦r ✉s✱ t❤❡ ❝❤❛♥❣❡s ✐♥✈♦❧✈❡❞ ❛♣♣❡❛r ❜✐❣✳ ❲❡ ❛r❡ ♣❛②✐♥❣

♦✛ ♦✉r ❧♦❛♥ ✻ ②❡❛rs ❡❛r❧✐❡r ❜② ♣❛②✐♥❣ ❥✉st ❛ ❜✐t ♠♦r❡ ❛ ♠♦♥t❤✳ ❚♦ ❛ ❜❛♥❦✱ ❤♦✇❡✈❡r✱ ✐t ❞♦❡s♥✬t ♠❛tt❡r ♠✉❝❤

❡✐t❤❡r ✇❛②✳ ■♥ ❛❧❧ ❧✐❦❡❧✐❤♦♦❞✱ ✐t ❞♦❡s♥✬t ❛✛❡❝t t❤❡✐r ♣r♦✜t ♠❛r❣✐♥s ♦♥❡ ❜✐t✦

❘❡♠❡♠❜❡r t❤❛t t❤❡ ❜❛♥❦ ❝❛❧❝✉❧❛t❡s r❡♣❛②♠❡♥t ❛♠♦✉♥ts ✉s✐♥❣ t❤❡ s❛♠❡ ♠❡t❤♦❞s ❛s ✇❡✬✈❡ ❜❡❡♥ ❧❡❛r♥✐♥❣✳

❚❤❡② ❞❡❝✐❞❡ ♦♥ t❤❡ ❝♦rr❡❝t r❡♣❛②♠❡♥t ❛♠♦✉♥ts ❢♦r ❛ ❣✐✈❡♥ ✐♥t❡r❡st r❛t❡ ❛♥❞ s❡t ♦❢ t❡r♠s✳ ❙♠❛❧❧❡r r❡♣❛②♠❡♥t

❛♠♦✉♥ts ✇✐❧❧ ♠❛❦❡ t❤❡ ❜❛♥❦ ♠♦r❡ ♠♦♥❡②✱ ❜❡❝❛✉s❡ ✐t ✇✐❧❧ t❛❦❡ ②♦✉ ❧♦♥❣❡r t♦ ♣❛② ♦✛ t❤❡ ❧♦❛♥ ❛♥❞ ♠♦r❡

✐♥t❡r❡st ✇✐❧❧ ❛❝✉♠✉❧❛t❡✳ ▲❛r❣❡r r❡♣❛②♠❡♥t ❛♠♦✉♥ts ♠❡❛♥ t❤❛t ②♦✉ ✇✐❧❧ ♣❛② ♦✛ t❤❡ ❧♦❛♥ ❢❛st❡r✱ s♦ ②♦✉ ✇✐❧❧

❛❝❝✉♠✉❧❛t❡ ❧❡ss ✐♥t❡r❡st ✐✳❡✳ t❤❡ ❜❛♥❦ ✇✐❧❧ ♠❛❦❡ ❧❡ss ♠♦♥❡② ♦✛ ♦❢ ②♦✉✳ ■t✬s ❛ s✐♠♣❧❡ ♠❛tt❡r ♦❢ ❧❡ss ♠♦♥❡②

♥♦✇ ♦r ♠♦r❡ ♠♦♥❡② ❧❛t❡r✳ ❇❛♥❦s ❣❡♥❡r❛❧❧② ✉s❡ ❛ ✷✵ ②❡❛r r❡♣❛②♠❡♥t ♣❡r✐♦❞ ❜② ❞❡❢❛✉❧t✳

▲❡❛r♥✐♥❣ ❛❜♦✉t ✜♥❛♥❝✐❛❧ ♠❛t❤❡♠❛t✐❝s ❡♥❛❜❧❡s ②♦✉ t♦ ❞✉♣❧✐❝❛t❡ t❤❡s❡ ❝❛❧❝✉❧❛t✐♦♥s ❢♦r ②♦✉rs❡❧❢✳ ❚❤✐s ✇❛②✱

②♦✉ ❝❛♥ ❞❡❝✐❞❡ ✇❤❛t✬s ❜❡st ❢♦r ②♦✉✳ ❨♦✉ ❝❛♥ ❞❡❝✐❞❡ ❤♦✇ ♠✉❝❤ ②♦✉ ✇❛♥t t♦ r❡♣❛② ❡❛❝❤ ♠♦♥t❤ ❛♥❞ ②♦✉✬❧❧

❦♥♦✇ ♦❢ ✐ts ❡✛❡❝ts✳ ❆ ❜❛♥❦ ✇♦✉❧❞♥✬t ❝❛r❡ ♠✉❝❤ ❡✐t❤❡r ✇❛②✱ s♦ ②♦✉ s❤♦✉❧❞ ♣✐❝❦ s♦♠❡t❤✐♥❣ t❤❛t s✉✐ts ②♦✉✳

❊①❡r❝✐s❡ ✷✳✹✿ ▼♦♥t❤❧② P❛②♠❡♥ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✷✳✮

❙t❡❢❛♥ ❛♥❞ ▼❛r♥❛ ✇❛♥t t♦ ❜✉② ❛ ❤♦✉s❡ t❤❛t ❝♦sts ❘ ✶ ✷✵✵ ✵✵✵✳ ❚❤❡✐r ♣❛r❡♥ts ♦✛❡r t♦ ♣✉t ❞♦✇♥ ❛

✷✵✪ ♣❛②♠❡♥t t♦✇❛r❞s t❤❡ ❝♦st ♦❢ t❤❡ ❤♦✉s❡✳ ❚❤❡② ♥❡❡❞ t♦ ❣❡t ❛ ♠♦r❛t❛❣❡ ❢♦r t❤❡ ❜❛❧❛♥❝❡✳ ❲❤❛t

❛r❡ t❤❡✐r ♠♦♥t❤❧② r❡♣❛②♠❡♥ts ✐❢ t❤❡ t❡r♠ ♦❢ t❤❡ ❤♦♠❡ ❧♦❛♥ ✐s ✸✵ ②❡❛rs ❛♥❞ t❤❡ ✐♥t❡r❡st ✐s ✼✱✺✪✱

❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②❄

✷✳✷✳✶✳✷ ❊①❡r❝✐s❡s ✲ ■♥✈❡st♠❡♥ts ❛♥❞ ▲♦❛♥s

✶✳ ❆ ♣r♦♣❡rt② ❝♦sts ❘✶ ✽✵✵ ✵✵✵✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥ts ✐❢ t❤❡ ✐♥t❡r❡st r❛t❡ ✐s ✶✹✪ ♣✳❛✳

❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧② ❛♥❞ t❤❡ ❧♦❛♥ ♠✉st ❜❡ ♣❛✐❞ ♦✛ ✐♥ ✷✵ ②❡❛rs t✐♠❡✳

✷✳ ❆ ❧♦❛♥ ♦❢ ❘ ✹ ✷✵✵ ✐s t♦ ❜❡ r❡t✉r♥❡❞ ✐♥ t✇♦ ❡q✉❛❧ ❛♥♥✉❛❧ ✐♥st❛❧♠❡♥ts✳ ■❢ t❤❡ r❛t❡ ♦❢ ✐♥t❡r❡st ♦❢ ✶✵✪ ♣❡r

❛♥♥✉♠✱ ❝♦♠♣♦✉♥❞❡❞ ❛♥♥✉❛❧❧②✱ ❝❛❧❝✉❧❛t❡ t❤❡ ❛♠♦✉♥t ♦❢ ❡❛❝❤ ✐♥st❛❧♠❡♥t✳

✷✳✷✳✶✳✸ ❈❛❧❝✉❧❛t✐♥❣ ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣

❆s ❞❡✜♥❡❞ ✐♥ ✧▲♦❛♥ ❙❝❤❡❞✉❧❡s✧ ✭❙❡❝t✐♦♥ ✷✳✷✳✶✳✶✿ ▲♦❛♥ ❙❝❤❡❞✉❧❡s✮✱ ❈❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣ ✐s t❤❡ ❛♠♦✉♥t ✇❡ st✐❧❧

♦✇❡ t❤❡ ♣❡♦♣❧❡ ✇❡ ❜♦rr♦✇❡❞ ♠♦♥❡② ❢r♦♠ ❛t ❛ ❣✐✈❡♥ ♠♦♠❡♥t ✐♥ t✐♠❡✳ ❲❡ ❛❧s♦ s❛✇ ❤♦✇ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤✐s

✉s✐♥❣ ❧♦❛♥ s❝❤❡❞✉❧❡s✳ ❍♦✇❡✈❡r✱ t❤❡r❡ ✐s ❛ s✐❣♥✐✜❝❛♥t ❞✐s❛❞✈❛♥t❛❣❡ t♦ t❤✐s ♠❡t❤♦❞✿ ✐t ✐s ✈❡r② t✐♠❡ ❝♦♥s✉♠✐♥❣✳

❋♦r ❡①❛♠♣❧❡✱ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ ❤♦✇ ♠✉❝❤ ❝❛♣✐t❛❧ ✐s st✐❧❧ ♦✉tst❛♥❞✐♥❣ ❛t t✐♠❡ ✶✷ ✉s✐♥❣ t❤❡ ❧♦❛♥ s❝❤❡❞✉❧❡✱

✇❡✬❧❧ ❤❛✈❡ t♦ ✜rst ❝❛❧❝✉❧❛t❡ ❤♦✇ ♠✉❝❤ ❝❛♣✐t❛❧ ✐s ♦✉tst❛♥❞✐♥❣ ❛t t✐♠❡ ✶ t❤r♦✉❣❤ t♦ ✶✶ ❛s ✇❡❧❧✳ ❚❤✐s ✐s ❛❧r❡❛❞②

✸✾

q✉✐t❡ ❛ ❜✐t ♠♦r❡ ✇♦r❦ t❤❛♥ ✇❡✬❞ ❧✐❦❡ t♦ ❞♦✳ ❈❛♥ ②♦✉ ✐♠❛❣✐♥❡ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❛♠♦✉♥t ♦✉tst❛♥❞✐♥❣ ❛❢t❡r ✶✵

②❡❛rs ✭t✐♠❡ ✶✷✵✮❄

❋♦rt✉♥❛t❡❧②✱ t❤❡r❡ ✐s ❛♥ ❡❛s✐❡r ♠❡t❤♦❞✳ ❍♦✇❡✈❡r✱ ✐t ✐s ♥♦t ✐♠♠❡❞✐❛t❡❧② ❝❧❡❛r ✇❤② t❤✐s ✇♦r❦s✱ s♦ ❧❡t✬s t❛❦❡

s♦♠❡ t✐♠❡ t♦ ❡①❛♠✐♥❡ t❤❡ ❝♦♥❝❡♣t✳

✷✳✷✳✶✳✸✳✶ Pr♦s♣❡❝t✐✈❡ ♠❡t❤♦❞ ❢♦r ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣

▲❡t✬s s❛② t❤❛t ❛❢t❡r ❛ ❝❡rt❛✐♥ ♥✉♠❜❡r ♦❢ ②❡❛rs✱ ❥✉st ❛❢t❡r ✇❡ ♠❛❞❡ ❛ r❡♣❛②♠❡♥t✱ ✇❡ st✐❧❧ ♦✇❡ ❛♠♦✉♥t Y ✳ ❲❤❛t

❞♦ ✇❡ ❦♥♦✇ ❛❜♦✉t Y ❄ ❲❡ ❦♥♦✇ t❤❛t ✉s✐♥❣ t❤❡ ❧♦❛♥ s❝❤❡❞✉❧❡✱ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ ✇❤❛t ✐t ❡q✉❛❧s t♦✱ ❜✉t t❤❛t

✐s ❛ ❧♦t ♦❢ r❡♣❡t✐t✐✈❡ ✇♦r❦✳ ❲❡ ❛❧s♦ ❦♥♦✇ t❤❛t Y ✐s t❤❡ ❛♠♦✉♥t t❤❛t ✇❡ ❛r❡ st✐❧❧ ❣♦✐♥❣ t♦ ♣❛② ♦✛✳ ■♥ ♦t❤❡r

✇♦r❞s✱ ❛❧❧ t❤❡ r❡♣❛②♠❡♥ts ✇❡ ❛r❡ st✐❧❧ ❣♦✐♥❣ t♦ ♠❛❦❡ ✐♥ t❤❡ ❢✉t✉r❡ ✇✐❧❧ ❡①❛❝t❧② ♣❛② ♦✛ Y ✳ ❚❤✐s ✐s tr✉❡ ❜❡❝❛✉s❡

✐♥ t❤❡ ❡♥❞✱ ❛❢t❡r ❛❧❧ t❤❡ r❡♣❛②♠❡♥ts✱ ✇❡ ✇♦♥✬t ❜❡ ♦✇✐♥❣ ❛♥②t❤✐♥❣✳

❚❤❡r❡❢♦r❡✱ t❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ ❛❧❧ ♦✉tst❛♥❞✐♥❣ ❢✉t✉r❡ ♣❛②♠❡♥ts ❡q✉❛❧ t❤❡ ♣r❡s❡♥t ❛♠♦✉♥t ♦✉tst❛♥❞✐♥❣✳

❚❤✐s ✐s t❤❡ ♣r♦s♣❡❝t✐✈❡ ♠❡t❤♦❞ ❢♦r ❝❛❧❝✉❧❛t✐♥❣ ❝❛♣✐t❛❧ ♦✉tst❛♥❞✐♥❣✳

▲❡t✬s r❡t✉r♥ t♦ ❛ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✳ ❘❡❝❛❧❧ t❤❡ ❝❛s❡ ✇❤❡r❡ ✇❡ ✇❡r❡ tr②✐♥❣ t♦ r❡♣❛② ❛ ❧♦❛♥ ♦❢ ❘✷✵✵ ✵✵✵

♦✈❡r ✷✵ ②❡❛rs✳ ❆ ❘✶✵ ✵✵✵ ❞❡♣♦s✐t ✇❛s ♣✉t ❞♦✇♥✱ s♦ t❤❡ ❛♠♦✉♥t ❜❡✐♥❣ ♣❛②❡❞ ♦✛ ✇❛s ❘✶✾✵ ✵✵✵✳ ❆t ❛♥ ✐♥t❡r❡st

r❛t❡ ♦❢ ✾✪ ❝♦♠♣♦✉♥❞❡❞ ♠♦♥t❤❧②✱ t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥t ✇❛s ❘✶ ✼✵✾✱✹✽✳ ■♥ ❚❛❜❧❡ ✷✳✷✱ ✇❡ ❝❛♥ s❡❡ t❤❛t ❛❢t❡r

✶✷ ♠♦♥t❤s✱ t❤❡ ❛♠♦✉♥t ♦✉tst❛♥❞✐♥❣ ✇❛s ❘✶✽✻ ✹✹✶✱✽✹✳ ▲❡t✬s tr② t♦ ✇♦r❦ t❤✐s ♦✉t ✉s✐♥❣ t❤❡ t❤❡ ♣r♦s♣❡❝t✐✈❡

♠❡t❤♦❞✳

❆❢t❡r t✐♠❡ ✶✷✱ t❤❡r❡ ❛r❡ st✐❧❧ 19 × 12 = 228 r❡♣❛②♠❡♥ts ❧❡❢t ♦❢ ❘✶ ✼✵✾✱✹✽ ❡❛❝❤✳ ❚❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ✐s✿

n

=

228

i

=

0, 75%

✭✷✳✷✵✮

Y

=

R1 709, 48 × 1−1,0075−228

0,0075

=

R186 441, 92

❖♦♣s✦ ❚❤✐s s❡❡♠s t♦ ❜❡ ❛❧♠♦st r✐❣❤t✱ ❜✉t ♥♦t q✉✐t❡✳ ❲❡ s❤♦✉❧❞ ❤❛✈❡ ❣♦t ❘✶✽✻ ✹✹✶✱✽✹✳ ❲❡ ❛r❡ ✽ ❝❡♥ts ♦✉t✳

❍♦✇❡✈❡r✱ t❤✐s ✐s ✐♥ ❢❛❝t ♥♦t ❛ ♠✐st❛❦❡✳ ❘❡♠❡♠❜❡r t❤❛t ✇❤❡♥ ✇❡ ✇♦r❦❡❞ ♦✉t t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥ts✱ ✇❡

r♦✉♥❞❡❞ t♦ t❤❡ ♥❡❛r❡st ❝❡♥ts ❛♥❞ ❛rr✐✈❡❞ ❛t ❘✶ ✼✵✾✱✹✽✳ ❚❤✐s ✇❛s ❜❡❝❛✉s❡ ♦♥❡ ❝❛♥♥♦t ♠❛❦❡ ❛ ♣❛②♠❡♥t ❢♦r ❛

❢r❛❝t✐♦♥ ♦❢ ❛ ❝❡♥t✳ ❚❤❡r❡❢♦r❡✱ t❤❡ r♦✉♥❞✐♥❣ ♦✛ ❡rr♦r ✇❛s ❝❛rr✐❡❞ t❤r♦✉❣❤✳ ❚❤❛t✬s ✇❤② t❤❡ t✇♦ ✜❣✉r❡s ❞♦♥✬t

♠❛t❝❤ ❡①❛❝t❧②✳ ■♥ ✜♥❛♥❝✐❛❧ ♠❛t❤❡♠❛t✐❝s✱ t❤✐s ✐s ❧❛r❣❡❧② ✉♥❛✈♦✐❞❛❜❧❡✳

✷✳✷✳✷ ❋♦r♠✉❧❛❡ ❙❤❡❡t

❆s ❛♥ ❡❛s② r❡❢❡r❡♥❝❡✱ ❤❡r❡ ❛r❡ t❤❡ ❦❡② ❢♦r♠✉❧❛❡ t❤❛t ✇❡ ❞❡r✐✈❡❞ ❛♥❞ ✉s❡❞ ❞✉r✐♥❣ t❤✐s ❝❤❛♣t❡r✳ ❲❤✐❧❡

♠❡♠♦r✐s✐♥❣ t❤❡♠ ✐s ♥✐❝❡ ✭t❤❡r❡ ❛r❡ ♥♦t ♠❛♥②✮✱ ✐t ✐s t❤❡ ❛♣♣❧✐❝❛t✐♦♥ t❤❛t ✐s ✉s❡❢✉❧✳ ❋✐♥❛♥❝✐❛❧ ❡①♣❡rts ❛r❡ ♥♦t

♣❛✐❞ ❛ s❛❧❛r② ✐♥ ♦r❞❡r t♦ r❡❝✐t❡ ❢♦r♠✉❧❛❡✱ t❤❡② ❛r❡ ♣❛✐❞ ❛ s❛❧❛r② t♦ ✉s❡ t❤❡ r✐❣❤t ♠❡t❤♦❞s t♦ s♦❧✈❡ ✜♥❛♥❝✐❛❧

♣r♦❜❧❡♠s✳

✷✳✷✳✷✳✶ ❉❡✜♥✐t✐♦♥s

P

Pr✐♥❝✐♣❛❧ ✭t❤❡ ❛♠♦✉♥t ♦❢ ♠♦♥❡② ❛t t❤❡ st❛rt✐♥❣ ♣♦✐♥t ♦❢ t❤❡ ❝❛❧❝✉❧❛t✐♦♥✮

i

✐♥t❡r❡st r❛t❡✱ ♥♦r♠❛❧❧② t❤❡ ❡✛❡❝t✐✈❡ r❛t❡ ♣❡r ❛♥♥✉♠

n

♣❡r✐♦❞ ❢♦r ✇❤✐❝❤ t❤❡ ✐♥✈❡st♠❡♥t ✐s ♠❛❞❡

iT

t❤❡ ✐♥t❡r❡st r❛t❡ ♣❛✐❞ T t✐♠❡s ♣❡r ❛♥♥✉♠✱ ✐✳❡✳ iT = NominalInterestRate

T

❚❛❜❧❡ ✷✳✹

✹✵

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

✷✳✷✳✷✳✷ ❊q✉❛t✐♦♥s

PresentValue − simple

FutureValue − simple } = P (1 + i · n) ✭✷✳✷✶✮

Solvefori

Solveforn

PresentValue − compound

FutureValue − compound } = P(1 + i)n✭✷✳✷✷✮

Solvefori

Solveforn

t✐♣✿ ❆❧✇❛②s ❦❡❡♣ t❤❡ ✐♥t❡r❡st ❛♥❞ t❤❡ t✐♠❡ ♣❡r✐♦❞ ✐♥ t❤❡ s❛♠❡ ✉♥✐ts ♦❢ t✐♠❡ ✭❡✳❣✳ ❜♦t❤ ✐♥ ②❡❛rs✱ ♦r

❜♦t❤ ✐♥ ♠♦♥t❤s ❡t❝✳✮✳

✷✳✷✳✸ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ ❚❤❛❜♦ ✐s ❛❜♦✉t t♦ ✐♥✈❡st ❤✐s ❘✽ ✺✵✵ ❜♦♥✉s ✐♥ ❛ s♣❡❝✐❛❧ ❜❛♥❦✐♥❣ ♣r♦❞✉❝t ✇❤✐❝❤ ✇✐❧❧ ♣❛② ✶✪ ♣❡r ❛♥♥✉♠

❢♦r ✶ ♠♦♥t❤✱ t❤❡♥ ✷✪ ♣❡r ❛♥♥✉♠ ❢♦r t❤❡ ♥❡①t ✷ ♠♦♥t❤s✱ t❤❡♥ ✸✪ ♣❡r ❛♥♥✉♠ ❢♦r t❤❡ ♥❡①t ✸ ♠♦♥t❤s✱

✹✪ ♣❡r ❛♥♥✉♠ ❢♦r t❤❡ ♥❡①t ✹ ♠♦♥t❤s✱ ❛♥❞ ✵✪ ❢♦r t❤❡ r❡st ♦❢ t❤❡ ②❡❛r✳ ❚❤❡ ❛r❡ ❣♦✐♥❣ t♦ ❝❤❛r❣❡ ❤✐♠

❘✶✵✵ t♦ s❡t ✉♣ t❤❡ ❛❝❝♦✉♥t✳ ❍♦✇ ♠✉❝❤ ❝❛♥ ❤❡ ❡①♣❡❝t t♦ ❣❡t ❜❛❝❦ ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ♣❡r✐♦❞❄

✷✳ ❆ s♣❡❝✐❛❧ ❜❛♥❦ ❛❝❝♦✉♥t ♣❛②s s✐♠♣❧❡ ✐♥t❡r❡st ♦❢ ✽✪ ♣❡r ❛♥♥✉♠✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♦♣❡♥✐♥❣ ❜❛❧❛♥❝❡ r❡q✉✐r❡❞

t♦ ❣❡♥❡r❛t❡ ❛ ❝❧♦s✐♥❣ ❜❛❧❛♥❝❡ ♦❢ ❘✺ ✵✵✵ ❛❢t❡r ✷ ②❡❛rs✳

✸✳ ❆ ❞✐✛❡r❡♥t ❜❛♥❦ ❛❝❝♦✉♥t ♣❛②s ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st ♦❢ ✽✪ ♣❡r ❛♥♥✉♠✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♦♣❡♥✐♥❣ ❜❛❧❛♥❝❡

r❡q✉✐r❡❞ t♦ ❣❡♥❡r❛t❡ ❛ ❝❧♦s✐♥❣ ❜❛❧❛♥❝❡ ♦❢ ❘✺ ✵✵✵ ❛❢t❡r ✷ ②❡❛rs✳

✹✳ ❲❤✐❝❤ ♦❢ t❤❡ t✇♦ ❛♥s✇❡rs ❛❜♦✈❡ ✐s ❧♦✇❡r✱ ❛♥❞ ✇❤②❄

✺✳ ✼ ▼♦♥t❤s ❛❢t❡r ❛♥ ✐♥✐t✐❛❧ ❞❡♣♦s✐t✱ t❤❡ ✈❛❧✉❡ ♦❢ ❛ ❜❛♥❦ ❛❝❝♦✉♥t ✇❤✐❝❤ ♣❛②s ❝♦♠♣♦✉♥❞ ✐♥t❡r❡st ♦❢ ✼✱✺✪

♣❡r ❛♥♥✉♠ ✐s ❘✸ ✻✺✵✱✽✶✳ ❲❤❛t ✇❛s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✐♥✐t✐❛❧ ❞❡♣♦s✐t❄

✻✳ ❚❤❛❜❛♥✐ ❛♥❞ ▲✉♥❣❡❧♦ ❛r❡ ❜♦t❤ ✉s✐♥❣ ❯❑❩◆ ❇❛♥❦ ❢♦r t❤❡✐r s❛✈✐♥❣✳ ❙✉♣♣♦s❡ ▲✉♥❣❡❧♦ ♠❛❦❡s ❛ ❞❡♣♦s✐t

♦❢ X t♦❞❛② ❛t ✐♥t❡r❡st r❛t❡ ♦❢ i ❢♦r s✐① ②❡❛rs✳ ❚❤❛❜❛♥✐ ♠❛❦❡s ❛ ❞❡♣♦s✐t ♦❢ 3X ❛t ❛♥ ✐♥t❡r❡st r❛t❡ ♦❢

0.05✳ ❚❤❛❜❛♥✐ ♠❛❞❡ ❤✐s ❞❡♣♦s✐t ✸ ②❡❛rs ❛❢t❡r ▲✉♥❣❡❧♦ ♠❛❞❡ ❤✐s ✜rst ❞❡♣♦s✐t✳ ■❢ ❛❢t❡r ✻ ②❡❛rs✱ t❤❡✐r

✐♥✈❡st♠❡♥ts ❛r❡ ❡q✉❛❧✱ ❝❛❧❝✉❧❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ i ❛♥❞ ✜♥❞ X✳ ■❢ t❤❡ s✉♠ ♦❢ t❤❡✐r ✐♥✈❡st♠❡♥t ✐s ❘✷✵ ✵✵✵✱

✉s❡ X ②♦✉ ❣♦t t♦ ✜♥❞ ♦✉t ❤♦✇ ♠✉❝❤ ❚❤❛❜❛♥✐ ❣♦t ✐♥ ✻ ②❡❛rs✳

✼✳ ❙✐♣❤♦ ✐♥✈❡sts ❘✺✵✵ ❛t ❛♥ ✐♥t❡r❡st r❛t❡ ♦❢ log (1, 12) ❢♦r ✺ ②❡❛rs✳ ❚❤❡♠❜❛✱ ❙✐♣❤♦✬s s✐st❡r ✐♥✈❡st❡❞ ❘✷✵✵

❛t ✐♥t❡r❡st r❛t❡ i ❢♦r ✶✵ ②❡❛rs ♦♥ t❤❡ s❛♠❡ ❞❛t❡ t❤❛t ❤❡r ❜r♦t❤❡r ♠❛❞❡ ❤✐s ✜rst ❞❡♣♦s✐t✳ ■❢ ❛❢t❡r ✺ ②❡❛rs✱

❚❤❡♠❜❛✬s ❛❝❝✉♠✉❧❛t✐♦♥ ❡q✉❛❧s ❙✐♣❤♦✬s✱ ✜♥❞ t❤❡ ✐♥t❡r❡st r❛t❡ i ❛♥❞ ✜♥❞ ♦✉t ✇❤❡t❤❡r ❚❤❡♠❜❛ ✇✐❧❧ ❜❡

❛❜❧❡ t♦ ❜✉② ❤❡r ❢❛✈♦r✐t❡ ❝❡❧❧ ♣❤♦♥❡ ❛❢t❡r ✶✵ ②❡❛rs ✇❤✐❝❤ ❝♦sts ❘✷ ✵✵✵✳

✽✳ ❈❛❧❝✉❧❛t❡ t❤❡ r❡❛❧ ❝♦st ♦❢ ❛ ❧♦❛♥ ♦❢ ❘✶✵ ✵✵✵ ❢♦r ✺ ②❡❛rs ❛t ✺✪ ❝❛♣✐t❛❧✐s❡❞ ♠♦♥t❤❧②✳ ❘❡♣❡❛t t❤✐s ❢♦r t❤❡

❝❛s❡ ✇❤❡r❡ ✐t ✐s ❝❛♣✐t❛❧✐s❡❞ ❤❛❧❢ ②❡❛r❧② ✐✳❡✳ ❊✈❡r② ✻ ♠♦♥t❤s✳

✾✳ ❉❡t❡r♠✐♥❡ ❤♦✇ ❧♦♥❣✱ ✐♥ ②❡❛rs✱ ✐t ✇✐❧❧ t❛❦❡ ❢♦r t❤❡ ✈❛❧✉❡ ♦❢ ❛ ♠♦t♦r ✈❡❤✐❝❧❡ t♦ ❞❡❝r❡❛s❡ t♦ ✷✺✪ ♦❢ ✐ts

♦r✐❣✐♥❛❧ ✈❛❧✉❡ ✐❢ t❤❡ r❛t❡ ♦❢ ❞❡♣r❡❝✐❛t✐♦♥✱ ❜❛s❡❞ ♦♥ t❤❡ r❡❞✉❝✐♥❣✲❜❛❧❛♥❝❡ ♠❡t❤♦❞✱ ✐s ✷✶✪ ♣❡r ❛♥♥✉♠✳

✹✶

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✷

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✷✳✶ ✭♣✳ ✸✵✮

❙t❡♣ ✶✳

• P ❂❘✸ ✺✵✵

• i❂✼✱✺✪

• A❂❘✹ ✵✹✹✱✻✾

❲❡ ❛r❡ r❡q✉✐r❡❞ t♦ ✜♥❞ n✳

❙t❡♣ ✷✳ ❲❡ ❦♥♦✇ t❤❛t✿

A

=

P (1 + i)n

(1 + i)n

=

(A/P )

log ((1 + i)n)

=

log (A/P )

✭✷✳✷✸✮

nlog (1 + i)

=

log (A/P )

n

=

log (A/P ) /log (1 + i)

❙t❡♣ ✸✳

n

=

log (A/P ) /log (1 + i)

log( 4 044,69 )

=

3 500

Rememberthat : 7.5% = 7.5 = 0.075

✭✷✳✷✹✮

log(1+0.075)

100

=

2.0

❙t❡♣ ✹✳ ❚❤❡ ❘✸ ✺✵✵ ✇❛s ✐♥✈❡st❡❞ ❢♦r ✷ ②❡❛rs✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✷✳✷ ✭♣✳ ✸✸✮

❙t❡♣ ✶✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❣✐✈❡♥✿

• ❉❡♣♦s✐t ❛♠♦✉♥t ❂ ❘✶✺ ✵✵✵

• Pr✐❝❡ ♦❢ ✢❛t ❂ ❘✷✺✵ ✵✵✵

• ✐♥t❡r❡st r❛t❡✱ i = 11%

❲❡ ❛r❡ r❡q✉✐r❡❞ t♦ ✜♥❞ t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥t ❢♦r ❛ ✸✵✲②❡❛r ♠♦rt❣❛❣❡✳

❙t❡♣ ✷✳ ❲❡ ❦♥♦✇ t❤❛t✿

M

X =

✭✷✳✷✺✮

(1−(1+i)−n)

i

■♥ ♦r❞❡r t♦ ✉s❡ t❤✐s ❡q✉❛t✐♦♥✱ ✇❡ ♥❡❡❞ t♦ ❝❛❧❝✉❧❛t❡ M✱ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ♠♦rt❣❛❣❡ ❜♦♥❞✱ ✇❤✐❝❤ ✐s t❤❡

♣✉r❝❤❛s❡ ♣r✐❝❡ ♦❢ ♣r♦♣❡rt② ❧❡ss t❤❡ ❞❡♣♦s✐t ✇❤✐❝❤ ❙❛♠ ♣❛②s ✉♣✲❢r♦♥t✳

M

=

R250 000 − R15 000

✭✷✳✷✻✮

=

R235 000

◆♦✇ ❜❡❝❛✉s❡ ✇❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ♠♦♥t❤❧② r❡♣❛②♠❡♥ts✱ ❜✉t ✇❡ ❤❛✈❡ ❜❡❡♥ ❣✐✈❡♥ ❛♥ ❛♥♥✉❛❧ ✐♥t❡r❡st r❛t❡✱

✇❡ ♥❡❡❞ t♦ ❝♦♥✈❡rt t❤✐s t♦ ❛ ♠♦♥t❤❧② ✐♥t❡r❡st r❛t❡✱ i12✳ ✭■❢ ②♦✉ ❛r❡ ♥♦t ❝❧❡❛r ♦♥ t❤✐s✱ ❣♦ ❜❛❝❦ ❛♥❞ r❡✈✐s❡

✳✮

(1 + i12)12

=

(1 + i)

(1 + i12)12

=

1, 11

✭✷✳✷✼✮

i12

=

0, 873459%

❲❡ ❦♥♦✇ t❤❛t t❤❡ ♠♦rt❣❛❣❡ ❜♦♥❞ ✐s ❢♦r ✸✵ ②❡❛rs✱ ✇❤✐❝❤ ❡q✉❛t❡s t♦ ✸✻✵ ♠♦♥t❤s✳

✹✷

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❙t❡♣ ✸✳ ◆♦✇ ✐t ✐s ❡❛s②✱ ✇❡ ❝❛♥ ❥✉st ♣❧✉❣ t❤❡ ♥✉♠❜❡rs ✐♥ t❤❡ ❢♦r♠✉❧❛✱ ❜✉t ❞♦ ♥♦t ❢♦r❣❡t t❤❛t ②♦✉ ❝❛♥ ❛❧✇❛②s

❞❡❞✉❝❡ t❤❡ ❢♦r♠✉❧❛ ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s ❛s ✇❡❧❧✦

X

=

M

» (

–

1−(1+i)−n )

i

=

R235 000

✭✷✳✷✽✮

» (

–

1−(1.00876459)−360 )

0,008734594

=

R2 146, 39

❙t❡♣ ✹✳ ❚❤❛t ♠❡❛♥s t❤❛t t♦ ❜✉② ❛ ✢❛t ❢♦r ❘✷✺✵ ✵✵✵✱ ❛❢t❡r ❙❛♠ ♣❛②s ❛ ❘✶✺ ✵✵✵ ❞❡♣♦s✐t✱ ❤❡ ✇✐❧❧ ♠❛❦❡ r❡♣❛②♠❡♥ts

t♦ t❤❡ ❜❛♥❦ ❡❛❝❤ ♠♦♥t❤ ❢♦r t❤❡ ♥❡①t ✸✵ ②❡❛rs ❡q✉❛❧ t♦ ❘✷ ✶✹✻✱✸✾✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✷✳✸ ✭♣✳ ✸✸✮

❙t❡♣ ✶✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❣✐✈❡♥✿

• ❉❡♣♦s✐t ❛♠♦✉♥t ❂ ❘✶✵ ✵✵✵

• Pr✐❝❡ ♦❢ ✢❛t ❂ ❘✷✵✵ ✵✵✵

• ✐♥t❡r❡st r❛t❡✱ i = 9%

❲❡ ❛r❡ r❡q✉✐r❡❞ t♦ ✜♥❞ t❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥t ❢♦r ❛ ✷✵✲②❡❛r ♠♦rt❣❛❣❡✳

❙t❡♣ ✷✳ ❲❡ ❛r❡ ❝♦♥s✐❞❡r✐♥❣ ♠♦♥t❤❧② ♠♦rt❣❛❣❡ r❡♣❛②♠❡♥ts✱ s♦ ✐t ♠❛❦❡s s❡♥s❡ t♦ ✉s❡ ♠♦♥t❤s ❛s ♦✉r t✐♠❡ ♣❡r✐♦❞✳

❚❤❡ ✐♥t❡r❡st r❛t❡ ✇❛s q✉♦t❡❞ ❛s ✾✪ ♣❡r ❛♥♥✉♠ ♣❛②❛❜❧❡ ♠♦♥t❤❧②✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ ♠♦♥t❤❧② ❡✛❡❝t✐✈❡

r❛t❡ = 9% = 0, 75% ♣❡r ♠♦♥t❤✳ ❖♥❝❡ ✇❡ ❤❛✈❡ ❝♦♥✈❡rt❡❞ ✷✵ ②❡❛rs ✐♥t♦ ✷✹✵ ♠♦♥t❤s✱ ✇❡ ❛r❡ r❡❛❞② t♦ ❞♦

12

t❤❡ ❝❛❧❝✉❧❛t✐♦♥s✦

❋✐rst ✇❡ ♥❡❡❞ t♦ ❝❛❧❝✉❧❛t❡ M✱ t❤❡ ❛♠♦✉♥t ♦❢ t❤❡ ♠♦rt❣❛❣❡ ❜♦♥❞✱ ✇❤✐❝❤ ✐s t❤❡ ♣✉r❝❤❛s❡ ♣r✐❝❡ ♦❢ ♣r♦♣❡rt②

♠✐♥✉s t❤❡ ❞❡♣♦s✐t ✇❤✐❝❤ ❙❛♠ ♣❛②s ✉♣✲❢r♦♥t✳

M

=

R200 000 − R10 000

✭✷✳✷✾✮

=

R190 000

❚❤❡ ♣r❡s❡♥t ✈❛❧✉❡ ♦❢ ♦✉r ♠♦rt❣❛❣❡ ♣❛②♠❡♥ts ❳ ✭✇❤✐❝❤ ✐♥❝❧✉❞❡s ✐♥t❡r❡st✮✱ ♠✉st ❡q✉❛t❡ t♦ t❤❡ ♣r❡s❡♥t

♠♦rt❣❛❣❡ ❛♠♦✉♥t

−

M = X × (1 + 0, 75%) 1

+

−

X × (1 + 0, 75%) 2

+

−

X × (1 + 0, 75%) 3

+

✭✷✳✸✵✮

−

X × (1 + 0, 75%) 4

+

...

−

−

X × (1 + 0, 75%) 239 + X × (1 + 0, 75%) 240

❇✉t ✐t ✐s ❝❧❡❛r❧② ♠✉❝❤ ❡❛s✐❡r t♦ ✉s❡ ♦✉r ❢♦r♠✉❧❛ t❤❛♥ ✇♦r❦ ♦✉t ✷✹✵ ❢❛❝t♦rs ❛♥❞ ❛❞❞ t❤❡♠ ❛❧❧ ✉♣✦

❙t❡♣ ✸✳

X × 1−(1+0,75%)−240

=

R190 000

0,75%

X × 111, 14495

=

R190 000

✭✷✳✸✶✮

X

=

R1 709, 48

❙t❡♣ ✹✳ ❙♦ t♦ r❡♣❛② ❛ ❘✶✾✵ ✵✵✵ ♠♦rt❣❛❣❡ ♦✈❡r ✷✵ ②❡❛rs✱ ❛t ✾✪ ✐♥t❡r❡st ♣❛②❛❜❧❡ ♠♦♥t❤❧②✱ ✇✐❧❧ ❝♦st ②♦✉ ❘✶ ✼✵✾✱✹✽

♣❡r ♠♦♥t❤ ❢♦r ✷✹✵ ♠♦♥t❤s✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✷✳✹ ✭♣✳ ✸✽✮

❙t❡♣ ✶✳ R1 200 00 − R240 000 = R960 000

✹✸

❙t❡♣ ✷✳ ❯s❡ t❤❡ ❢♦r♠✉❧❛✿

−

x 1 − (1 + i) n

P =

✭✷✳✸✷✮

i

❲❤❡r❡

P = 960 000

n = 30 × 12 = 360months

i = 0, 075 ÷ 12 = 0, 00625

❙t❡♣ ✸✳

x[1−(1+0,00625)−360]

R960 000

=

0,00625

=

x (143, 0176273)

✭✷✳✸✸✮

x

=

R6 712, 46

❙t❡♣ ✹✳ ❚❤❡ ♠♦♥t❤❧② r❡♣❛②♠❡♥ts = R6 712, 46

✹✹

❈❍❆P❚❊❘ ✷✳ ❋■◆❆◆❈❊

❈❤❛♣t❡r ✸

❋❛❝t♦r✐s✐♥❣ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧s

✸✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❋❛❝t♦r ❚❤❡♦r❡♠✱ ❋❛❝t♦r✐s✐♥❣ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s✶

✸✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

■♥ ❣r❛❞❡s ✶✵ ❛♥❞ ✶✶✱ ②♦✉ ❧❡❛r♥t ❤♦✇ t♦ s♦❧✈❡ ❞✐✛❡r❡♥t t②♣❡s ♦❢ ❡q✉❛t✐♦♥s✳ ▼♦st ♦❢ t❤❡ s♦❧✉t✐♦♥s✱ r❡❧✐❡❞ ♦♥

❜❡✐♥❣ ❛❜❧❡ t♦ ❢❛❝t♦r✐s❡ s♦♠❡ ❡①♣r❡ss✐♦♥ ❛♥❞ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ ♦❢ q✉❛❞r❛t✐❝s ✇❛s st✉❞✐❡❞ ✐♥ ❞❡t❛✐❧✳ ❚❤✐s ❝❤❛♣t❡r

❢♦❝✉ss❡s ♦♥ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ ♦❢ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧s✱ t❤❛t ✐s ❡①♣r❡ss✐♦♥s ✇✐t❤ t❤❡ ❤✐❣❤❡st ♣♦✇❡r ❡q✉❛❧ t♦ ✸✳

✸✳✶✳✷ ❚❤❡ ❋❛❝t♦r ❚❤❡♦r❡♠

❚❤❡ ❢❛❝t♦r t❤❡♦r❡♠ ❞❡s❝r✐❜❡s t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ r♦♦t ♦❢ ❛ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ ❛ ❢❛❝t♦r ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧✳

❉❡✜♥✐t✐♦♥ ✸✳✶✿ ❋❛❝t♦r ❚❤❡♦r❡♠

❋♦r ❛♥② ♣♦❧②♥♦♠✐❛❧✱ f (x)✱ ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢ a ✇❤✐❝❤ s❛t✐s❢② f (a) = 0✱ (x − a) ✐s ❛ ❢❛❝t♦r ♦❢ f (x)✳

❖r✱ ♠♦r❡ ❝♦♥❝✐s❡❧②✿

f (x) = (x − a) q (x)

✭✸✳✶✮

✐s ❛ ♣♦❧②♥♦♠✐❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✿ ■❢ t❤❡ r❡♠❛✐♥❞❡r ✇❤❡♥ ❞✐✈✐❞✐♥❣ f (x) ❜② (x − a) ✐s ③❡r♦✱ t❤❡♥ (x − a) ✐s ❛ ❢❛❝t♦r

♦❢ f (x)✳

❙♦ ✐❢ f − b = 0✱ t❤❡♥ (ax + b) ✐s ❛ ❢❛❝t♦r ♦❢ f (x)✳

a

❊①❡r❝✐s❡ ✸✳✶✿ ❋❛❝t♦r ❚❤❡♦r❡♠

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✾✳✮

❯s❡ t❤❡ ❋❛❝t♦r ❚❤❡♦r❡♠ t♦ ❞❡t❡r♠✐♥❡ ✇❤❡t❤❡r y − 1 ✐s ❛ ❢❛❝t♦r ♦❢ f (y) = 2y4 + 3y2 − 5y + 7✳

❊①❡r❝✐s❡ ✸✳✷✿ ❋❛❝t♦r ❚❤❡♦r❡♠

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✾✳✮

❯s✐♥❣ t❤❡ ❋❛❝t♦r ❚❤❡♦r❡♠✱ ✈❡r✐❢② t❤❛t y + 4 ✐s ❛ ❢❛❝t♦r ♦❢ g (y) = 5y4 + 16y3 − 15y2 + 8y + 16✳

✸✳✶✳✸ ❋❛❝t♦r✐s❛t✐♦♥ ♦❢ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s

❆ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠

ax3 + bx2 + cx + d

✭✸✳✷✮

✇❤❡r❡ ❛ ✐s ♥♦♥③❡r♦✳ ❲❡ ❤❛✈❡ s❡❡♥ ✐♥ ●r❛❞❡ ✶✵ t❤❛t t❤❡ s✉♠ ❛♥❞ ❞✐✛❡r❡♥❝❡ ♦❢ ❝✉❜❡s ✐s ❢❛❝t♦r✐s❡❞ ❛s ❢♦❧❧♦✇s✳✿

(x + y) x2 − xy + y2 = x3 + y3

✭✸✳✸✮

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✾✴✶✳✶✴❃✳

✹✺

✹✻

❈❍❆P❚❊❘ ✸✳ ❋❆❈❚❖❘■❙■◆● ❈❯❇■❈ P❖▲❨◆❖▼■❆▲❙

❛♥❞

(x − y) x2 + xy + y2 = x3 − y3

✭✸✳✹✮

❲❡ ❛❧s♦ s❛✇ t❤❛t t❤❡ q✉❛❞r❛t✐❝ t❡r♠ ❞♦❡s ♥♦t ❤❛✈❡ r❛t✐♦♥❛❧ r♦♦ts✳

❚❤❡r❡ ❛r❡ ♠❛♥② ♠❡t❤♦❞s ♦❢ ❢❛❝t♦r✐s✐♥❣ ❛ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧✳ ❚❤❡ ❣❡♥❡r❛❧ ♠❡t❤♦❞ ✐s s✐♠✐❧❛r t♦ t❤❛t ✉s❡❞

t♦ ❢❛❝t♦r✐s❡ q✉❛❞r❛t✐❝ ❡q✉❛t✐♦♥s✳ ■❢ ②♦✉ ❤❛✈❡ ❛ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ♦❢ t❤❡ ❢♦r♠✿

f (x) = ax3 + bx2 + cx + d

✭✸✳✺✮

t❤❡♥ ✐♥ ❛♥ ✐❞❡❛❧ ✇♦r❧❞ ②♦✉ ✇♦✉❧❞ ❣❡t ❢❛❝t♦rs ♦❢ t❤❡ ❢♦r♠✿

(Ax + B) (Cx + D) (Ex + F ) .

✭✸✳✻✮

❇✉t s♦♠❡t✐♠❡s ②♦✉ ✇✐❧❧ ❣❡t ❢❛❝t♦rs ♦❢ t❤❡ ❢♦r♠✿

(Ax + B) Cx2 + Ex + D

✭✸✳✼✮

❲❡ ✇✐❧❧ ❞❡❛❧ ✇✐t❤ s✐♠♣❧❡st ❝❛s❡ ✜rst✳ ❲❤❡♥ a = 1✱ t❤❡♥ A = C = E = 1✱ ❛♥❞ ②♦✉ ♦♥❧② ❤❛✈❡ t♦ ❞❡t❡r♠✐♥❡

B✱ D ❛♥❞ F ✳ ❋♦r ❡①❛♠♣❧❡✱ ✜♥❞ t❤❡ ❢❛❝t♦rs ♦❢✿

x3 − 2x2 − 5x + 6.

✭✸✳✽✮

■♥ t❤✐s ❝❛s❡ ✇❡ ❤❛✈❡

a

=

1

b

=

−2

✭✸✳✾✮

c

=

−5

d

=

6

❚❤❡ ❢❛❝t♦rs ✇✐❧❧ ❤❛✈❡ t❤❡ ❣❡♥❡r❛❧ ❢♦r♠ s❤♦✇♥ ✐♥ ✭✸✳✻✮✱ ✇✐t❤ A = C = E = 1✳ ❲❡ ❝❛♥ t❤❡♥ ✉s❡ ✈❛❧✉❡s ❢♦r a✱

b✱ c ❛♥❞ d t♦ ❞❡t❡r♠✐♥❡ ✈❛❧✉❡s ❢♦r B✱ D ❛♥❞ F ✳ ❲❡ ❝❛♥ r❡✲✇r✐t❡ ✭✸✳✻✮ ✇✐t❤ A = C = E = 1 ❛s✿

(x + B) (x + D) (x + F ) .

✭✸✳✶✵✮

■❢ ✇❡ ♠✉❧t✐♣❧② t❤✐s ♦✉t ✇❡ ❣❡t✿

(x + B) (x + D) (x + F )

=

(x + B) x2 + Dx + F x + DF

=

x3 + Dx2 + F x2 + Bx2 + DF x + BDx + BF x + BDF

✭✸✳✶✶✮

=

x3 + (D + F + B) x2 + (DF + BD + BF ) x + BDF

❲❡ ❝❛♥ t❤❡r❡❢♦r❡ ✇r✐t❡✿

b

=

−2 = D + F + B

c

=

−5 = DF + BD + BF

✭✸✳✶✷✮

d

=

6 = BDF.

❚❤✐s ✐s ❛ s❡t ♦❢ t❤r❡❡ ❡q✉❛t✐♦♥s ✐♥ t❤r❡❡ ✉♥❦♥♦✇♥s✳ ❍♦✇❡✈❡r✱ ✇❡ ❦♥♦✇ t❤❛t B✱ D ❛♥❞ F ❛r❡ ❢❛❝t♦rs ♦❢ ✻

❜❡❝❛✉s❡ BDF = 6✳ ❚❤❡r❡❢♦r❡ ✇❡ ❝❛♥ ✉s❡ ❛ tr✐❛❧ ❛♥❞ ❡rr♦r ♠❡t❤♦❞ t♦ ✜♥❞ B✱ D ❛♥❞ F ✳

❚❤✐s ❝❛♥ ❜❡❝♦♠❡ ❛ ✈❡r② t❡❞✐♦✉s ♠❡t❤♦❞✱ t❤❡r❡❢♦r❡ t❤❡ ❋❛❝t♦r ❚❤❡♦r❡♠ ❝❛♥ ❜❡ ✉s❡❞ t♦ ✜♥❞ t❤❡ ❢❛❝t♦rs

♦❢ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧s✳

❊①❡r❝✐s❡ ✸✳✸✿ ❋❛❝t♦r✐s❛t✐♦♥ ♦❢ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✾✳✮

❋❛❝t♦r✐s❡ f (x) = x3 + x2 − 9x − 9 ✐♥t♦ t❤r❡❡ ❧✐♥❡❛r ❢❛❝t♦rs✳

✹✼

■♥ ❣❡♥❡r❛❧✱ t♦ ❢❛❝t♦r✐s❡ ❛ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧✱ ②♦✉ ✜♥❞ ♦♥❡ ❢❛❝t♦r ❜② tr✐❛❧ ❛♥❞ ❡rr♦r✳ ❯s❡ t❤❡ ❢❛❝t♦r t❤❡♦r❡♠

t♦ ❝♦♥✜r♠ t❤❛t t❤❡ ❣✉❡ss ✐s ❛ r♦♦t✳ ❚❤❡♥ ❞✐✈✐❞❡ t❤❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧ ❜② t❤❡ ❢❛❝t♦r t♦ ♦❜t❛✐♥ ❛ q✉❛❞r❛t✐❝✳

❖♥❝❡ ②♦✉ ❤❛✈❡ t❤❡ q✉❛❞r❛t✐❝✱ ②♦✉ ❝❛♥ ❛♣♣❧② t❤❡ st❛♥❞❛r❞ ♠❡t❤♦❞s t♦ ❢❛❝t♦r✐s❡ t❤❡ q✉❛❞r❛t✐❝✳

❋♦r ❡①❛♠♣❧❡ t❤❡ ❢❛❝t♦rs ♦❢ x3 − 2x2 − 5x + 6 ❝❛♥ ❜❡ ❢♦✉♥❞ ❛s ❢♦❧❧♦✇s✿ ❚❤❡r❡ ❛r❡ t❤r❡❡ ❢❛❝t♦rs ✇❤✐❝❤ ✇❡

❝❛♥ ✇r✐t❡ ❛s

(x − a) (x − b) (x − c) .

✭✸✳✶✸✮

❊①❡r❝✐s❡ ✸✳✹✿ ❋❛❝t♦r✐s❛t✐♦♥ ♦❢ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✹✾✳✮

❯s❡ t❤❡ ❋❛❝t♦r ❚❤❡♦r❡♠ t♦ ❢❛❝t♦r✐s❡

x3 − 2x2 − 5x + 6.

✭✸✳✶✹✮

✸✳✷ ❋❛❝t♦r ❚❤❡♦r❡♠ ❊①❡r❝✐s❡s✱ ❙♦❧✈✐♥❣ ❈✉❜✐❝ ❊q✉❛t✐♦♥s✷

✸✳✷✳✶ ❊①❡r❝✐s❡s ✲ ❯s✐♥❣ ❋❛❝t♦r ❚❤❡♦r❡♠

✶✳ ❋✐♥❞ t❤❡ r❡♠❛✐♥❞❡r ✇❤❡♥ 4x3 − 4x2 + x − 5 ✐s ❞✐✈✐❞❡❞ ❜② (x + 1)✳

✷✳ ❯s❡ t❤❡ ❢❛❝t♦r t❤❡♦r❡♠ t♦ ❢❛❝t♦r✐s❡ x3 − 3x2 + 4 ❝♦♠♣❧❡t❡❧②✳

✸✳ f (x) = 2x3 + x2 − 5x + 2

❛✳ ❋✐♥❞ f (1)✳

❜✳ ❋❛❝t♦r✐s❡ f (x) ❝♦♠♣❧❡t❡❧②

✹✳ ❯s❡ t❤❡ ❋❛❝t♦r ❚❤❡♦r❡♠ t♦ ❞❡t❡r♠✐♥❡ ❛❧❧ t❤❡ ❢❛❝t♦rs ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①♣r❡ss✐♦♥✿

x3 + x2 − 17x + 15

✭✸✳✶✺✮

✺✳ ❈♦♠♣❧❡t❡✿ ■❢ f (x) ✐s ❛ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ p ✐s ❛ ♥✉♠❜❡r s✉❝❤ t❤❛t f (p) = 0✱ t❤❡♥ (x − p) ✐s ✳✳✳✳✳

✸✳✷✳✷ ❙♦❧✈✐♥❣ ❈✉❜✐❝ ❊q✉❛t✐♦♥s

❖♥❝❡ ②♦✉ ❦♥♦✇ ❤♦✇ t♦ ❢❛❝t♦r✐s❡ ❝✉❜✐❝ ♣♦❧②♥♦♠✐❛❧s✱ ✐t ✐s ❛❧s♦ ❡❛s② t♦ s♦❧✈❡ ❝✉❜✐❝ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❦✐♥❞

ax3 + bx2 + cx + d = 0

✭✸✳✶✻✮

❊①❡r❝✐s❡ ✸✳✺✿ ❙♦❧✉t✐♦♥ ♦❢ ❈✉❜✐❝ ❊q✉❛t✐♦♥s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✺✵✳✮

❙♦❧✈❡

6x3 − 5x2 − 17x + 6 = 0.

✭✸✳✶✼✮

❙♦♠❡t✐♠❡s ✐t ✐s ♥♦t ♣♦ss✐❜❧❡ t♦ ❢❛❝t♦r✐s❡ t❤❡ tr✐♥♦♠✐❛❧ ✭✧s❡❝♦♥❞ ❜r❛❝❦❡t✧✮✳ ❚❤✐s ✐s ✇❤❡♥ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛

√

−b ±

b2 − 4ac

x =

✭✸✳✶✽✮

2a

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✻✴✶✳✶✴❃✳

✹✽

❈❍❆P❚❊❘ ✸✳ ❋❆❈❚❖❘■❙■◆● ❈❯❇■❈ P❖▲❨◆❖▼■❆▲❙

❝❛♥ ❜❡ ✉s❡❞ t♦ s♦❧✈❡ t❤❡ ❝✉❜✐❝ ❡q✉❛t✐♦♥ ❢✉❧❧②✳

❋♦r ❡①❛♠♣❧❡✿

❊①❡r❝✐s❡ ✸✳✻✿ ❙♦❧✉t✐♦♥ ♦❢ ❈✉❜✐❝ ❊q✉❛t✐♦♥s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✺✵✳✮

❙♦❧✈❡ ❢♦r x✿ x3 − 2x2 − 6x + 4 = 0.

✸✳✷✳✷✳✶ ❊①❡r❝✐s❡s ✲ ❙♦❧✈✐♥❣ ♦❢ ❈✉❜✐❝ ❊q✉❛t✐♦♥s

✶✳ ❙♦❧✈❡ ❢♦r x✿ x3 + x2 − 5x + 3 = 0

✷✳ ❙♦❧✈❡ ❢♦r y✿ y3 − 3y2 − 16y − 12 = 0

✸✳ ❙♦❧✈❡ ❢♦r m✿ m3 − m2 − 4m − 4 = 0

✹✳ ❙♦❧✈❡ ❢♦r x✿ x3 − x2 = 3 (3x + 2)

t✐♣✿ ✿

❘❡♠♦✈❡ ❜r❛❝❦❡ts ❛♥❞ ✇r✐t❡ ❛s ❛♥ ❡q✉❛t✐♦♥ ❡q✉❛❧ t♦ ③❡r♦✳

✺✳ ❙♦❧✈❡ ❢♦r x ✐❢ 2x3 − 3x2 − 8x = 3

✸✳✷✳✸ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ ❙♦❧✈❡ ❢♦r x✿ 16 (x + 1) = x2 (x + 1)

✷✳

❛✳ ❙❤♦✇ t❤❛t x − 2 ✐s ❛ ❢❛❝t♦r ♦❢ 3x3 − 11x2 + 12x − 4

❜✳ ❍❡♥❝❡✱ ❜② ❢❛❝t♦r✐s✐♥❣ ❝♦♠♣❧❡t❡❧②✱ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥

3x3 − 11x2 + 12x − 4 = 0

✭✸✳✶✾✮

✸✳ 2x3 − x2 − 2x + 2 = Q (x) . (2x − 1) + R ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢ x✳ ❲❤❛t ✐s t❤❡ ✈❛❧✉❡ ♦❢ R❄

✹✳

❛✳ ❯s❡ t❤❡ ❢❛❝t♦r t❤❡♦r❡♠ t♦ s♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❢♦r m✿

8m3 + 7m2 − 17m + 2 = 0

✭✸✳✷✵✮

❜✳ ❍❡♥❝❡✱ ♦r ♦t❤❡r✇✐s❡✱ s♦❧✈❡ ❢♦r x✿

23x+3 + 7 · 22x + 2 = 17 · 2x

✭✸✳✷✶✮

✺✳ ❆ ❝❤❛❧❧❡♥❣❡✿ ❉❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡s ♦❢ p ❢♦r ✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥

f (x) = 3p3 − (3p − 7) x2 + 5x − 3

✭✸✳✷✷✮

❧❡❛✈❡s ❛ r❡♠❛✐♥❞❡r ♦❢ ✾ ✇❤❡♥ ✐t ✐s ❞✐✈✐❞❡❞ ❜② (x − p)✳

✹✾

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✸

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✸✳✶ ✭♣✳ ✹✺✮

❙t❡♣ ✶✳ ■♥ ♦r❞❡r ❢♦r y − 1 t♦ ❜❡ ❛ ❢❛❝t♦r✱ f (1) ♠✉st ❜❡ ✵✳

❙t❡♣ ✷✳

f (y)

=

2y4 + 3y2 − 5y + 7

∴ f (1) = 2(1)4 + 3(1)2 − 5 (1) + 7

✭✸✳✷✸✮

=

2 + 3 − 5 + 7

=

7

❙t❡♣ ✸✳ ❙✐♥❝❡ f (1) = 0✱ y − 1 ✐s ♥♦t ❛ ❢❛❝t♦r ♦❢ f (y) = 2y4 + 3y2 − 5y + 7✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✸✳✷ ✭♣✳ ✹✺✮

❙t❡♣ ✶✳ ■♥ ♦r❞❡r ❢♦r y + 4 t♦ ❜❡ ❛ ❢❛❝t♦r✱ g (−4) ♠✉st ❜❡ ✵✳

❙t❡♣ ✷✳

g (y)

=

5y4 + 16y3 − 15y2 + 8y + 16

∴ g (−4) = 5(−4)4 + 16(−4)3 − 15(−4)2 + 8 (−4) + 16

=

5 (256) + 16 (−64) − 15 (16) + 8 (−4) + 16

✭✸✳✷✹✮

=

1280 − 1024 − 240 − 32 + 16

=

0

❙t❡♣ ✸✳ ❙✐♥❝❡ g (−4) = 0✱ y + 4 ✐s ❛ ❢❛❝t♦r ♦❢ g (y) = 5y4 + 16y3 − 15y2 + 8y + 16✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✸✳✸ ✭♣✳ ✹✻✮

❙t❡♣ ✶✳ ❚r②

f (1) = (1)3 + (1)2 − 9 (1) − 9 = 1 + 1 − 9 − 9 = −16

✭✸✳✷✺✮

❚❤❡r❡❢♦r❡ (x − 1) ✐s ♥♦t ❛ ❢❛❝t♦r

❚r②

f (−1) = (−1)3 + (−1)2[U+0096]9 (−1) [U+0096]9 = [U+0096]1 + 1 + 9[U+0096]9 = 0

✭✸✳✷✻✮

❚❤✉s (x + 1) ✐s ❛ ❢❛❝t♦r✱ ❜❡❝❛✉s❡ f (−1) = 0✳

◆♦✇ ❞✐✈✐❞❡ f (x) ❜② (x + 1) ✉s✐♥❣ ❞✐✈✐s✐♦♥ ❜② ✐♥s♣❡❝t✐♦♥✿

❲r✐t❡ x3 + x2 − 9x − 9 = (x + 1) (

)

❚❤❡ ✜rst t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ x2 t♦ ❣✐✈❡ x3 ✐❢ ♦♥❡ ✇♦r❦s ❜❛❝❦✇❛r❞s✳

❚❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ −9 ❜❡❝❛✉s❡ +1 × −9 = −9✳

❙♦ ✇❡ ❤❛✈❡ x3 + x2 − 9x − 9 = (x + 1) x2+?x − 9 ✳

◆♦✇✱ ✇❡ ♠✉st ✜♥❞ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ♠✐❞❞❧❡ t❡r♠ ✭x✮✳

(+1) x2 ❣✐✈❡s t❤❡ x2 ✐♥ t❤❡ ♦r✐❣✐♥❛❧ ♣♦❧②♥♦♠✐❛❧✳ ❙♦✱ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ x✲t❡r♠ ♠✉st ❜❡ ✵✳

❙♦ f (x) = (x + 1) x2 − 9 ✳

❙t❡♣ ✷✳ x2 − 9 ❝❛♥ ❜❡ ❢✉rt❤❡r ❢❛❝t♦r✐s❡❞ t♦ (x − 3) (x + 3)✱

❛♥❞ ✇❡ ❛r❡ ♥♦✇ ❧❡❢t ✇✐t❤ f (x) = (x + 1) (x − 3) (x + 3)

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✸✳✹ ✭♣✳ ✹✼✮

❙t❡♣ ✶✳ ❚r②

f (1) = (1)3 − 2(1)2 − 5 (1) + 6 = 1 − 2 − 5 + 6 = 0

✭✸✳✷✼✮

❚❤❡r❡❢♦r❡ (x − 1) ✐s ❛ ❢❛❝t♦r✳

✺✵

❈❍❆P❚❊❘ ✸✳ ❋❆❈❚❖❘■❙■◆● ❈❯❇■❈ P❖▲❨◆❖▼■❆▲❙

❙t❡♣ ✷✳ x3 − 2x2 − 5x + 6 = (x − 1) (

)

❚❤❡ ✜rst t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ x2 t♦ ❣✐✈❡ x3 ✐❢ ♦♥❡ ✇♦r❦s ❜❛❝❦✇❛r❞s✳

❚❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ −6 ❜❡❝❛✉s❡ −1 × −6 = +6✳

❙♦ ✇❡ ❤❛✈❡ x3 − 2x2 − 5x + 6 = (x − 1) x2+?x − 6 ✳

◆♦✇✱ ✇❡ ♠✉st ✜♥❞ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ♠✐❞❞❧❡ t❡r♠ ✭x✮✳

(−1) x2 ❣✐✈❡s −x2✳ ❙♦✱ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ x✲t❡r♠ ♠✉st ❜❡ −1✳

❙♦ f (x) = (x − 1) x2 − x − 6 ✳

❙t❡♣ ✸✳ x2 − x − 6 ❝❛♥ ❜❡ ❢✉rt❤❡r ❢❛❝t♦r✐s❡❞ t♦ (x − 3) (x + 2)✱

❛♥❞ ✇❡ ❛r❡ ♥♦✇ ❧❡❢t ✇✐t❤ x3 − 2x2 − 5x + 6 = (x − 1) (x − 3) (x + 2)

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✸✳✺ ✭♣✳ ✹✼✮

❙t❡♣ ✶✳ ❚r②

f (1) = 6(1)3 − 5(1)2 − 17 (1) + 6 = 6 − 5 − 17 + 6 = −10

✭✸✳✷✽✮

❚❤❡r❡❢♦r❡ (x − 1) ✐s ◆❖❚ ❛ ❢❛❝t♦r✳

❚r②

f (2) = 6(2)3 − 5(2)2 − 17 (2) + 6 = 48 − 20 − 34 + 6 = 0

✭✸✳✷✾✮

❚❤❡r❡❢♦r❡ (x − 2) ■❙ ❛ ❢❛❝t♦r✳

❙t❡♣ ✷✳ 6x3 − 5x2 − 17x + 6 = (x − 2) (

)

❚❤❡ ✜rst t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ 6x2 t♦ ❣✐✈❡ 6x3 ✐❢ ♦♥❡ ✇♦r❦s ❜❛❝❦✇❛r❞s✳

❚❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ −3 ❜❡❝❛✉s❡ −2 × −3 = +6✳

❙♦ ✇❡ ❤❛✈❡ 6x3 − 5x2 − 17x + 6 = (x − 2) 6x2+?x − 3 ✳

◆♦✇✱ ✇❡ ♠✉st ✜♥❞ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ♠✐❞❞❧❡ t❡r♠ ✭x✮✳

(−2) 6x2 ❣✐✈❡s −12x2✳ ❙♦✱ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ x✲t❡r♠ ♠✉st ❜❡ ✼✳

❙♦✱ 6x3 − 5x2 − 17x + 6 = (x − 2) 6x2 + 7x − 3 ✳

❙t❡♣ ✸✳ 6x2 + 7x − 3 ❝❛♥ ❜❡ ❢✉rt❤❡r ❢❛❝t♦r✐s❡❞ t♦ (2x + 3) (3x − 1)✱

❛♥❞ ✇❡ ❛r❡ ♥♦✇ ❧❡❢t ✇✐t❤ 6x3 − 5x2 − 17x + 6 = (x − 2) (2x + 3) (3x − 1)

❙t❡♣ ✹✳

6x3 − 5x2 − 17x + 6

=

0

(x − 2) (2x + 3) (3x − 1)

=

0

✭✸✳✸✵✮

x

=

2; 1 ; − 3

3

2

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✸✳✻ ✭♣✳ ✹✽✮

❙t❡♣ ✶✳ ❚r②

f (1) = (1)3 − 2(1)2 − 6 (1) + 4 = 1 − 2 − 6 + 4 = −1

✭✸✳✸✶✮

❚❤❡r❡❢♦r❡ (x − 1) ✐s ◆❖❚ ❛ ❢❛❝t♦r✳

❚r②

f (2) = (2)3 − 2(2)2 − 6 (2) + 4 = 8 − 8 − 12 + 4 = −8

✭✸✳✸✷✮

❚❤❡r❡❢♦r❡ (x − 2) ✐s ◆❖❚ ❛ ❢❛❝t♦r✳

f (−2) = (−2)3 − 2(−2)2 − 6 (−2) + 4 = −8 − 8 + 12 + 4 = 0

✭✸✳✸✸✮

❚❤❡r❡❢♦r❡ (x + 2) ■❙ ❛ ❢❛❝t♦r✳

✺✶

❙t❡♣ ✷✳ x3 − 2x2 − 6x + 4 = (x + 2) (

)

❚❤❡ ✜rst t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ x2 t♦ ❣✐✈❡ x3✳

❚❤❡ ❧❛st t❡r♠ ✐♥ t❤❡ s❡❝♦♥❞ ❜r❛❝❦❡t ♠✉st ❜❡ ✷ ❜❡❝❛✉s❡ 2 × 2 = +4✳

❙♦ ✇❡ ❤❛✈❡ x3 − 2x2 − 6x + 4 = (x + 2) x2+?x + 2 ✳

◆♦✇✱ ✇❡ ♠✉st ✜♥❞ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ ♠✐❞❞❧❡ t❡r♠ ✭x✮✳

(2) x2 ❣✐✈❡s 2x2✳ ❙♦✱ t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ t❤❡ x✲t❡r♠ ♠✉st ❜❡ −4✳ ✭2x2 − 4x2 = −2x2✮

❙♦ x3 − 2x2 − 6x + 4 = (x + 2) x2 − 4x + 2 ✳

x2 − 4x + 2 ❝❛♥♥♦t ❜❡ ❢❛❝t♦r✐s❡❞ ❛♥② ❢✉t❤❡r ❛♥❞ ✇❡ ❛r❡ ♥♦✇ ❧❡❢t ✇✐t❤

(x + 2) x2 − 4x + 2 = 0

❙t❡♣ ✸✳

(x + 2) x2 − 4x + 2

=

0

✭✸✳✸✹✮

(x + 2) = 0

or

x2 − 4x + 2 = 0

❙t❡♣ ✹✳ ❆❧✇❛②s ✇r✐t❡ ❞♦✇♥ t❤❡ ❢♦r♠✉❧❛ ✜rst ❛♥❞ t❤❡♥ s✉❜st✐t✉t❡ t❤❡ ✈❛❧✉❡s ♦❢ a, b ❛♥❞ c✳

√

x

=

−b± b2−4ac

2a

√

−(−4)±

(−4)2−4(1)(2)

=

2(1)

√

✭✸✳✸✺✮

=

4±

8

2√

=

2 ±

2

√

❙t❡♣ ✺✳ x = −2 or x = 2 ± 2

✺✷

❈❍❆P❚❊❘ ✸✳ ❋❆❈❚❖❘■❙■◆● ❈❯❇■❈ P❖▲❨◆❖▼■❆▲❙

❈❤❛♣t❡r ✹

❋✉♥❝t✐♦♥s ❛♥❞ ❣r❛♣❤s

✹✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❉❡✜♥✐t✐♦♥✱ ◆♦t❛t✐♦♥✶

✹✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

■♥ ❣r❛❞❡s ✶✵ ❛♥❞ ✶✶ ②♦✉ ❤❛✈❡ ❧❡❛r♥t ❛❜♦✉t ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ❛♥❞ q✉❛❞r❛t✐❝ ❢✉♥❝t✐♦♥s ❛s ✇❡❧❧ ❛s t❤❡ ❤②♣❡r❜♦❧✐❝

❢✉♥❝t✐♦♥s ❛♥❞ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♠❛♥② ♠♦r❡✳ ■♥ ❣r❛❞❡ ✶✷ ②♦✉ ❛r❡ ❡①♣❡❝t❡❞ t♦ ❞❡♠♦♥str❛t❡ t❤❡ ❛❜✐❧✐t②

t♦ ✇♦r❦ ✇✐t❤ ✈❛r✐♦✉s t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ r❡❧❛t✐♦♥s ✐♥❝❧✉❞✐♥❣ t❤❡ ✐♥✈❡rs❡s ♦❢ s♦♠❡ ❢✉♥❝t✐♦♥s ❛♥❞ ❣❡♥❡r❛t❡

❣r❛♣❤s ♦❢ t❤❡ ✐♥✈❡rs❡ r❡❧❛t✐♦♥s ♦❢ ❢✉♥❝t✐♦♥s✱ ✐♥ ♣❛rt✐❝✉❧❛r t❤❡ ✐♥✈❡rs❡s ♦❢✿

y = ax + q

✭✹✳✶✮

y = ax2

✭✹✳✷✮

y = ax; a > 0

✭✹✳✸✮

✳

✹✳✶✳✷ ❉❡✜♥✐t✐♦♥ ♦❢ ❛ ❋✉♥❝t✐♦♥

❆ ❢✉♥❝t✐♦♥ ✐s ❛ r❡❧❛t✐♦♥ ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ✈❛❧✉❡ ♦❢ y ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❛♥② ✈❛❧✉❡ ♦❢ x✳ ❲❡ s♦♠❡t✐♠❡s

✇r✐t❡ y = f (x)✱ ✇❤✐❝❤ ✐s ♥♦t❛t✐♦♥ ♠❡❛♥✐♥❣ ✬y ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ x✬✳ ❚❤✐s ❞❡✜♥✐t✐♦♥ ♠❛❦❡s ❝♦♠♣❧❡t❡ s❡♥s❡ ✇❤❡♥

❝♦♠♣❛r❡❞ t♦ ♦✉r r❡❛❧ ✇♦r❧❞ ❡①❛♠♣❧❡s ✖ ❡❛❝❤ ♣❡rs♦♥ ❤❛s ♦♥❧② ♦♥❡ ❤❡✐❣❤t✱ s♦ ❤❡✐❣❤t ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ ♣❡♦♣❧❡❀

♦♥ ❡❛❝❤ ❞❛②✱ ✐♥ ❛ s♣❡❝✐✜❝ t♦✇♥✱ t❤❡r❡ ✐s ♦♥❧② ♦♥❡ ❛✈❡r❛❣❡ t❡♠♣❡r❛t✉r❡✳

❍♦✇❡✈❡r✱ s♦♠❡ ✈❡r② ❝♦♠♠♦♥ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♥str✉❝t✐♦♥s ❛r❡ ♥♦t ❢✉♥❝t✐♦♥s✳ ❋♦r ❡①❛♠♣❧❡✱ ❝♦♥s✐❞❡r t❤❡

r❡❧❛t✐♦♥ x2 + y2 = 4✳ ❚❤✐s r❡❧❛t✐♦♥ ❞❡s❝r✐❜❡s ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s ✷ ❝❡♥tr❡❞ ❛t t❤❡ ♦r✐❣✐♥✱ ❛s ✐♥ ❋✐❣✉r❡ ✹✳✶✳ ■❢

✇❡ ❧❡t x = 0✱ ✇❡ s❡❡ t❤❛t y2 = 4 ❛♥❞ t❤✉s ❡✐t❤❡r y = 2 ♦r y = −2✳ ❙✐♥❝❡ t❤❡r❡ ❛r❡ t✇♦ y ✈❛❧✉❡s ✇❤✐❝❤ ❛r❡

♣♦ss✐❜❧❡ ❢♦r t❤❡ s❛♠❡ x ✈❛❧✉❡✱ t❤❡ r❡❧❛t✐♦♥ x2 + y2 = 4 ✐s ♥♦t t❤❡ ❣r❛♣❤ ❛ ❢✉♥❝t✐♦♥✳

❚❤❡r❡ ✐s ❛ s✐♠♣❧❡ t❡st t♦ ❝❤❡❝❦ ✐❢ ❛ r❡❧❛t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥✱ ❜② ❧♦♦❦✐♥❣ ❛t ✐ts ❣r❛♣❤✳ ❚❤✐s t❡st ✐s ❝❛❧❧❡❞ t❤❡

✈❡rt✐❝❛❧ ❧✐♥❡ t❡st✳ ■❢ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❞r❛✇ ❛♥② ✈❡rt✐❝❛❧ ❧✐♥❡ ✭❛ ❧✐♥❡ ♦❢ ❝♦♥st❛♥t x✮ ✇❤✐❝❤ ❝r♦ss❡s t❤❡ ❣r❛♣❤ ♦❢

t❤❡ r❡❧❛t✐♦♥ ♠♦r❡ t❤❛♥ ♦♥❝❡✱ t❤❡♥ t❤❡ r❡❧❛t✐♦♥ ✐s ♥♦t ❛ ❢✉♥❝t✐♦♥✳ ■❢ ♠♦r❡ t❤❛♥ ♦♥❡ ✐♥t❡rs❡❝t✐♦♥ ♣♦✐♥t ❡①✐sts✱

t❤❡♥ t❤❡ ✐♥t❡rs❡❝t✐♦♥s ❝♦rr❡s♣♦♥❞ t♦ ♠✉❧t✐♣❧❡ ✈❛❧✉❡s ♦❢ y ❢♦r ❛ s✐♥❣❧❡ ✈❛❧✉❡ ♦❢ x✳

❲❡ ❝❛♥ s❡❡ t❤✐s ✇✐t❤ ♦✉r ♣r❡✈✐♦✉s ❡①❛♠♣❧❡ ♦❢ t❤❡ ❝✐r❝❧❡ ❜② ❧♦♦❦✐♥❣ ❛t ✐ts ❣r❛♣❤ ❛❣❛✐♥ ✐♥ ❋✐❣✉r❡ ✹✳✶✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✽✻✴✶✳✶✴❃✳

✺✸

✺✹

❈❍❆P❚❊❘ ✹✳ ❋❯◆❈❚■❖◆❙ ❆◆❉ ●❘❆P❍❙

❋✐❣✉r❡ ✹✳✶✿ ●r❛♣❤ ♦❢ x2 + y2 = 4

❲❡ s❡❡ t❤❛t ✇❡ ❝❛♥ ❞r❛✇ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡✱ ❢♦r ❡①❛♠♣❧❡ t❤❡ ❞♦tt❡❞ ❧✐♥❡ ✐♥ t❤❡ ❞r❛✇✐♥❣✱ ✇❤✐❝❤ ❝✉ts t❤❡ ❝✐r❝❧❡

♠♦r❡ t❤❛♥ ♦♥❝❡✳ ❚❤❡r❡❢♦r❡ t❤✐s ✐s ♥♦t ❛ ❢✉♥❝t✐♦♥✳

✹✳✶✳✷✳✶ ❊①❡r❝✐s❡s

✶✳ ❙t❛t❡ ✇❤❡t❤❡r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s ❛r❡ ❢✉♥❝t✐♦♥s ♦r ♥♦t✿

❛✳ x + y = 4

❜✳ y = x4

❝✳ y = 2x

❞✳ x2 + y2 = 4

✷✳ ❚❤❡ t❛❜❧❡ ❣✐✈❡s t❤❡ ❛✈❡r❛❣❡ ♣❡r ❝❛♣✐t❛ ✐♥❝♦♠❡✱ d✱ ✐♥ ❛ r❡❣✐♦♥ ♦❢ t❤❡ ❝♦✉♥tr② ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ u✱ t❤❡

♣❡r❝❡♥t❛❣❡ ✉♥❡♠♣❧♦②❡❞✳ ❲r✐t❡ ❞♦✇♥ t❤❡ ❡q✉❛t✐♦♥ t♦ s❤♦✇ t❤❛t ✐♥❝♦♠❡ ✐s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♣❡rs❡♥t

✉♥❡♠♣❧♦②❡❞✳

u

✶

✷

✸

✹

d

✷✷✺✵✵ ✷✷✵✵✵ ✷✶✺✵✵ ✷✶✵✵✵

❚❛❜❧❡ ✹✳✶

✹✳✶✳✸ ◆♦t❛t✐♦♥ ✉s❡❞ ❢♦r ❋✉♥❝t✐♦♥s

■♥ ❣r❛❞❡ ✶✵ ②♦✉ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ t♦ t❤❡ ♥♦t❛t✐♦♥ ✉s❡❞ t♦ ✧♥❛♠❡✧ ❛ ❢✉♥❝t✐♦♥✳ ■♥ ❛ ❢✉♥❝t✐♦♥ y = f (x)✱ y ✐s

❝❛❧❧❡❞ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ ❜❡❝❛✉s❡ t❤❡ ✈❛❧✉❡ ♦❢ y ❞❡♣❡♥❞s ♦♥ ✇❤❛t ②♦✉ ❝❤♦♦s❡ ❛s x✳ ❲❡ s❛② x ✐s t❤❡

✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✱ s✐♥❝❡ ✇❡ ❝❛♥ ❝❤♦♦s❡ x t♦ ❜❡ ❛♥② ♥✉♠❜❡r✳ ❙✐♠✐❧❛r❧② ✐❢ g (t) = 2t + 1✱ t❤❡♥ t ✐s t❤❡

✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❛♥❞ g ✐s t❤❡ ❢✉♥❝t✐♦♥ ♥❛♠❡✳ ■❢ f (x) = 3x − 5 ❛♥❞ ②♦✉ ❛r❡ ❛s❦ t♦ ❞❡t❡r♠✐♥❡ f (3)✱ t❤❡♥

②♦✉ ❤❛✈❡ t♦ ✇♦r❦ ♦✉t t❤❡ ✈❛❧✉❡ ❢♦r f (x) ✇❤❡♥ x = 3✳ ❋♦r ❡①❛♠♣❧❡✱

f (x)

=

3x − 5

f (3)

=

3 (3) − 5

✭✹✳✹✮

=

4

✹✳✷ ●r❛♣❤s ♦❢ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥s✷

✹✳✷✳✶ ●r❛♣❤s ♦❢ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥s

■♥ ❡❛r❧✐❡r ❣r❛❞❡s✱ ②♦✉ st✉❞✐❡❞ ✈❛r✐♦✉s t②♣❡s ♦❢ ❢✉♥❝t✐♦♥s ❛♥❞ ✉♥❞❡rst♦♦❞ t❤❡ ❡✛❡❝t ♦❢ ✈❛r✐♦✉s ♣❛r❛♠❡t❡rs ✐♥

t❤❡ ❣❡♥❡r❛❧ ❡q✉❛t✐♦♥✳ ■♥ t❤✐s s❡❝t✐♦♥✱ ✇❡ ✇✐❧❧ ❝♦♥s✐❞❡r ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s✳

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✽✷✴✶✳✶✴❃✳

✺✺

❆♥ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✐s ❛ ❢✉♥❝t✐♦♥ ✇❤✐❝❤ ✧❞♦❡s t❤❡ r❡✈❡rs❡✧ ♦❢ ❛ ❣✐✈❡♥ ❢✉♥❝t✐♦♥✳ ▼♦r❡ ❢♦r♠❛❧❧②✱ ✐❢ f ✐s ❛

❢✉♥❝t✐♦♥ ✇✐t❤ ❞♦♠❛✐♥ X✱ t❤❡♥ f−1 ✐s ✐ts ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❢♦r ❡✈❡r② x ∈ X ✇❡ ❤❛✈❡✿

f −1 (f (x)) == x

✭✹✳✺✮

❆ s✐♠♣❧❡ ✇❛② t♦ t❤✐♥❦ ❛❜♦✉t t❤✐s ✐s t❤❛t ❛ ❢✉♥❝t✐♦♥✱ s❛② y = f (x)✱ ❣✐✈❡s ②♦✉ ❛ y✲✈❛❧✉❡ ✐❢ ②♦✉ s✉❜st✐t✉t❡ ❛♥

x✲✈❛❧✉❡ ✐♥t♦ f (x)✳ ❚❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ t❡❧❧s ②♦✉ t❡❧❧s ②♦✉ ✇❤✐❝❤ x✲✈❛❧✉❡ ✇❛s ✉s❡❞ t♦ ❣❡t ❛ ♣❛rt✐❝✉❧❛r y✲✈❛❧✉❡

✇❤❡♥ ②♦✉ s✉❜st✐t✉❡ t❤❡ y✲✈❛❧✉❡ ✐♥t♦ f−1 (x)✳ ❚❤❡r❡ ❛r❡ s♦♠❡ t❤✐♥❣s ✇❤✐❝❤ ❝❛♥ ❝♦♠♣❧✐❝❛t❡ t❤✐s ❢♦r ❡①❛♠♣❧❡✱

t❤✐♥❦ ❛❜♦✉t ❛ sin ❢✉♥❝t✐♦♥✱ t❤❡r❡ ❛r❡ ♠❛♥② x✲✈❛❧✉❡s t❤❛t ❣✐✈❡ ②♦✉ ❛ ♣❡❛❦ ❛s t❤❡ ❢✉♥❝t✐♦♥ ♦s❝✐❧❧❛t❡s✳ ❚❤✐s

♠❡❛♥s t❤❛t t❤❡ ✐♥✈❡rs❡ ♦❢ ❛ sin ❢✉♥❝t✐♦♥ ✇♦✉❧❞ ❜❡ tr✐❝❦② t♦ ❞❡✜♥❡ ❜❡❝❛✉s❡ ✐❢ ②♦✉ s✉❜st✐t✉t❡ t❤❡ ♣❡❛❦ y✲✈❛❧✉❡

✐♥t♦ ✐t ②♦✉ ✇♦♥✬t ❦♥♦✇ ✇❤✐❝❤ ♦❢ t❤❡ x✲✈❛❧✉❡s ✇❛s ✉s❡❞ t♦ ❣❡t t❤❡ ♣❡❛❦✳

y

=

f (x)

we have a function

y1

=

f (x1)

we substitute a specific x − value into the function to get a specific y − value

consider the inverse function

x

=

f −1 (y)

✭✹✳✻✮

x

=

f −1 (y)

substituting the specific y − value into the inverse should return the specific x − value

=

f −1 (y1)

=

x1

❚❤✐s ✇♦r❦s ❜♦t❤ ✇❛②s✱ ✐❢ ✇❡ ❞♦♥✬t ❤❛✈❡ ❛♥② ❝♦♠♣❧✐❝❛t✐♦♥s ❧✐❦❡ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ sin ❢✉♥❝t✐♦♥✱ s♦ ✇❡ ❝❛♥

✇r✐t❡✿

f −1 (f (x)) = f f −1 (x) = x

✭✹✳✼✮

❋♦r ❡①❛♠♣❧❡✱ ✐❢ t❤❡ ❢✉♥❝t✐♦♥ x → 3x + 2 ✐s ❣✐✈❡♥✱ t❤❡♥ ✐ts ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✐s x → (x−2)✳ ❚❤✐s ✐s ✉s✉❛❧❧②

3

✇r✐tt❡♥ ❛s✿

f

:

x → 3x + 2

✭✹✳✽✮

f −1

:

x → (x−2)

3

❚❤❡ s✉♣❡rs❝r✐♣t ✧✲✶✧ ✐s ♥♦t ❛♥ ❡①♣♦♥❡♥t✳

■❢ ❛ ❢✉♥❝t✐♦♥ f ❤❛s ❛♥ ✐♥✈❡rs❡ t❤❡♥ f ✐s s❛✐❞ t♦ ❜❡ ✐♥✈❡rt✐❜❧❡✳

■❢ f ✐s ❛ r❡❛❧✲✈❛❧✉❡❞ ❢✉♥❝t✐♦♥✱ t❤❡♥ ❢♦r f t♦ ❤❛✈❡ ❛ ✈❛❧✐❞ ✐♥✈❡rs❡✱ ✐t ♠✉st ♣❛ss t❤❡ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ t❡st✱

t❤❛t ✐s ❛ ❤♦r✐③♦♥t❛❧ ❧✐♥❡ y = k ♣❧❛❝❡❞ ❛♥②✇❤❡r❡ ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ♠✉st ♣❛ss t❤r♦✉❣❤ f ❡①❛❝t❧② ♦♥❝❡ ❢♦r ❛❧❧

r❡❛❧ k✳

■t ✐s ♣♦ss✐❜❧❡ t♦ ✇♦r❦ ❛r♦✉♥❞ t❤✐s ❝♦♥❞✐t✐♦♥✱ ❜② ❞❡✜♥✐♥❣ ❛ ✏♠✉❧t✐✲✈❛❧✉❡❞✏ ❢✉♥❝t✐♦♥ ❛s ❛♥ ✐♥✈❡rs❡✳

■❢ ♦♥❡ r❡♣r❡s❡♥ts t❤❡ ❢✉♥❝t✐♦♥ f ❣r❛♣❤✐❝❛❧❧② ✐♥ ❛ xy✲❝♦♦r❞✐♥❛t❡ s②st❡♠✱ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❡q✉❛t✐♦♥

♦❢ ❛ str❛✐❣❤t ❧✐♥❡✱ f−1✱ ✐s t❤❡ r❡✢❡❝t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f ❛❝r♦ss t❤❡ ❧✐♥❡ y = x✳

❆❧❣❡❜r❛✐❝❛❧❧②✱ ♦♥❡ ❝♦♠♣✉t❡s t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ f ❜② s♦❧✈✐♥❣ t❤❡ ❡q✉❛t✐♦♥

y = f (x)

✭✹✳✾✮

❢♦r x✱ ❛♥❞ t❤❡♥ ❡①❝❤❛♥❣✐♥❣ y ❛♥❞ x t♦ ❣❡t

y = f −1 (x)

✭✹✳✶✵✮

✺✻

❈❍❆P❚❊❘ ✹✳ ❋❯◆❈❚■❖◆❙ ❆◆❉ ●❘❆P❍❙

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❲✽✹❧❖❜♠❖♣✽▼✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✹✳✷

✹✳✷✳✶✳✶ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥ ♦❢ y = ax + q

❚❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ y = ax + q ✐s ❞❡t❡r♠✐♥❡❞ ❜② s♦❧✈✐♥❣ ❢♦r x ❛s✿

y

=

ax + q

ax

=

y − q

✭✹✳✶✶✮

x

=

y−q

a

=

1 y − q

a

a

❚❤❡r❡❢♦r❡ t❤❡ ✐♥✈❡rs❡ ♦❢ y = ax + q ✐s y = 1 x − q ✳

a

a

❚❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐s ❛❧s♦ ❛ str❛✐❣❤t ❧✐♥❡✱ ❡①❝❡♣t ❢♦r t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ str❛✐❣❤t ❧✐♥❡

✐s ❛ ♣❡r❢❡❝t❧② ❤♦r✐③♦♥t❛❧ ❧✐♥❡✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ t❤❡ ✐♥✈❡rs❡ ✐s ✉♥❞❡✜♥❡❞✳

❋♦r ❡①❛♠♣❧❡✱ t❤❡ str❛✐❣❤t ❧✐♥❡ ❡q✉❛t✐♦♥ ❣✐✈❡♥ ❜② y = 2x − 3 ❤❛s ❛s ✐♥✈❡rs❡ t❤❡ ❢✉♥❝t✐♦♥✱ y = 1x + 3✳ ❚❤❡

2

2

❣r❛♣❤s ♦❢ t❤❡s❡ ❢✉♥❝t✐♦♥s ❛r❡ s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✹✳✸✳ ■t ❝❛♥ ❜❡ s❡❡♥ t❤❛t t❤❡ t✇♦ ❣r❛♣❤s ❛r❡ r❡✢❡❝t✐♦♥s ♦❢ ❡❛❝❤

♦t❤❡r ❛❝r♦ss t❤❡ ❧✐♥❡ y = x✳

❋✐❣✉r❡ ✹✳✸✿ ❚❤❡ ❣r❛♣❤s ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = 2x − 3 ❛♥❞ ✐ts ✐♥✈❡rs❡ f−1 (x) = 1 x + 3 ✳ ❚❤❡ ❧✐♥❡ y = x

2

2

✐s s❤♦✇♥ ❛s ❛ ❞❛s❤❡❞ ❧✐♥❡✳

✹✳✷✳✶✳✶✳✶ ❉♦♠❛✐♥ ❛♥❞ ❘❛♥❣❡

❲❡ ❤❛✈❡ s❡❡♥ t❤❛t t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠ y = ax+q ✐s {x : x ∈ R} ❛♥❞ t❤❡ r❛♥❣❡ ✐s {y : y ∈ R}✳

❙✐♥❝❡ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡ ✐s ❛❧s♦ ❛ str❛✐❣❤t ❧✐♥❡✱ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ✇✐❧❧ ❤❛✈❡ t❤❡ s❛♠❡

❞♦♠❛✐♥ ❛♥❞ r❛♥❣❡ ❛s t❤❡ ♦r✐❣✐♥❛❧ ❢✉♥❝t✐♦♥✳

✹✳✷✳✶✳✶✳✷ ■♥t❡r❝❡♣ts

❚❤❡ ❣❡♥❡r❛❧ ❢♦r♠ ♦❢ t❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ❢♦r♠ y = ax + q ✐s y = 1 x − q ✳

a

a

❇② s❡tt✐♥❣ x = 0 ✇❡ ❤❛✈❡ t❤❛t t❤❡ y✲✐♥t❡r❝❡♣t ✐s yint = − q ✳ ❙✐♠✐❧❛r❧②✱ ❜② s❡tt✐♥❣ y = 0 ✇❡ ❤❛✈❡ t❤❛t t❤❡

a

x✲✐♥t❡r❝❡♣t ✐s xint = q✳

■t ✐s ✐♥t❡r❡st✐♥❣ t♦ ♥♦t❡ t❤❛t ✐❢ f (x) = ax + q✱ t❤❡♥ f−1 (x) = 1 x − q ❛♥❞ t❤❡ y✲✐♥t❡r❝❡♣t ♦❢ f (x) ✐s t❤❡

a

a

x✲✐♥t❡r❝❡♣t ♦❢ f −1 (x) ❛♥❞ t❤❡ x✲✐♥t❡r❝❡♣t ♦❢ f (x) ✐s t❤❡ y✲✐♥t❡r❝❡♣t ♦❢ f −1 (x)✳

✺✼

✹✳✷✳✶✳✷ ❊①❡r❝✐s❡s

✶✳ ●✐✈❡♥ f (x) = 2x − 3✱ ✜♥❞ f−1 (x)

✷✳ ❈♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x − 7✳

❛✳ ■s t❤❡ r❡❧❛t✐♦♥ ❛ ❢✉♥❝t✐♦♥❄

❜✳ ■❢ ✐t ✐s ❛ ❢✉♥❝t✐♦♥✱ ✐❞❡♥t✐❢② t❤❡ ❞♦♠❛✐♥ ❛♥❞ r❛♥❣❡✳

✸✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = 3x − 1 ❛♥❞ ✐ts ✐♥✈❡rs❡ ♦♥ t❤❡ s❛♠❡ s❡t ♦❢ ❛①❡s✳

✹✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s f−1 (x) = 2x − 4✱ ✇❤❛t ✐s t❤❡ ❢✉♥❝t✐♦♥ f (x)❄

✹✳✷✳✶✳✸ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥ ♦❢ y = ax2

❚❤❡ ✐♥✈❡rs❡ r❡❧❛t✐♦♥✱ ♣♦ss✐❜❧② ❛ ❢✉♥❝t✐♦♥✱ ♦❢ y = ax2 ✐s ❞❡t❡r♠✐♥❡❞ ❜② s♦❧✈✐♥❣ ❢♦r x ❛s✿

y

=

ax2

x2

=

y

✭✹✳✶✷✮

a

x

=

±

y

a

√

❋✐❣✉r❡ ✹✳✹✿ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = x2 ❛♥❞ ✐ts ✐♥✈❡rs❡ f−1 (x) = ± x✳ ❚❤❡ ❧✐♥❡ y = x ✐s s❤♦✇♥ ❛s ❛

❞❛s❤❡❞ ❧✐♥❡✳

❲❡ s❡❡ t❤❛t t❤❡ ✐♥✈❡rs❡ ✑❢✉♥❝t✐♦♥✑ ♦❢ y = ax2 ✐s ♥♦t ❛ ❢✉♥❝t✐♦♥ ❜❡❝❛✉s❡ ✐t ❢❛✐❧s t❤❡ ✈❡rt✐❝❛❧ ❧✐♥❡ t❡st✳ ■❢

√

✇❡ ❞r❛✇ ❛ ✈❡rt✐❝❛❧ ❧✐♥❡ t❤r♦✉❣❤ t❤❡ ❣r❛♣❤ ♦❢ f−1 (x) = ± x✱ t❤❡ ❧✐♥❡ ✐♥t❡rs❡❝ts t❤❡ ❣r❛♣❤ ♠♦r❡ t❤❛♥ ♦♥❝❡✳

❚❤❡r❡ ❤❛s t♦ ❜❡ ❛ r❡str✐❝t✐♦♥ ♦♥ t❤❡ ❞♦♠❛✐♥ ♦❢ ❛ ♣❛r❛❜♦❧❛ ❢♦r t❤❡ ✐♥✈❡rs❡ t♦ ❛❧s♦ ❜❡ ❛ ❢✉♥❝t✐♦♥✳ ❈♦♥s✐❞❡r t❤❡

❢✉♥❝t✐♦♥ f (x) = −x2 + 9✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ f ❝❛♥ ❜❡ ❢♦✉♥❞ ❜② ✇✐t✐♥❣ f (y) = x✳ ❚❤❡♥

x

=

−y2 + 9

y2

=

9 − x

✭✹✳✶✸✮

√

y

=

± 9 − x

√

■❢ ✇❡ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ♦❢ f (x) t♦ ❜❡ x ≥ 0✱ t❤❡♥ 9 − x ✐s ❛ ❢✉♥❝t✐♦♥✳ ■❢ t❤❡ r❡str✐❝t✐♦♥ ♦♥ t❤❡ ❞♦♠❛✐♥

√

♦❢ f ✐s x ≤ 0 t❤❡♥ − 9 − x ✇♦✉❧❞ ❜❡ ❛ ❢✉♥❝t✐♦♥✱ ✐♥✈❡rs❡ t♦ f✳

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s ✲ ✷

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❛❡②❋❜✷❡❱❍✶❝✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✹✳✺

✺✽

❈❍❆P❚❊❘ ✹✳ ❋❯◆❈❚■❖◆❙ ❆◆❉ ●❘❆P❍❙

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥s ✲ ✸

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❇q✾❝q✾❋❩✉◆▼✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✹✳✻

✹✳✷✳✶✳✹ ❊①❡r❝✐s❡s

✶✳ ❚❤❡ ❣r❛♣❤ ♦❢ f−1 ✐s s❤♦✇♥✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ f✱ ❣✐✈❡♥ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ f ✐s ❛ ♣❛r❛❜♦❧❛✳ ✭❉♦ ♥♦t

s✐♠♣❧✐❢② ②♦✉r ❛♥s✇❡r✮

❋✐❣✉r❡ ✹✳✼

✷✳ f (x) = 2x2✳

❛✳ ❉r❛✇ t❤❡ ❣r❛♣❤ ♦❢ f ❛♥❞ st❛t❡ ✐ts ❞♦♠❛✐♥ ❛♥❞ r❛♥❣❡✳

❜✳ ❋✐♥❞ f−1 ❛♥❞✱ ✐❢ ✐t ❡①✐sts✱ st❛t❡ t❤❡ ❞♦♠❛✐♥ ❛♥❞ r❛♥❣❡✳

❝✳ ❲❤❛t ♠✉st t❤❡ ❞♦♠❛✐♥ ♦❢ f ❜❡✱ s♦ t❤❛t f−1 ✐s ❛ ❢✉♥❝t✐♦♥ ❄

✸✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ x = − 10 − y2✳ ▲❛❜❡❧ ❛ ♣♦✐♥t ♦♥ t❤❡ ❣r❛♣❤ ♦t❤❡r t❤❛♥ t❤❡ ✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡

❛①❡s✳

✹✳

❛✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ❧❛❜❡❧❧✐♥❣ ❛ ♣♦✐♥t ♦t❤❡r t❤❛♥ t❤❡ ♦r✐❣✐♥ ♦♥ ②♦✉r ❣r❛♣❤✳

❜✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❛❜♦✈❡ ❣r❛♣❤ ✐♥ t❤❡ ❢♦r♠ y = ...✳

√

❝✳ ◆♦✇ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ y = x✳ √

❞✳ ❚❤❡ t❛♥❣❡♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ y = x ❛t t❤❡ ♣♦✐♥t ❆✭✾❀✸✮ ✐♥t❡rs❡❝ts t❤❡ x✲❛①✐s ❛t ❇✳ ❋✐♥❞ t❤❡

❡q✉❛t✐♦♥ ♦❢ t❤✐s t❛♥❣❡♥t ❛♥❞ ❤❡♥❝❡ ♦r ♦t❤❡r✇✐s❡ ♣r♦✈❡ t❤❛t t❤❡ y✲❛①✐s ❜✐s❡❝ts t❤❡ str❛✐❣❤t ❧✐♥❡ ❆❇✳

√

✺✳ ●✐✈❡♥✿ g (x) = −1 + x✱ ✜♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ g (x) ✐♥ t❤❡ ❢♦r♠ g−1 (x) = ...✳

✹✳✷✳✶✳✺ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥ ♦❢ y = ax

❚❤❡ ✐♥✈❡rs❡ ❢✉♥❝t✐♦♥ ♦❢ y = ax2 ✐s ❞❡t❡r♠✐♥❡❞ ❜② s♦❧✈✐♥❣ ❢♦r x ❛s ❢♦❧❧♦✇s✿

y

=

ax

log (y)

=

log (ax)

✭✹✳✶✹✮

=

xlog (a)

∴

x

=

log(y)

log(a)

❚❤❡ ✐♥✈❡rs❡ ♦❢ y = 10x ✐s x = 10y✱ ✇❤✐❝❤ ✇❡ ✇r✐t❡ ❛s y = log (x)✳ ❚❤❡r❡❢♦r❡✱ ✐❢ f (x) = 10x✱ t❤❡♥

f −1 = log (x)✳

✺✾

❋✐❣✉r❡ ✹✳✽✿ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = 10x ❛♥❞ ✐ts ✐♥✈❡rs❡ f−1 (x) = log (x)✳ ❚❤❡ ❧✐♥❡ y = x ✐s s❤♦✇♥ ❛s ❛

❞❛s❤❡❞ ❧✐♥❡✳

❚❤❡ ❡①♣♦♥❡♥t✐❛❧ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❧♦❣❛r✐t❤♠✐❝ ❢✉♥❝t✐♦♥ ❛r❡ ✐♥✈❡rs❡s ♦❢ ❡❛❝❤ ♦t❤❡r❀ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♦♥❡ ✐s

t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ♦t❤❡r✱ r❡✢❡❝t❡❞ ✐♥ t❤❡ ❧✐♥❡ y = x✳ ❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡q✉❛❧ t♦ t❤❡ r❛♥❣❡ ♦❢ t❤❡

✐♥✈❡rs❡✳ ❚❤❡ r❛♥❣❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐s ❡q✉❛❧ t♦ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ✐♥✈❡rs❡✳

✹✳✷✳✶✳✻ ❊①❡r❝✐s❡s

✶✳ ●✐✈❡♥ t❤❛t

x

f (x) = 1

✱ s❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ f ❛♥❞ f−1 ♦♥ t❤❡ s❛♠❡ s②st❡♠ ♦❢ ❛①❡s ✐♥❞✐❝❛t✐♥❣ ❛ ♣♦✐♥t

5

♦♥ ❡❛❝❤ ❣r❛♣❤ ✭♦t❤❡r t❤❛♥ t❤❡ ✐♥t❡r❝❡♣ts✮ ❛♥❞ s❤♦✇✐♥❣ ❝❧❡❛r❧② ✇❤✐❝❤ ✐s f ❛♥❞ ✇❤✐❝❤ ✐s f−1✳

✷✳ ●✐✈❡♥ t❤❛t f (x) = 4−x✱

❛✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ f ❛♥❞ f−1 ♦♥ t❤❡ s❛♠❡ s②st❡♠ ♦❢ ❛①❡s ✐♥❞✐❝❛t✐♥❣ ❛ ♣♦✐♥t ♦♥ ❡❛❝❤ ❣r❛♣❤

✭♦t❤❡r t❤❛♥ t❤❡ ✐♥t❡r❝❡♣ts✮ ❛♥❞ s❤♦✇✐♥❣ ❝❧❡❛r❧② ✇❤✐❝❤ ✐s f ❛♥❞ ✇❤✐❝❤ ✐s f−1✳

❜✳ ❲r✐t❡ f−1 ✐♥ t❤❡ ❢♦r♠ y = ...✳

√

✸✳ ●✐✈❡♥ g (x) = −1 + x✱ ✜♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ g (x) ✐♥ t❤❡ ❢♦r♠ g−1 (x) = ...

✹✳ ❆♥s✇❡r t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥s✿

❛✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ y = x2✱ ❧❛❜❡❧✐♥❣ ❛ ♣♦✐♥t ♦t❤❡r t❤❛♥ t❤❡ ♦r✐❣✐♥ ♦♥ ②♦✉r ❣r❛♣❤✳

❜✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❛❜♦✈❡ ❣r❛♣❤ ✐♥ t❤❡ ❢♦r♠ y = ...

√

❝✳ ◆♦✇✱ s❦❡t❝❤ y = x✳

√

❞✳ ❚❤❡ t❛♥❣❡♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ y = x ❛t t❤❡ ♣♦✐♥t A (9; 3) ✐♥t❡rs❡❝ts t❤❡ x✲❛①✐s ❛t B✳ ❋✐♥❞ t❤❡

❡q✉❛t✐♦♥ ♦❢ t❤✐s t❛♥❣❡♥t✱ ❛♥❞ ❤❡♥❝❡✱ ♦r ♦t❤❡r✇✐s❡✱ ♣r♦✈❡ t❤❛t t❤❡ y✲❛①✐s ❜✐s❡❝ts t❤❡ str❛✐❣❤t ❧✐♥❡

AB✳

✹✳✷✳✷ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ x = − 10 − y2✳ ■s t❤✐s ❣r❛♣❤ ❛ ❢✉♥❝t✐♦♥ ❄ ❱❡r✐❢② ②♦✉r ❛♥s✇❡r✳

✷✳ f (x) = 1 ✱

x−5

❛✳ ❞❡t❡r♠✐♥❡ t❤❡ y✲✐♥t❡r❝❡♣t ♦❢ f (x)

❜✳ ❞❡t❡r♠✐♥❡ x ✐❢ f (x) = −1✳

✸✳ ❇❡❧♦✇✱ ②♦✉ ❛r❡ ❣✐✈❡♥ ✸ ❣r❛♣❤s ❛♥❞ ✺ ❡q✉❛t✐♦♥s✳

❋✐❣✉r❡ ✹✳✾

❛✳ y = log x

3

❜✳ y = −log x

3

❝✳ y = log (−x)

3

✻✵

❈❍❆P❚❊❘ ✹✳ ❋❯◆❈❚■❖◆❙ ❆◆❉ ●❘❆P❍❙

❞✳ y = 3−x

❡✳ y = 3x

❲r✐t❡ t❤❡ ❡q✉❛t✐♦♥ t❤❛t ❜❡st ❞❡s❝r✐❜❡s ❡❛❝❤ ❣r❛♣❤✳

√

✹✳ ●✐✈❡♥ g (x) = −1 + x✱ ✜♥❞ t❤❡ ✐♥✈❡rs❡ ♦❢ g (x) ✐♥ t❤❡ ❢♦r♠ g−1 (x) = ...

✺✳ ❈♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ h (x) = 3x

❛✳ ❲r✐t❡ ❞♦✇♥ t❤❡ ✐♥✈❡rs❡ ✐♥ t❤❡ ❢♦r♠ h−1 (x) = ...

❜✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ h (x) ❛♥❞ h−1 (x) ♦♥ t❤❡ s❛♠❡ s❡t ♦❢ ❛①❡s✱ ❧❛❜❡❧❧✐♥❣ t❤❡ ✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡

❛①❡s✳

❝✳ ❋♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ x ✐s h−1 (x) ✉♥❞❡✜♥❡❞ ❄

✻✳

❛✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ y = 2x2 + 1✱ ❧❛❜❡❧❧✐♥❣ ❛ ♣♦✐♥t ♦t❤❡r t❤❛♥ t❤❡ ♦r✐❣✐♥ ♦♥ ②♦✉r ❣r❛♣❤✳

❜✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ✐♥✈❡rs❡ ♦❢ t❤❡ ❛❜♦✈❡ ❣r❛♣❤ ✐♥ t❤❡ ❢♦r♠ y = ...

√

❝✳ ◆♦✇✱ s❦❡t❝❤ y = x✳

√

❞✳ ❚❤❡ t❛♥❣❡♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ y = x ❛t t❤❡ ♣♦✐♥t A (9; 3) ✐♥t❡rs❡❝ts t❤❡ x✲❛①✐s ❛t B✳ ❋✐♥❞ t❤❡

❡q✉❛t✐♦♥ ♦❢ t❤✐s t❛♥❣❡♥t✱ ❛♥❞ ❤❡♥❝❡✱ ♦r ♦t❤❡r✇✐s❡✱ ♣r♦✈❡ t❤❛t t❤❡ y✲❛①✐s ❜✐s❡❝ts t❤❡ str❛✐❣❤t ❧✐♥❡

AB✳

❈❤❛♣t❡r ✺

❉✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s

✺✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ▲✐♠✐ts✱ ❆✈❡r❛❣❡ ●r❛❞✐❡♥t✶

✺✳✶✳✶ ❲❤② ❞♦ ■ ❤❛✈❡ t♦ ❧❡❛r♥ t❤✐s st✉✛❄

❈❛❧❝✉❧✉s ✐s ♦♥❡ ♦❢ t❤❡ ❝❡♥tr❛❧ ❜r❛♥❝❤❡s ♦❢ ♠❛t❤❡♠❛t✐❝s ❛♥❞ ✇❛s ❞❡✈❡❧♦♣❡❞ ❢r♦♠ ❛❧❣❡❜r❛ ❛♥❞ ❣❡♦♠❡tr②✳

❈❛❧❝✉❧✉s ✐s ❜✉✐❧t ♦♥ t❤❡ ❝♦♥❝❡♣t ♦❢ ❧✐♠✐ts✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✳ ❈❛❧❝✉❧✉s ❝♦♥s✐sts ♦❢ t✇♦

❝♦♠♣❧❡♠❡♥t❛r② ✐❞❡❛s✿ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❛♥❞ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s✳ ❲❡ ✇✐❧❧ ♦♥❧② ❜❡ ❞❡❛❧✐♥❣ ✇✐t❤ ❞✐✛❡r❡♥t✐❛❧

❝❛❧❝✉❧✉s ✐♥ t❤✐s t❡①t✳ ❉✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ q✉❛♥t✐t✐❡s

✇✐t❤ r❡s♣❡❝t t♦ ♦t❤❡r q✉❛♥t✐t✐❡s✱ ♦r ♠♦r❡ ♣r❡❝✐s❡❧②✱ t❤❡ ❧♦❝❛❧ ❜❡❤❛✈✐♦✉r ♦❢ ❢✉♥❝t✐♦♥s✳ ❚❤✐s ❝❛♥ ❜❡ ✐❧❧✉str❛t❡❞

❜② t❤❡ s❧♦♣❡ ♦❢ ❛ ❢✉♥❝t✐♦♥✬s ❣r❛♣❤✳ ❊①❛♠♣❧❡s ♦❢ t②♣✐❝❛❧ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠s ✐♥❝❧✉❞❡✿ ✜♥❞✐♥❣ t❤❡

❛❝❝❡❧❡r❛t✐♦♥ ❛♥❞ ✈❡❧♦❝✐t② ♦❢ ❛ ❢r❡❡✲❢❛❧❧✐♥❣ ❜♦❞② ❛t ❛ ♣❛rt✐❝✉❧❛r ♠♦♠❡♥t ❛♥❞ ✜♥❞✐♥❣ t❤❡ ♦♣t✐♠❛❧ ♥✉♠❜❡r ♦❢

✉♥✐ts ❛ ❝♦♠♣❛♥② s❤♦✉❧❞ ♣r♦❞✉❝❡ t♦ ♠❛①✐♠✐③❡ ✐ts ♣r♦✜t✳

❈❛❧❝✉❧✉s ✐s ❢✉♥❞❛♠❡♥t❛❧❧② ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ♠❛t❤❡♠❛t✐❝s t❤❛t ②♦✉ ❤❛✈❡ st✉❞✐❡❞ ♣r❡✈✐♦✉s❧②✳ ❈❛❧❝✉❧✉s ✐s

♠♦r❡ ❞②♥❛♠✐❝ ❛♥❞ ❧❡ss st❛t✐❝✳ ■t ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ ❝❤❛♥❣❡ ❛♥❞ ♠♦t✐♦♥✳ ■t ❞❡❛❧s ✇✐t❤ q✉❛♥t✐t✐❡s t❤❛t ❛♣♣r♦❛❝❤

♦t❤❡r q✉❛♥t✐t✐❡s✳ ❋♦r t❤❛t r❡❛s♦♥ ✐t ♠❛② ❜❡ ✉s❡❢✉❧ t♦ ❤❛✈❡ ❛♥ ♦✈❡r✈✐❡✇ ♦❢ t❤❡ s✉❜❥❡❝t ❜❡❢♦r❡ ❜❡❣✐♥♥✐♥❣ ✐ts

✐♥t❡♥s✐✈❡ st✉❞②✳

❈❛❧❝✉❧✉s ✐s ❛ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ♠❛♥② ♣❤❡♥♦♠❡♥❛✱ ❜♦t❤ ♥❛t✉r❛❧ ❛♥❞ ♠❛♥✲♠❛❞❡✱ ❧✐❦❡ ❤♦✇ t❤❡ ✇✐♥❞ ❜❧♦✇s✱

❤♦✇ ✇❛t❡r ✢♦✇s✱ ❤♦✇ ❧✐❣❤t tr❛✈❡❧s✱ ❤♦✇ s♦✉♥❞ tr❛✈❡❧s✱ ❤♦✇ t❤❡ ♣❧❛♥❡ts ♠♦✈❡ ❛♥❞ ❡✈❡♥ ❡❝♦♥♦♠✐❝s✳

■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❣✐✈❡ ❛ ❣❧✐♠♣s❡ ♦❢ s♦♠❡ ♦❢ t❤❡ ♠❛✐♥ ✐❞❡❛s ♦❢ ❝❛❧❝✉❧✉s ❜② s❤♦✇✐♥❣ ❤♦✇ ❧✐♠✐ts ❛r✐s❡ ✇❤❡♥

✇❡ ❛tt❡♠♣t t♦ s♦❧✈❡ ❛ ✈❛r✐❡t② ♦❢ ♣r♦❜❧❡♠s✳

✺✳✶✳✶✳✶ ■♥t❡❣r❛❧ ❈❛❧❝✉❧✉s

■♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♦❢ q✉❛♥t✐t✐❡s✱ s✉❝❤ ❛s ❛r❡❛s ✉♥❞❡r ❛ ❝✉r✈❡✱ ❧✐♥❡❛r ❞✐st❛♥❝❡

tr❛✈❡❧❡❞✱ ♦r ✈♦❧✉♠❡ ❞✐s♣❧❛❝❡❞✳ ❉✐✛❡r❡♥t✐❛❧ ❛♥❞ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ❛❝t ✐♥✈❡rs❡❧② t♦ ❡❛❝❤ ♦t❤❡r✳ ❊①❛♠♣❧❡s ♦❢

t②♣✐❝❛❧ ✐♥t❡❣r❛❧ ❝❛❧❝✉❧✉s ♣r♦❜❧❡♠s ✐♥❝❧✉❞❡ ✜♥❞✐♥❣ ❛r❡❛s ❛♥❞ ✈♦❧✉♠❡s✱ ✜♥❞✐♥❣ t❤❡ ❛♠♦✉♥t ♦❢ ✇❛t❡r ♣✉♠♣❡❞

❜② ❛ ♣✉♠♣ ✇✐t❤ ❛ s❡t ♣♦✇❡r ✐♥♣✉t ❜✉t ✈❛r②✐♥❣ ❝♦♥❞✐t✐♦♥s ♦❢ ♣✉♠♣✐♥❣ ❧♦ss❡s ❛♥❞ ♣r❡ss✉r❡ ❛♥❞ ✜♥❞✐♥❣ t❤❡

❛♠♦✉♥t ♦❢ r❛✐♥ t❤❛t ❢❡❧❧ ✐♥ ❛ ❝❡rt❛✐♥ ❛r❡❛ ✐❢ t❤❡ r❛✐♥ ❢❡❧❧ ❛t ❛ s♣❡❝✐✜❝ r❛t❡✳

♥♦t❡✿ ❇♦t❤ ■s❛❛❝ ◆❡✇t♦♥ ✭✹ ❏❛♥✉❛r② ✶✻✹✸ ✕ ✸✶ ▼❛r❝❤ ✶✼✷✼✮ ❛♥❞ ●♦tt❢r✐❡❞ ▲❡✐❜♥✐t③ ✭✶ ❏✉❧② ✶✻✹✻

✕ ✶✹ ◆♦✈❡♠❜❡r ✶✼✶✻ ✭❍❛♥♦✈❡r✱ ●❡r♠❛♥②✮✮ ❛r❡ ❝r❡❞✐t❡❞ ✇✐t❤ t❤❡ ❵✐♥✈❡♥t✐♦♥✬ ♦❢ ❝❛❧❝✉❧✉s✳ ◆❡✇t♦♥

✇❛s t❤❡ ✜rst t♦ ❛♣♣❧② ❝❛❧❝✉❧✉s t♦ ❣❡♥❡r❛❧ ♣❤②s✐❝s✱ ✇❤✐❧❡ ▲❡✐❜♥✐t③ ❞❡✈❡❧♦♣❡❞ ♠♦st ♦❢ t❤❡ ♥♦t❛t✐♦♥

t❤❛t ✐s st✐❧❧ ✐♥ ✉s❡ t♦❞❛②✳

❲❤❡♥ ◆❡✇t♦♥ ❛♥❞ ▲❡✐❜♥✐③ ✜rst ♣✉❜❧✐s❤❡❞ t❤❡✐r r❡s✉❧ts✱ t❤❡r❡ ✇❛s s♦♠❡ ❝♦♥tr♦✈❡rs② ♦✈❡r ✇❤❡t❤❡r

▲❡✐❜♥✐③✬s ✇♦r❦ ✇❛s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ◆❡✇t♦♥✬s✳ ❲❤✐❧❡ ◆❡✇t♦♥ ❞❡r✐✈❡❞ ❤✐s r❡s✉❧ts ②❡❛rs ❜❡❢♦r❡ ▲❡✐❜♥✐③✱

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✵✴✶✳✶✴❃✳

✻✶

✻✷

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

✐t ✇❛s ♦♥❧② s♦♠❡ t✐♠❡ ❛❢t❡r ▲❡✐❜♥✐③ ♣✉❜❧✐s❤❡❞ ✐♥ ✶✻✽✹ t❤❛t ◆❡✇t♦♥ ♣✉❜❧✐s❤❡❞✳ ▲❛t❡r✱ ◆❡✇t♦♥

✇♦✉❧❞ ❝❧❛✐♠ t❤❛t ▲❡✐❜♥✐③ ❣♦t t❤❡ ✐❞❡❛ ❢r♦♠ ◆❡✇t♦♥✬s ♥♦t❡s ♦♥ t❤❡ s✉❜❥❡❝t❀ ❤♦✇❡✈❡r ❡①❛♠✐♥❛t✐♦♥ ♦❢

t❤❡ ♣❛♣❡rs ♦❢ ▲❡✐❜♥✐③ ❛♥❞ ◆❡✇t♦♥ s❤♦✇ t❤❡② ❛rr✐✈❡❞ ❛t t❤❡✐r r❡s✉❧ts ✐♥❞❡♣❡♥❞❡♥t❧②✱ ✇✐t❤ ▲❡✐❜♥✐③

st❛rt✐♥❣ ✜rst ✇✐t❤ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ ◆❡✇t♦♥ ✇✐t❤ ❞✐✛❡r❡♥t✐❛t✐♦♥✳ ❚❤✐s ❝♦♥tr♦✈❡rs② ❜❡t✇❡❡♥ ▲❡✐❜♥✐③

❛♥❞ ◆❡✇t♦♥ ❞✐✈✐❞❡❞ ❊♥❣❧✐s❤✲s♣❡❛❦✐♥❣ ♠❛t❤❡♠❛t✐❝✐❛♥s ❢r♦♠ t❤♦s❡ ✐♥ ❊✉r♦♣❡ ❢♦r ♠❛♥② ②❡❛rs✱ ✇❤✐❝❤

s❧♦✇❡❞ t❤❡ ❞❡✈❡❧♦♣♠❡♥t ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❛♥❛❧②s✐s✳ ❚♦❞❛②✱ ❜♦t❤ ◆❡✇t♦♥ ❛♥❞ ▲❡✐❜♥✐③ ❛r❡ ❣✐✈❡♥ ❝r❡❞✐t

❢♦r ✐♥❞❡♣❡♥❞❡♥t❧② ❞❡✈❡❧♦♣✐♥❣ ❝❛❧❝✉❧✉s✳ ■t ✐s ▲❡✐❜♥✐③✱ ❤♦✇❡✈❡r✱ ✇❤♦ ✐s ❝r❡❞✐t❡❞ ✇✐t❤ ❣✐✈✐♥❣ t❤❡ ♥❡✇

❞✐s❝✐♣❧✐♥❡ t❤❡ ♥❛♠❡ ✐t ✐s ❦♥♦✇♥ ❜② t♦❞❛②✿ ✧❝❛❧❝✉❧✉s✧✳ ◆❡✇t♦♥✬s ♥❛♠❡ ❢♦r ✐t ✇❛s ✧t❤❡ s❝✐❡♥❝❡ ♦❢

✢✉①✐♦♥s✧✳

✺✳✶✳✷ ▲✐♠✐ts

✺✳✶✳✷✳✶ ❆ ❚❛❧❡ ♦❢ ❆❝❤✐❧❧❡s ❛♥❞ t❤❡ ❚♦rt♦✐s❡

♥♦t❡✿ ❩❡♥♦ ✭❝✐r❝❛ ✹✾✵ ❇❈ ✲ ❝✐r❝❛ ✹✸✵ ❇❈✮ ✇❛s ❛ ♣r❡✲❙♦❝r❛t✐❝ ●r❡❡❦ ♣❤✐❧♦s♦♣❤❡r ♦❢ s♦✉t❤❡r♥ ■t❛❧②

✇❤♦ ✐s ❢❛♠♦✉s ❢♦r ❤✐s ♣❛r❛❞♦①❡s✳

❖♥❡ ♦❢ ❩❡♥♦✬s ♣❛r❛❞♦①❡s ❝❛♥ ❜❡ s✉♠♠❛r✐s❡❞ ❜②✿

❆❝❤✐❧❧❡s ❛♥❞ ❛ t♦rt♦✐s❡ ❛❣r❡❡ t♦ ❛ r❛❝❡✱ ❜✉t t❤❡ t♦rt♦✐s❡ ✐s ✉♥❤❛♣♣② ❜❡❝❛✉s❡ ❆❝❤✐❧❧❡s ✐s ✈❡r② ❢❛st✳

❙♦✱ t❤❡ t♦rt♦✐s❡ ❛s❦s ❆❝❤✐❧❧❡s ❢♦r ❛ ❤❡❛❞✲st❛rt✳ ❆❝❤✐❧❧❡s ❛❣r❡❡s t♦ ❣✐✈❡ t❤❡ t♦rt♦✐s❡ ❛ ✶ ✵✵✵ ♠ ❤❡❛❞

st❛rt✳ ❉♦❡s ❆❝❤✐❧❧❡s ♦✈❡rt❛❦❡ t❤❡ t♦rt♦✐s❡❄

❲❡ ❦♥♦✇ ❤♦✇ t♦ s♦❧✈❡ t❤✐s ♣r♦❜❧❡♠✳ ❲❡ st❛rt ❜② ✇r✐t✐♥❣✿

xA

=

vAt

✭✺✳✶✮

xt

=

1000m + vtt

✇❤❡r❡

xA

❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❜② ❆❝❤✐❧❧❡s

vA

❆❝❤✐❧❧❡s✬ s♣❡❡❞

t

t✐♠❡ t❛❦❡♥ ❜② ❆❝❤✐❧❧❡s t♦ ♦✈❡rt❛❦❡ t♦rt♦✐s❡

xt

❞✐st❛♥❝❡ ❝♦✈❡r❡❞ ❜② t❤❡ t♦rt♦✐s❡

vt

t❤❡ t♦rt♦✐s❡✬s s♣❡❡❞

❚❛❜❧❡ ✺✳✶

■❢ ✇❡ ❛ss✉♠❡ t❤❛t ❆❝❤✐❧❧❡s r✉♥s ❛t ✷ ♠·s−1❛♥❞ t❤❡ t♦rt♦✐s❡ r✉♥s ❛t ✵✱✷✺ ♠·s−1t❤❡♥ ❆❝❤✐❧❧❡s ✇✐❧❧ ♦✈❡rt❛❦❡

t❤❡ t♦rt♦✐s❡ ✇❤❡♥ ❜♦t❤ ♦❢ t❤❡♠ ❤❛✈❡ ❝♦✈❡r❡❞ t❤❡ s❛♠❡ ❞✐st❛♥❝❡✳ ❚❤✐s ♠❡❛♥s t❤❛t ❆❝❤✐❧❧❡s ♦✈❡rt❛❦❡s t❤❡

✻✸

t♦rt♦✐s❡ ❛t ❛ t✐♠❡ ❝❛❧❝✉❧❛t❡❞ ❛s✿

xA

=

xt

vAt

=

1000 + vtt

2m · s−1 t

=

1000m + 0, 25m · s−1 t

2m · s−1 − 0, 25m · s−1 t

=

1000m

t

=

1000m

✭✺✳✷✮

1 3 m·s−1

4

=

1000m

7 m·s−1

4

=

(4)(1000) s

7

=

4000 s

7

=

571 3 s

7

❍♦✇❡✈❡r✱ ❩❡♥♦ ✭t❤❡ ●r❡❡❦ ♣❤✐❧♦s♦♣❤❡r ✇❤♦ t❤♦✉❣❤t ✉♣ t❤✐s ♣r♦❜❧❡♠✮ ❧♦♦❦❡❞ ❛t ✐t ❛s ❢♦❧❧♦✇s✿ ❆❝❤✐❧❧❡s t❛❦❡s

t = 1000 = 500s t♦ tr❛✈❡❧ t❤❡ ✶ ✵✵✵ ♠ ❤❡❛❞ st❛rt t❤❛t t❤❡ t♦rt♦✐s❡ ❤❛❞✳ ❍♦✇❡✈❡r✱ ✐♥ t❤✐s ✺✵✵ s✱ t❤❡ t♦rt♦✐s❡

2

❤❛s tr❛✈❡❧❧❡❞ ❛ ❢✉rt❤❡r x = (500) (0, 25) = 125m. ❆❝❤✐❧❧❡s t❤❡♥ t❛❦❡s ❛♥♦t❤❡r t = 125 = 62, 5s t♦ tr❛✈❡❧ t❤❡

2

✶✷✺ ♠✳ ■♥ t❤✐s ✻✷✱✺ s✱ t❤❡ t♦rt♦✐s❡ tr❛✈❡❧s ❛ ❢✉rt❤❡r x = (62, 5) (0, 25) = 15, 625m. ❩❡♥♦ s❛✇ t❤❛t ❆❝❤✐❧❧❡s

✇♦✉❧❞ ❛❧✇❛②s ❣❡t ❝❧♦s❡r ❜✉t ✇♦✉❧❞♥✬t ❛❝t✉❛❧❧② ♦✈❡rt❛❦❡ t❤❡ t♦rt♦✐s❡✳

✺✳✶✳✷✳✷ ❙❡q✉❡♥❝❡s✱ ❙❡r✐❡s ❛♥❞ ❋✉♥❝t✐♦♥s

❙♦ ✇❤❛t ❞♦❡s ❩❡♥♦✱ ❆❝❤✐❧❧❡s ❛♥❞ t❤❡ t♦rt♦✐s❡ ❤❛✈❡ t♦ ❞♦ ✇✐t❤ ❝❛❧❝✉❧✉s❄

❲❡❧❧✱ ✐♥ ●r❛❞❡s ✶✵ ❛♥❞ ✶✶ ②♦✉ st✉❞✐❡❞ s❡q✉❡♥❝❡s✳ ❋♦r t❤❡ s❡q✉❡♥❝❡ 0, 1, 2, 3, 4, ... ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❜② t❤❡

2

3

4

5

❡①♣r❡ss✐♦♥ an = 1 − 1 t❤❡ t❡r♠s ❣❡t ❝❧♦s❡r t♦ ✶ ❛s n ❣❡ts ❧❛r❣❡r✳ ❙✐♠✐❧❛r❧②✱ ❢♦r t❤❡ s❡q✉❡♥❝❡ 1, 1, 1, 1, 1, ...

n

2

3

4

5

✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❜② t❤❡ ❡①♣r❡ss✐♦♥ an = 1 t❤❡ t❡r♠s ❣❡t ❝❧♦s❡r t♦ ✵ ❛s n ❣❡ts ❧❛r❣❡r✳ ❲❡ ❤❛✈❡ ❛❧s♦ s❡❡♥ t❤❛t

n

t❤❡ ✐♥✜♥✐t❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ❤❛s ❛ ✜♥✐t❡ t♦t❛❧✳ ❚❤❡ ✐♥✜♥✐t❡ ❣❡♦♠❡tr✐❝ s❡r✐❡s ✐s

∞

S∞ =

a

for

r = −1

✭✺✳✸✮

i=1

1.ri−1 =

a1

1−r

✇❤❡r❡ a1 ✐s t❤❡ ✜rst t❡r♠ ♦❢ t❤❡ s❡r✐❡s ❛♥❞ r ✐s t❤❡ ❝♦♠♠♦♥ r❛t✐♦✳

❲❡ s❡❡ t❤❛t t❤❡r❡ ❛r❡ s♦♠❡ ❢✉♥❝t✐♦♥s ✇❤❡r❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❣❡ts ❝❧♦s❡ t♦ ♦r ❛♣♣r♦❛❝❤❡s ❛

❝❡rt❛✐♥ ✈❛❧✉❡✳

❙✐♠✐❧❛r❧②✱ ❢♦r t❤❡ ❢✉♥❝t✐♦♥✿ y = x2+4x−12 ❚❤❡ ♥✉♠❡r❛t♦r ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❝❛♥ ❜❡ ❢❛❝t♦r✐s❡❞ ❛s✿ y =

x+6

(x+6)(x−2) . ❚❤❡♥ ✇❡ ❝❛♥ ❝❛♥❝❡❧ t❤❡ x − 6 ❢r♦♠ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r ❛♥❞ ✇❡ ❛r❡ ❧❡❢t ✇✐t❤✿ y = x − 2.

x+6

❍♦✇❡✈❡r✱ ✇❡ ❛r❡ ♦♥❧② ❛❜❧❡ t♦ ❝❛♥❝❡❧ t❤❡ x + 6 t❡r♠ ✐❢ x = −6✳ ■❢ x = −6✱ t❤❡♥ t❤❡ ❞❡♥♦♠✐♥❛t♦r ❜❡❝♦♠❡s ✵

❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♥♦t ❞❡✜♥❡❞✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ✐♥❝❧✉❞❡ x = −6✳ ❇✉t

✇❡ ❝❛♥ ❡①❛♠✐♥❡ ✇❤❛t ❤❛♣♣❡♥s t♦ t❤❡ ✈❛❧✉❡s ❢♦r y ❛s x ❣❡ts ❝❧♦s❡ t♦ ✲✻✳ ❚❤❡s❡ ✈❛❧✉❡s ❛r❡ ❧✐st❡❞ ✐♥ ❚❛❜❧❡ ✺✳✷

✇❤✐❝❤ s❤♦✇s t❤❛t ❛s x ❣❡ts ❝❧♦s❡r t♦ ✲✻✱ y ❣❡ts ❝❧♦s❡ t♦ ✽✳

✻✹

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

x

y = (x+6)(x−2)

x+6

✲✾

✲✶✶

✲✽

✲✶✵

✲✼

✲✾

✲✻✳✺

✲✽✳✺

✲✻✳✹

✲✽✳✹

✲✻✳✸

✲✽✳✸

✲✻✳✷

✲✽✳✷

✲✻✳✶

✲✽✳✶

✲✻✳✵✾ ✲✽✳✵✾

✲✻✳✵✽ ✲✽✳✵✽

✲✻✳✵✶ ✲✽✳✵✶

✲✺✳✾

✲✼✳✾

✲✺✳✽

✲✼✳✽

✲✺✳✼

✲✼✳✼

✲✺✳✻

✲✼✳✻

✲✺✳✺

✲✼✳✺

✲✺

✲✼

✲✹

✲✻

✲✸

✲✺

❚❛❜❧❡ ✺✳✷✿ ❱❛❧✉❡s ❢♦r t❤❡ ❢✉♥❝t✐♦♥ y = (x+6)(x−2) ❛s x ❣❡ts ❝❧♦s❡ t♦ ✲✻✳

x+6

❚❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺✳✶✳ ❚❤❡ ❣r❛♣❤ ✐s ❛ str❛✐❣❤t ❧✐♥❡ ✇✐t❤ s❧♦♣❡ ✶ ❛♥❞ ✐♥t❡r❝❡♣t

✲✷✱ ❜✉t ✇✐t❤ ❛ ♠✐ss✐♥❣ s❡❝t✐♦♥ ❛t x = −6✳

❋✐❣✉r❡ ✺✳✶✿ ●r❛♣❤ ♦❢ y = (x+6)(x−2) ✳

x+6

❲❡ s❛② t❤❛t ❛ ❢✉♥❝t✐♦♥ ✐s ❝♦♥t✐♥✉♦✉s ✐❢ t❤❡r❡ ❛r❡ ♥♦ ✈❛❧✉❡s ♦❢ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❢♦r ✇❤✐❝❤ t❤❡

❢✉♥❝t✐♦♥ ✐s ✉♥❞❡✜♥❡❞✳

✺✳✶✳✷✳✸ ▲✐♠✐ts

❲❡ ❝❛♥ ♥♦✇ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ♥♦t❛t✐♦♥✳ ❋♦r t❤❡ ❢✉♥❝t✐♦♥ y = (x+6)(x−2)✱ ✇❡ ❝❛♥ ✇r✐t❡✿ lim

(x+6)(x−2) =

x+6

x→−6

x+6

−8. ❚❤✐s ✐s r❡❛❞✿ t❤❡ ❧✐♠✐t ♦❢ (x+6)(x−2) ❛s x t❡♥❞s t♦ ✲✻ ✐s ✽✳

x+6

✻✺

✺✳✶✳✷✳✸✳✶ ■♥✈❡st✐❣❛t✐♦♥ ✿ ▲✐♠✐ts

■❢ f (x) = x + 1✱ ❞❡t❡r♠✐♥❡✿

❢✭✲✵✳✶✮

❢✭✲✵✳✵✺✮

❢✭✲✵✳✵✹✮

❢✭✲✵✳✵✸✮

❢✭✲✵✳✵✷✮

❢✭✲✵✳✵✶✮

❢✭✵✳✵✵✮

❢✭✵✳✵✶✮

❢✭✵✳✵✷✮

❢✭✵✳✵✸✮

❢✭✵✳✵✹✮

❢✭✵✳✵✺✮

❢✭✵✳✶✮

❚❛❜❧❡ ✺✳✸

❲❤❛t ❞♦ ②♦✉ ♥♦t✐❝❡ ❛❜♦✉t t❤❡ ✈❛❧✉❡ ♦❢ f (x) ❛s x ❣❡ts ❝❧♦s❡ t♦ ✵✳

❊①❡r❝✐s❡ ✺✳✶✿ ▲✐♠✐ts ◆♦t❛t✐♦♥

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✼✳✮

❙✉♠♠❛r✐s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥ ❜② ✉s✐♥❣ ❧✐♠✐t ♥♦t❛t✐♦♥✿ ❆s x ❣❡ts ❝❧♦s❡ t♦ ✶✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡

❢✉♥❝t✐♦♥ y = x + 2 ❣❡ts ❝❧♦s❡ t♦ ✸✳

❲❡ ❝❛♥ ❛❧s♦ ❤❛✈❡ t❤❡ s✐t✉❛t✐♦♥ ✇❤❡r❡ ❛ ❢✉♥❝t✐♦♥ ❤❛s ❛ ❞✐✛❡r❡♥t ✈❛❧✉❡ ❞❡♣❡♥❞✐♥❣ ♦♥ ✇❤❡t❤❡r x ❛♣♣r♦❛❝❤❡s

❢r♦♠ t❤❡ ❧❡❢t ♦r t❤❡ r✐❣❤t✳ ❆♥ ❡①❛♠♣❧❡ ♦❢ t❤✐s ✐s s❤♦✇♥ ✐♥ ❋✐❣✉r❡ ✺✳✷✳

❋✐❣✉r❡ ✺✳✷✿ ●r❛♣❤ ♦❢ y = 1 ✳

x

❆s x → 0 ❢r♦♠ t❤❡ ❧❡❢t✱ y = 1 ❛♣♣r♦❛❝❤❡s −∞✳ ❆s x → 0 ❢r♦♠ t❤❡ r✐❣❤t✱ y = 1 ❛♣♣r♦❛❝❤❡s +∞✳ ❚❤✐s

x

x

✐s ✇r✐tt❡♥ ✐♥ ❧✐♠✐ts ♥♦t❛t✐♦♥ ❛s✿ lim

1

1

x→0−

= −∞ ❢♦r x ❛♣♣r♦❛❝❤✐♥❣ ③❡r♦ ❢r♦♠ t❤❡ ❧❡❢t ❛♥❞ lim

= ∞

x

x→0+ x

❢♦r x ❛♣♣r♦❛❝❤✐♥❣ ③❡r♦ ❢r♦♠ t❤❡ r✐❣❤t✳ ❨♦✉ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ ❧✐♠✐t ♦❢ ♠❛♥② ❞✐✛❡r❡♥t ❢✉♥❝t✐♦♥s ✉s✐♥❣ ❛ s❡t

♠❡t❤♦❞✳

▼❡t❤♦❞✿

▲✐♠✐ts✿ ■❢ ②♦✉ ❛r❡ r❡q✉✐r❡❞ t♦ ❝❛❧❝✉❧❛t❡ ❛ ❧✐♠✐t ❧✐❦❡ limx→a t❤❡♥✿

✶✳ ❙✐♠♣❧✐❢② t❤❡ ❡①♣r❡ss✐♦♥ ❝♦♠♣❧❡t❡❧②✳

✷✳ ■❢ ✐t ✐s ♣♦ss✐❜❧❡✱ ❝❛♥❝❡❧ ❛❧❧ ❝♦♠♠♦♥ t❡r♠s✳

✸✳ ▲❡t x ❛♣♣r♦❛❝❤ a✳

✻✻

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❊①❡r❝✐s❡ ✺✳✷✿ ▲✐♠✐ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✼✳✮

❉❡t❡r♠✐♥❡ limx→110

❊①❡r❝✐s❡ ✺✳✸✿ ▲✐♠✐ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✼✳✮

❉❡t❡r♠✐♥❡ limx→2x

❊①❡r❝✐s❡ ✺✳✹✿ ▲✐♠✐ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✼✳✮

❉❡t❡r♠✐♥❡ lim

x2−100

x→10 x−10

✺✳✶✳✷✳✹ ❆✈❡r❛❣❡ ●r❛❞✐❡♥t ❛♥❞ ●r❛❞✐❡♥t ❛t ❛ P♦✐♥t

■♥ ●r❛❞❡ ✶✵ ②♦✉ ❧❡❛r♥t ❛❜♦✉t ❛✈❡r❛❣❡ ❣r❛❞✐❡♥ts ♦♥ ❛ ❝✉r✈❡✳ ❚❤❡ ❛✈❡r❛❣❡ ❣r❛❞✐❡♥t ❜❡t✇❡❡♥ ❛♥② t✇♦ ♣♦✐♥ts ♦♥

❛ ❝✉r✈❡ ✐s ❣✐✈❡♥ ❜② t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡ t❤❛t ♣❛ss❡s t❤r♦✉❣❤ ❜♦t❤ ♣♦✐♥ts✳ ■♥ ●r❛❞❡ ✶✶ ②♦✉ ✇❡r❡

✐♥tr♦❞✉❝❡❞ t♦ t❤❡ ✐❞❡❛ ♦❢ ❛ ❣r❛❞✐❡♥t ❛t ❛ s✐♥❣❧❡ ♣♦✐♥t ♦♥ ❛ ❝✉r✈❡✳ ❲❡ s❛✇ t❤❛t t❤✐s ✇❛s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡

t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ❛t t❤❡ ❣✐✈❡♥ ♣♦✐♥t✱ ❜✉t ✇❡ ❞✐❞ ♥♦t ❧❡❛r♥ ❤♦✇ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t✳

◆♦✇ ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ ♦❢ tr②✐♥❣ t♦ ✜♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ t❛♥❣❡♥t t t♦ ❛ ❝✉r✈❡ ✇✐t❤ ❡q✉❛t✐♦♥

y = f (x) ❛t ❛ ❣✐✈❡♥ ♣♦✐♥t P ✳

❋✐❣✉r❡ ✺✳✸

❲❡ ❦♥♦✇ ❤♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❛✈❡r❛❣❡ ❣r❛❞✐❡♥t ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ♦♥ ❛ ❝✉r✈❡✱ ❜✉t ✇❡ ♥❡❡❞ t✇♦ ♣♦✐♥ts✳

❚❤❡ ♣r♦❜❧❡♠ ♥♦✇ ✐s t❤❛t ✇❡ ♦♥❧② ❤❛✈❡ ♦♥❡ ♣♦✐♥t✱ ♥❛♠❡❧② P ✳ ❚♦ ❣❡t ❛r♦✉♥❞ t❤❡ ♣r♦❜❧❡♠ ✇❡ ✜rst ❝♦♥s✐❞❡r

❛ s❡❝❛♥t t♦ t❤❡ ❝✉r✈❡ t❤❛t ♣❛ss❡s t❤r♦✉❣❤ ♣♦✐♥t P ❛♥❞ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡ Q✳ ❲❡ ❝❛♥ ♥♦✇ ✜♥❞ t❤❡

❛✈❡r❛❣❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❝✉r✈❡ ❜❡t✇❡❡♥ ♣♦✐♥ts P ❛♥❞ Q✳

❋✐❣✉r❡ ✺✳✹

■❢ t❤❡ x✲❝♦♦r❞✐♥❛t❡ ♦❢ P ✐s a✱ t❤❡♥ t❤❡ y✲❝♦♦r❞✐♥❛t❡ ✐s f (a)✳ ❙✐♠✐❧❛r❧②✱ ✐❢ t❤❡ x✲❝♦♦r❞✐♥❛t❡ ♦❢ Q ✐s a − h✱

t❤❡♥ t❤❡ y✲❝♦♦r❞✐♥❛t❡ ✐s f (a − h)✳ ■❢ ✇❡ ❝❤♦♦s❡ a ❛s x2 ❛♥❞ a − h ❛s x1✱ t❤❡♥✿ y1 = f (a − h)y2 = f (a) . ❲❡

❝❛♥ ♥♦✇ ❝❛❧❝✉❧❛t❡ t❤❡ ❛✈❡r❛❣❡ ❣r❛❞✐❡♥t ❛s✿

y2−y1

=

f (a)−f (a−h)

x2−x1

a−(a−h)

✭✺✳✹✮

=

f (a)−f (a−h)

h

◆♦✇ ✐♠❛❣✐♥❡ t❤❛t Q ♠♦✈❡s ❛❧♦♥❣ t❤❡ ❝✉r✈❡ t♦✇❛r❞ P ✳ ❚❤❡ s❡❝❛♥t ❧✐♥❡ ❛♣♣r♦❛❝❤❡s t❤❡ t❛♥❣❡♥t ❧✐♥❡ ❛s ✐ts

❧✐♠✐t✐♥❣ ♣♦s✐t✐♦♥✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❛✈❡r❛❣❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ s❡❝❛♥t ❛♣♣r♦❛❝❤❡s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t

t♦ t❤❡ ❝✉r✈❡ ❛t P ✳ ■♥ ✇❡ s❡❡ t❤❛t ❛s ♣♦✐♥t Q ❛♣♣r♦❛❝❤❡s ♣♦✐♥t P ✱ h ❣❡ts ❝❧♦s❡r t♦ ✵✳ ❲❤❡♥ h = 0✱ ♣♦✐♥ts P

❛♥❞ Q ❛r❡ ❡q✉❛❧✳ ❲❡ ❝❛♥ ♥♦✇ ✉s❡ ♦✉r ❦♥♦✇❧❡❞❣❡ ♦❢ ❧✐♠✐ts t♦ ✇r✐t❡ t❤✐s ❛s✿

f (a) − f (a − h)

gradient at P = lim

.

✭✺✳✺✮

h→0

h

✻✼

❛♥❞ ✇❡ s❛② t❤❛t t❤❡ ❣r❛❞✐❡♥t ❛t ♣♦✐♥t P ✐s t❤❡ ❧✐♠✐t ♦❢ t❤❡ ❛✈❡r❛❣❡ ❣r❛❞✐❡♥t ❛s Q ❛♣♣r♦❛❝❤❡s P ❛❧♦♥❣ t❤❡

❝✉r✈❡✳

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ❝❛❧❝✉❧✉s ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❆◆②❱♣▼❙✸❍▲✹✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✺✳✺

❊①❡r❝✐s❡ ✺✳✺✿ ▲✐♠✐ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✼✳✮

❋♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = 2x2 − 5x✱ ❞❡t❡r♠✐♥❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ❛t t❤❡ ♣♦✐♥t

x = 2✳

❊①❡r❝✐s❡ ✺✳✻✿ ▲✐♠✐ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✼✳✮

❋♦r t❤❡ ❢✉♥❝t✐♦♥ f (x) = 5x2 − 4x + 1✱ ❞❡t❡r♠✐♥❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t t♦ ❝✉r✈❡ ❛t t❤❡ ♣♦✐♥t

x = a✳

✺✳✶✳✷✳✹✳✶ ▲✐♠✐ts

❉❡t❡r♠✐♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣

✶✳ lim

x2−9

x→3 x+3

✷✳ lim

x+3

x→3 x2+3x

✸✳ lim

3x2−4x

x→2

3−x

✹✳ lim

x2−x−12

x→4

x−4

✺✳ limx→23x + 13x

✺✳✷ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✭❋✐rst Pr✐♥❝✐♣❧❡s✱ ❘✉❧❡s✮ ❛♥❞ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s✷

✺✳✷✳✶ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❢r♦♠ ❋✐rst Pr✐♥❝✐♣❧❡s

❚❤❡ t❛♥❣❡♥t ♣r♦❜❧❡♠ ❤❛s ❣✐✈❡♥ r✐s❡ t♦ t❤❡ ❜r❛♥❝❤ ♦❢ ❝❛❧❝✉❧✉s ❝❛❧❧❡❞ ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❛♥❞ t❤❡ ❡q✉❛t✐♦♥✿

lim

f (x+h)−f (x)

h→0

❞❡✜♥❡s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x)✳ ❯s✐♥❣ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ❝❛❧❧❡❞

h

✜♥❞✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s✳

❉❡✜♥✐t✐♦♥ ✺✳✶✿ ❉❡r✐✈❛t✐✈❡

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f (x) ✐s ✇r✐tt❡♥ ❛s f✬ (x) ❛♥❞ ✐s ❞❡✜♥❡❞ ❜②✿

f (x + h) − f (x)

f ✬ (x) = lim

✭✺✳✻✮

h→0

h

❚❤❡r❡ ❛r❡ ❛ ❢❡✇ ❞✐✛❡r❡♥t ♥♦t❛t✐♦♥s ✉s❡❞ t♦ r❡❢❡r t♦ ❞❡r✐✈❛t✐✈❡s✳ ■❢ ✇❡ ✉s❡ t❤❡ tr❛❞✐t✐♦♥❛❧ ♥♦t❛t✐♦♥ y = f (x)

t♦ ✐♥❞✐❝❛t❡ t❤❛t t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s y ❛♥❞ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ✐s x✱ t❤❡♥ s♦♠❡ ❝♦♠♠♦♥ ❛❧t❡r♥❛t✐✈❡

♥♦t❛t✐♦♥s ❢♦r t❤❡ ❞❡r✐✈❛t✐✈❡ ❛r❡ ❛s ❢♦❧❧♦✇s✿ f✬ (x) = y✬ = dy = df = d f (x) = Df (x) = D

dx

dx

dx

xf (x)

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✶✸✴✶✳✶✴❃✳

✻✽

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❚❤❡ s②♠❜♦❧s D ❛♥❞ d ❛r❡ ❝❛❧❧❡❞ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦rs ❜❡❝❛✉s❡ t❤❡② ✐♥❞✐❝❛t❡ t❤❡ ♦♣❡r❛t✐♦♥ ♦❢ ❞✐✛❡r✲

dx

❡♥t✐❛t✐♦♥✱ ✇❤✐❝❤ ✐s t❤❡ ♣r♦❝❡ss ♦❢ ❝❛❧❝✉❧❛t✐♥❣ ❛ ❞❡r✐✈❛t✐✈❡✳ ■t ✐s ✈❡r② ✐♠♣♦rt❛♥t t❤❛t ②♦✉ ❧❡❛r♥ t♦ ✐❞❡♥t✐❢②

t❤❡s❡ ❞✐✛❡r❡♥t ✇❛②s ♦❢ ❞❡♥♦t✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡✱ ❛♥❞ t❤❛t ②♦✉ ❛r❡ ❝♦♥s✐st❡♥t ✐♥ ②♦✉r ✉s❛❣❡ ♦❢ t❤❡♠ ✇❤❡♥

❛♥s✇❡r✐♥❣ q✉❡st✐♦♥s✳

t✐♣✿ ❚❤♦✉❣❤ ✇❡ ❝❤♦♦s❡ t♦ ✉s❡ ❛ ❢r❛❝t✐♦♥❛❧ ❢♦r♠ ♦❢ r❡♣r❡s❡♥t❛t✐♦♥✱ dy ✐s ❛ ❧✐♠✐t ❛♥❞ ✐s ♥♦t ❛ ❢r❛❝t✐♦♥✱

dx

✐✳❡✳ dy ❞♦❡s ♥♦t ♠❡❛♥ dy ÷ dx✳ dy ♠❡❛♥s y ❞✐✛❡r❡♥t✐❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ x✳ ❚❤✉s✱ dp ♠❡❛♥s p

dx

dx

dx

❞✐✛❡r❡♥t✐❛t❡❞ ✇✐t❤ r❡s♣❡❝t t♦ x✳ ❚❤❡ ❵ d ✬ ✐s t❤❡ ✏♦♣❡r❛t♦r✧✱ ♦♣❡r❛t✐♥❣ ♦♥ s♦♠❡ ❢✉♥❝t✐♦♥ ♦❢ x✳

dx

❱✐❞❡♦ ♦♥ ❝❛❧❝✉❧✉s ✲ ✷

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴●♣▼❣▼❝❆✷❙❇❨❄✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✺✳✻

❊①❡r❝✐s❡ ✺✳✼✿ ❉❡r✐✈❛t✐✈❡s ✲ ❋✐rst Pr✐♥❝✐♣❧❡s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✽✳✮

❈❛❧❝✉❧❛t❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ g (x) = x − 1 ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s✳

✺✳✷✳✶✳✶ ❉❡r✐✈❛t✐✈❡s

✶✳ ●✐✈❡♥ g (x) = −x2

❛✳ ❞❡t❡r♠✐♥❡ g(x+h)−g(x)

h

❜✳ ❤❡♥❝❡✱ ❞❡t❡r♠✐♥❡ lim

g(x+h)−g(x)

h→0

h

❝✳ ❡①♣❧❛✐♥ t❤❡ ♠❡❛♥✐♥❣ ♦❢ ②♦✉r ❛♥s✇❡r ✐♥ ✭❜✮✳

✷✳ ❋✐♥❞ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = −2x2 + 3x ✉s✐♥❣ ✜rst ♣r✐♥❝✐♣❧❡s✳

✸✳ ❉❡t❡r♠✐♥❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ f (x) = 1 ✉s✐♥❣ ✜rst ♣r✐♥❝✐♣❧❡s✳

x−2

✹✳ ❉❡t❡r♠✐♥❡ f✬ (3) ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s ✐❢ f (x) = −5x2✳

✺✳ ■❢ h (x) = 4x2 − 4x✱ ❞❡t❡r♠✐♥❡ h✬ (x) ✉s✐♥❣ ✜rst ♣r✐♥❝✐♣❧❡s✳

✺✳✷✳✷ ❘✉❧❡s ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥

❈❛❧❝✉❧❛t✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s ✐s ✈❡r② ❧♦♥❣✱ ❛♥❞ ✐t ✐s ❡❛s② t♦ ♠❛❦❡ ♠✐st❛❦❡s✳

❋♦rt✉♥❛t❡❧②✱ t❤❡r❡ ❛r❡ r✉❧❡s ✇❤✐❝❤ ♠❛❦❡ ❝❛❧❝✉❧❛t✐♥❣ t❤❡ ❞❡r✐✈❛t✐✈❡ s✐♠♣❧❡✳

✺✳✷✳✷✳✶ ■♥✈❡st✐❣❛t✐♦♥ ✿ ❘✉❧❡s ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥

❋r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s✱ ❞❡t❡r♠✐♥❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✳ f (x) = b

✷✳ f (x) = x

✸✳ f (x) = x2

✹✳ f (x) = x3

✺✳ f (x) = 1/x

❨♦✉ s❤♦✉❧❞ ❤❛✈❡ ❢♦✉♥❞ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✻✾

f (x)

f ✬ (x)

b

✵

x

✶

x2

2x

x3

3x2

1/x = x−1

−x−2

❚❛❜❧❡ ✺✳✹

■❢ ✇❡ ❡①❛♠✐♥❡ t❤❡s❡ r❡s✉❧ts ✇❡ s❡❡ t❤❛t t❤❡r❡ ✐s ❛ ♣❛tt❡r♥✱ ✇❤✐❝❤ ❝❛♥ ❜❡ s✉♠♠❛r✐s❡❞ ❜②✿ d (xn) = nxn−1

dx

❚❤❡r❡ ❛r❡ t✇♦ ♦t❤❡r r✉❧❡s ✇❤✐❝❤ ♠❛❦❡ ❞✐✛❡r❡♥t✐❛t✐♦♥ s✐♠♣❧❡r✳ ❋♦r ❛♥② t✇♦ ❢✉♥❝t✐♦♥s f (x) ❛♥❞ g (x)✿

d [f (x) ± g (x)] = f ✬ (x) ± g✬ (x) ❚❤✐s ♠❡❛♥s t❤❛t ✇❡ ❞✐✛❡r❡♥t✐❛t❡ ❡❛❝❤ t❡r♠ s❡♣❛r❛t❡❧②✳

dx ❚❤❡ ✜♥❛❧ r✉❧❡ ❛♣♣❧✐❡s t♦ ❛ ❢✉♥❝t✐♦♥ f (x) t❤❛t ✐s ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ❝♦♥st❛♥t k✳ d [k.f (x)] = kf✬ (x)

dx

❱✐❞❡♦ ♦♥ ❝❛❧❝✉❧✉s ✲ ✸

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❙❥❡❝❇❣❑✲❙◆❨❄✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✺✳✼

❊①❡r❝✐s❡ ✺✳✽✿ ❘✉❧❡s ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✽✳✮

❉❡t❡r♠✐♥❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ x − 1 ✉s✐♥❣ t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✳

✺✳✷✳✷✳✷ ❙✉♠♠❛r② ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥ ❘✉❧❡s

●✐✈❡♥ t✇♦ ❢✉♥❝t✐♦♥s✱ f (x) ❛♥❞ g (X) ✇❡ ❦♥♦✇ t❤❛t✿

d b = 0

dx

d (xn) = nxn−1

dx

d (kf) = k df

dx

dx

d (f + g) = df + dg

dx

dx

dx

❚❛❜❧❡ ✺✳✺

✼✵

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

✺✳✷✳✷✳✷✳✶ ❘✉❧❡s ♦❢ ❉✐✛❡r❡♥t✐❛t✐♦♥

✶✳ ❋✐♥❞ f✬ (x) ✐❢ f (x) = x2−5x+6✳

x−2

✷✳ ❋✐♥❞

√

f ✬ (y) ✐❢ f (y) =

y✳

✸✳ ❋✐♥❞ f✬ (z) ✐❢ f (z) = (z − 1) (z + 1)✳

√

✹✳ ❉❡t❡r♠✐♥❡ dy ✐❢ y = x3+2 x−3✳

dx

x

√

✺✳ ❉❡t❡r♠✐♥❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ y = x3 + 1 ✳

3x3

✺✳✷✳✸ ❆♣♣❧②✐♥❣ ❉✐✛❡r❡♥t✐❛t✐♦♥ t♦ ❉r❛✇ ●r❛♣❤s

❚❤✉s ❢❛r ✇❡ ❤❛✈❡ ❧❡❛r♥t ❛❜♦✉t ❤♦✇ t♦ ❞✐✛❡r❡♥t✐❛t❡ ✈❛r✐♦✉s ❢✉♥❝t✐♦♥s✱ ❜✉t ■ ❛♠ s✉r❡ t❤❛t ②♦✉ ❛r❡ ❜❡❣✐♥♥✐♥❣ t♦

❛s❦✱ ❲❤❛t ✐s t❤❡ ♣♦✐♥t ♦❢ ❧❡❛r♥✐♥❣ ❛❜♦✉t ❞❡r✐✈❛t✐✈❡s❄ ❲❡❧❧✱ ✇❡ ❦♥♦✇ ♦♥❡ ✐♠♣♦rt❛♥t ❢❛❝t ❛❜♦✉t ❛ ❞❡r✐✈❛t✐✈❡✿ ✐t

✐s ❛ ❣r❛❞✐❡♥t✳ ❙♦✱ ❛♥② ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ♦❢ ❣r❛❞✐❡♥ts ♦r r❛t❡s ♦❢ ❝❤❛♥❣❡ ❝❛♥ ✉s❡ ❞❡r✐✈❛t✐✈❡s✳

❖♥❡ s✐♠♣❧❡ ❛♣♣❧✐❝❛t✐♦♥ ✐s t♦ ❞r❛✇ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s ❜② ✜rst❧② ❞❡t❡r♠✐♥❡ t❤❡ ❣r❛❞✐❡♥ts ♦❢ str❛✐❣❤t ❧✐♥❡s ❛♥❞

s❡❝♦♥❞❧② t♦ ❞❡t❡r♠✐♥❡ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤✳

✺✳✷✳✸✳✶ ❋✐♥❞✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❚❛♥❣❡♥ts t♦ ❈✉r✈❡s

■♥ ✧❆✈❡r❛❣❡ ●r❛❞✐❡♥t ❛♥❞ ●r❛❞✐❡♥t ❛t ❛ P♦✐♥t✧ ✇❡ s❛✇ t❤❛t ✜♥❞✐♥❣ t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ t❛♥❣❡♥t t♦ ❛ ❝✉r✈❡ ✐s

t❤❡ s❛♠❡ ❛s ✜♥❞✐♥❣ t❤❡ ❣r❛❞✐❡♥t ✭♦r s❧♦♣❡✮ ♦❢ t❤❡ s❛♠❡ ❝✉r✈❡ ❛t t❤❡ ♣♦✐♥t ♦❢ t❤❡ t❛♥❣❡♥t✳ ❲❡ ❛❧s♦ s❛✇ t❤❛t

t❤❡ ❣r❛❞✐❡♥t ♦❢ ❛ ❢✉♥❝t✐♦♥ ❛t ❛ ♣♦✐♥t ✐s ❥✉st ✐ts ❞❡r✐✈❛t✐✈❡✳

❙✐♥❝❡ ✇❡ ❤❛✈❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t ❛♥❞ t❤❡ ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡ t❤r♦✉❣❤ ✇❤✐❝❤ t❤❡ t❛♥❣❡♥t ♣❛ss❡s✱

✇❡ ❝❛♥ ✜♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t✳

❊①❡r❝✐s❡ ✺✳✾✿ ❋✐♥❞✐♥❣ t❤❡ ❊q✉❛t✐♦♥ ♦❢ ❛ ❚❛♥❣❡♥t t♦ ❛ ❈✉r✈❡

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✽✳✮

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ y = x2 ❛t t❤❡ ♣♦✐♥t ✭✶✱✶✮ ❛♥❞ ❞r❛✇ ❜♦t❤ ❢✉♥❝t✐♦♥s✳

✺✳✷✳✸✳✷ ❈✉r✈❡ ❙❦❡t❝❤✐♥❣

❉✐✛❡r❡♥t✐❛t✐♦♥ ❝❛♥ ❜❡ ✉s❡❞ t♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✱ ❜② ❤❡❧♣✐♥❣ ❞❡t❡r♠✐♥❡ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts✳ ❲❡

❦♥♦✇ t❤❛t ✐❢ ❛ ❣r❛♣❤ ✐s ✐♥❝r❡❛s✐♥❣ ♦♥ ❛♥ ✐♥t❡r✈❛❧ ❛♥❞ r❡❛❝❤❡s ❛ t✉r♥✐♥❣ ♣♦✐♥t✱ t❤❡♥ t❤❡ ❣r❛♣❤ ✇✐❧❧ st❛rt

❞❡❝r❡❛s✐♥❣ ❛❢t❡r t❤❡ t✉r♥✐♥❣ ♣♦✐♥t✳ ❚❤❡ t✉r♥✐♥❣ ♣♦✐♥t ✐s ❛❧s♦ ❦♥♦✇♥ ❛s ❛ st❛t✐♦♥❛r② ♣♦✐♥t ❜❡❝❛✉s❡ t❤❡

❣r❛❞✐❡♥t ❛t ❛ t✉r♥✐♥❣ ♣♦✐♥t ✐s ✵✳ ❲❡ ❝❛♥ t❤❡♥ ✉s❡ t❤✐s ✐♥❢♦r♠❛t✐♦♥ t♦ ❝❛❧❝✉❧❛t❡ t✉r♥✐♥❣ ♣♦✐♥ts✱ ❜② ❝❛❧❝✉❧❛t✐♥❣

t❤❡ ♣♦✐♥ts ❛t ✇❤✐❝❤ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✐s ✵✳

t✐♣✿ ■❢ x = a ✐s ❛ t✉r♥✐♥❣ ♣♦✐♥t ♦❢ f (x)✱ t❤❡♥✿ f✬ (a) = 0 ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ✵ ❛t ❛

t✉r♥✐♥❣ ♣♦✐♥t✳

❚❛❦❡ t❤❡ ❣r❛♣❤ ♦❢ y = x2 ❛s ❛♥ ❡①❛♠♣❧❡✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ❤❛s ❛ t✉r♥✐♥❣ ♣♦✐♥t ❛t

✭✵✱✵✮✱ ❜✉t ✇❡ ❝❛♥ ✉s❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿ y✬ = 2x ❛♥❞ s❡t ✐t ❡q✉❛❧ t♦ ✵ t♦ ✜♥❞ t❤❡ x✲✈❛❧✉❡ ❢♦r

✇❤✐❝❤ t❤❡ ❣r❛♣❤ ❤❛s ❛ t✉r♥✐♥❣ ♣♦✐♥t✳

2x

=

0

✭✺✳✼✮

x

=

0

❲❡ t❤❡♥ s✉❜st✐t✉t❡ t❤✐s ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❣r❛♣❤ ✭✐✳❡✳ y = x2✮ t♦ ❞❡t❡r♠✐♥❡ t❤❡ y✲❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡

t✉r♥✐♥❣ ♣♦✐♥t✿ f (0) = (0)2 = 0 ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ♣♦✐♥t t❤❛t ✇❡ ❤❛✈❡ ♣r❡✈✐♦✉s❧② ❝❛❧❝✉❧❛t❡❞✳

❊①❡r❝✐s❡ ✺✳✶✵✿ ❈❛❧❝✉❧❛t✐♦♥ ♦❢ ❚✉r♥✐♥❣ P♦✐♥ts

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✾✳✮

❈❛❧❝✉❧❛t❡ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f (x) = 2x3 − 9x2 + 12x − 15✳

❲❡ ❛r❡ ♥♦✇ r❡❛❞② t♦ s❦❡t❝❤ ❣r❛♣❤s ♦❢ ❢✉♥❝t✐♦♥s✳

▼❡t❤♦❞✿

✼✶

❙❦❡t❝❤✐♥❣ ●r❛♣❤s✿ ❙✉♣♣♦s❡ ✇❡ ❛r❡ ❣✐✈❡♥ t❤❛t f (x) = ax3 + bx2 + cx + d✱ t❤❡♥ t❤❡r❡ ❛r❡ ✜✈❡ st❡♣s t♦ ❜❡

❢♦❧❧♦✇❡❞ t♦ s❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✿

✶✳ ■❢ a > 0✱ t❤❡♥ t❤❡ ❣r❛♣❤ ✐s ✐♥❝r❡❛s✐♥❣ ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ❛♥❞ ♠❛② ❤❛✈❡ ❛ ♠❛①✐♠✉♠ ❛♥❞ ❛ ♠✐♥✐♠✉♠✳

❆s x ✐♥❝r❡❛s❡s✱ s♦ ❞♦❡s f (x)✳ ■❢ a < 0✱ t❤❡♥ t❤❡ ❣r❛♣❤ ❞❡❝r❡❛s✐♥❣ ✐s ❢r♦♠ ❧❡❢t t♦ r✐❣❤t✱ ❛♥❞ ❤❛s ✜rst ❛

♠✐♥✐♠✉♠ ❛♥❞ t❤❡♥ ❛ ♠❛①✐♠✉♠✳ f (x) ❞❡❝r❡❛s❡s ❛s x ✐♥❝r❡❛s❡s✳

✷✳ ❉❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ y✲✐♥t❡r❝❡♣t ❜② s✉❜st✐t✉t✐♥❣ x = 0 ✐♥t♦ f (x)

✸✳ ❉❡t❡r♠✐♥❡ t❤❡ x✲✐♥t❡r❝❡♣ts ❜② ❢❛❝t♦r✐s✐♥❣ ax3 + bx2 + cx + d = 0 ❛♥❞ s♦❧✈✐♥❣ ❢♦r x✳ ❋✐rst tr② t♦

❡❧✐♠✐♥❛t❡ ❝♦♥st❛♥t ❝♦♠♠♦♥ ❢❛❝t♦rs✱ ❛♥❞ t♦ ❣r♦✉♣ ❧✐❦❡ t❡r♠s t♦❣❡t❤❡r s♦ t❤❛t t❤❡ ❡①♣r❡ss✐♦♥ ✐s ❡①♣r❡ss❡❞

❛s ❡❝♦♥♦♠✐❝❛❧❧② ❛s ♣♦ss✐❜❧❡✳ ❯s❡ t❤❡ ❢❛❝t♦r t❤❡♦r❡♠ ✐❢ ♥❡❝❡ss❛r②✳

✹✳ ❋✐♥❞ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❜② ✇♦r❦✐♥❣ ♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ df ❛♥❞ s❡tt✐♥❣ ✐t t♦ ③❡r♦✱ ❛♥❞

dx

s♦❧✈✐♥❣ ❢♦r x✳

✺✳ ❉❡t❡r♠✐♥❡ t❤❡ y✲❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❜② s✉❜st✐t✉t✐♥❣ t❤❡ x ✈❛❧✉❡s ♦❜t❛✐♥❡❞ ✐♥ t❤❡ ♣r❡✈✐♦✉s

st❡♣✱ ✐♥t♦ t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r f (x)✳

✻✳ ❯s❡ t❤❡ ✐♥❢♦r♠❛t✐♦♥ ②♦✉✬r❡ ❣✐✈❡♥ t♦ ♣❧♦t t❤❡ ♣♦✐♥ts ❛♥❞ ❣❡t ❛ r♦✉❣❤ ✐❞❡❛ ♦❢ t❤❡ ❣r❛❞✐❡♥ts ❜❡t✇❡❡♥

♣♦✐♥ts✳ ❚❤❡♥ ✜❧❧ ✐♥ t❤❡ ♠✐ss✐♥❣ ♣❛rts ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ ❛ s♠♦♦t❤✱ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡✳

❊①❡r❝✐s❡ ✺✳✶✶✿ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✼✾✳✮

❉r❛✇ t❤❡ ❣r❛♣❤ ♦❢ g (x) = x2 − x + 2

❊①❡r❝✐s❡ ✺✳✶✷✿ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✽✵✳✮

❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ g (x) = −x3 + 6x2 − 9x + 4✳

✺✳✷✳✸✳✷✳✶ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s

✶✳ ●✐✈❡♥ f (x) = x3 + x2 − 5x + 3✿

❛✳ ❙❤♦✇ t❤❛t (x − 1) ✐s ❛ ❢❛❝t♦r ♦❢ f (x) ❛♥❞ ❤❡♥❝❡ ❢❛t♦r✐s❡ f (x) ❢✉❧❧②✳

❜✳ ❋✐♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡ ❛①❡s ❛♥❞ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛♥❞ s❦❡t❝❤ t❤❡ ❣r❛♣❤

✷✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f (x) = x3 − 4x2 − 11x + 30 s❤♦✇✐♥❣ ❛❧❧ t❤❡ r❡❧❛t✐✈❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛♥❞ ✐♥t❡r❝❡♣ts

✇✐t❤ t❤❡ ❛①❡s✳

✸✳

❛✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f (x) = x3 − 9x2 + 24x − 20✱ s❤♦✇✐♥❣ ❛❧❧ ✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡ ❛①❡s ❛♥❞ t✉r♥✐♥❣

♣♦✐♥ts✳

❜✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ f (x) ❛t x = 4✳

✺✳✷✳✸✳✸ ▲♦❝❛❧ ♠✐♥✐♠✉♠✱ ▲♦❝❛❧ ♠❛①✐♠✉♠ ❛♥❞ P♦✐♥t ♦❢ ■♥✢❡①t✐♦♥

■❢ t❤❡ ❞❡r✐✈❛t✐✈❡ ✭ dy ✮ ✐s ③❡r♦ ❛t ❛ ♣♦✐♥t✱ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t ❛t t❤❛t ♣♦✐♥t ✐s ③❡r♦✳ ■t ♠❡❛♥s t❤❛t ❛

dx

t✉r♥✐♥❣ ♣♦✐♥t ♦❝❝✉rs ❛s s❡❡♥ ✐♥ t❤❡ ♣r❡✈✐♦✉s ❡①❛♠♣❧❡✳

❋✐❣✉r❡ ✺✳✽

❋r♦♠ t❤❡ ❞r❛✇✐♥❣ t❤❡ ♣♦✐♥t ✭✶❀✵✮ r❡♣r❡s❡♥ts ❛ ❧♦❝❛❧ ♠✐♥✐♠✉♠ ❛♥❞ t❤❡ ♣♦✐♥t ✭✸❀✹✮ t❤❡ ❧♦❝❛❧ ♠❛①✐♠✉♠✳

❆ ❣r❛♣❤ ❤❛s ❛ ❤♦r✐③♦♥t❛❧ ♣♦✐♥t ♦❢ ✐♥✢❡①✐♦♥ ✇❤❡r❡ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ③❡r♦ ❜✉t t❤❡ s✐❣♥ ♦❢ t❤❡ s✐❣♥ ♦❢ t❤❡

❣r❛❞✐❡♥t ❞♦❡s ♥♦t ❝❤❛♥❣❡✳ ❚❤❛t ♠❡❛♥s t❤❡ ❣r❛♣❤ ❛❧✇❛②s ✐♥❝r❡❛s❡s ♦r ❛❧✇❛②s ❞❡❝r❡❛s❡s✳

✼✷

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❋✐❣✉r❡ ✺✳✾

❋r♦♠ t❤✐s ❞r❛✇✐♥❣✱ t❤❡ ♣♦✐♥t ✭✸❀✶✮ ✐s ❛ ❤♦r✐③♦♥t❛❧ ♣♦✐♥t ♦❢ ✐♥✢❡①✐♦♥✱ ❜❡❝❛✉s❡ t❤❡ s✐❣♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡

❞♦❡s ♥♦t ❝❤❛♥❣❡ ❢r♦♠ ♣♦s✐t✐✈❡ t♦ ♥❡❣❛t✐✈❡✳

✺✳✸ ❙♦❧✈✐♥❣ Pr♦❜❧❡♠s✸

✺✳✸✳✶ ❯s✐♥❣ ❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s t♦ ❙♦❧✈❡ Pr♦❜❧❡♠s

❲❡ ❤❛✈❡ s❡❡♥ t❤❛t ❞✐✛❡r❡♥t✐❛❧ ❝❛❧❝✉❧✉s ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ st❛t✐♦♥❛r② ♣♦✐♥ts ♦❢ ❢✉♥❝t✐♦♥s✱ ✐♥ ♦r❞❡r

t♦ s❦❡t❝❤ t❤❡✐r ❣r❛♣❤s✳ ❍♦✇❡✈❡r✱ ❞❡t❡r♠✐♥✐♥❣ st❛t✐♦♥❛r② ♣♦✐♥ts ❛❧s♦ ❧❡♥❞s ✐ts❡❧❢ t♦ t❤❡ s♦❧✉t✐♦♥ ♦❢ ♣r♦❜❧❡♠s

t❤❛t r❡q✉✐r❡ s♦♠❡ ✈❛r✐❛❜❧❡ t♦ ❜❡ ♦♣t✐♠✐s❡❞✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ❢✉❡❧ ✉s❡❞ ❜② ❛ ❝❛r ✐s ❞❡✜♥❡❞ ❜②✿

3

f (v) =

v2 − 6v + 245

✭✺✳✽✮

80

✇❤❡r❡ v ✐s t❤❡ tr❛✈❡❧❧✐♥❣ s♣❡❡❞✱ ✇❤❛t ✐s t❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ s♣❡❡❞ ✭t❤❛t ♠❡❛♥s t❤❡ s♣❡❡❞ t❤❛t ✉s❡s t❤❡

❧❡❛st ❢✉❡❧✮❄

■❢ ✇❡ ❞r❛✇ t❤❡ ❣r❛♣❤ ♦❢ t❤✐s ❢✉♥❝t✐♦♥ ✇❡ ✜♥❞ t❤❛t t❤❡ ❣r❛♣❤ ❤❛s ❛ ♠✐♥✐♠✉♠✳ ❚❤❡ s♣❡❡❞ ❛t t❤❡ ♠✐♥✐♠✉♠

✇♦✉❧❞ t❤❡♥ ❣✐✈❡ t❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ s♣❡❡❞✳

❋✐❣✉r❡ ✺✳✶✵

❲❡ ❤❛✈❡ s❡❡♥ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥t ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❜② ❞✐✛❡r❡♥t✐❛t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥

❛♥❞ ✜♥❞✐♥❣ t❤❡ x✲❝♦♦r❞✐♥❛t❡ ✭s♣❡❡❞ ✐♥ t❤❡ ❝❛s❡ ♦❢ t❤❡ ❡①❛♠♣❧❡✮ ❢♦r ✇❤✐❝❤ t❤❡ ❞❡r✐✈❛t✐✈❡ ✐s ✵✳

❉✐✛❡r❡♥t✐❛t✐♥❣ ✭✺✳✽✮✱ ✇❡ ❣❡t✿ f✬ (v) = 3 v − 6 ■❢ ✇❡ s❡t f✬ (v) = 0 ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ s♣❡❡❞ t❤❛t

40

❝♦rr❡s♣♦♥❞s t♦ t❤❡ t✉r♥✐♥❣ ♣♦✐♥t✳

f ✬ (v)

=

3 v − 6

40

0

=

3 v − 6

40

✭✺✳✾✮

v

=

6×40

3

=

80

❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ♠♦st ❡❝♦♥♦♠✐❝❛❧ s♣❡❡❞ ✐s ✽✵ ❦♠·❤r−1✳

✸❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✸✴✶✳✶✴❃✳

✼✸

❱✐❞❡♦ ♦♥ ❝❛❧❝✉❧✉s ✲ ✹

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴✶❚❑✻✾❙❱◗◆P❦❄✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✺✳✶✶

❊①❡r❝✐s❡ ✺✳✶✸✿ ❖♣t✐♠✐s❛t✐♦♥ Pr♦❜❧❡♠s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✽✶✳✮

❚❤❡ s✉♠ ♦❢ t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ✐s ✶✵✳ ❖♥❡ ♦❢ t❤❡ ♥✉♠❜❡rs ✐s ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ sq✉❛r❡ ♦❢ t❤❡

♦t❤❡r✳ ■❢ ❡❛❝❤ ♥✉♠❜❡r ✐s ❣r❡❛t❡r t❤❛♥ ✵✱ ✜♥❞ t❤❡ ♥✉♠❜❡rs t❤❛t ♠❛❦❡ t❤✐s ♣r♦❞✉❝t ❛ ♠❛①✐♠✉♠✳

❊①❡r❝✐s❡ ✺✳✶✹✿ ❖♣t✐♠✐s❛t✐♦♥ Pr♦❜❧❡♠s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✽✷✳✮

▼✐❝❤❛❡❧ ✇❛♥ts t♦ st❛rt ❛ ✈❡❣❡t❛❜❧❡ ❣❛r❞❡♥✱ ✇❤✐❝❤ ❤❡ ❞❡❝✐❞❡s t♦ ❢❡♥❝❡ ♦✛ ✐♥ t❤❡ s❤❛♣❡ ♦❢ ❛ r❡❝t❛♥❣❧❡

❢r♦♠ t❤❡ r❡st ♦❢ t❤❡ ❣❛r❞❡♥✳ ▼✐❝❤❛❡❧ ♦♥❧② ❤❛s ✶✻✵ ♠ ♦❢ ❢❡♥❝✐♥❣✱ s♦ ❤❡ ❞❡❝✐❞❡s t♦ ✉s❡ ❛ ✇❛❧❧ ❛s ♦♥❡

❜♦r❞❡r ♦❢ t❤❡ ✈❡❣❡t❛❜❧❡ ❣❛r❞❡♥✳ ❈❛❧❝✉❧❛t❡ t❤❡ ✇✐❞t❤ ❛♥❞ ❧❡♥❣t❤ ♦❢ t❤❡ ❣❛r❞❡♥ t❤❛t ❝♦rr❡s♣♦♥❞s t♦

❧❛r❣❡st ♣♦ss✐❜❧❡ ❛r❡❛ t❤❛t ▼✐❝❤❛❡❧ ❝❛♥ ❢❡♥❝❡ ♦✛✳

❋✐❣✉r❡ ✺✳✶✷

✺✳✸✳✶✳✶ ❙♦❧✈✐♥❣ ❖♣t✐♠✐s❛t✐♦♥ Pr♦❜❧❡♠s ✉s✐♥❣ ❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s

✶✳ ❚❤❡ s✉♠ ♦❢ t✇♦ ♣♦s✐t✐✈❡ ♥✉♠❜❡rs ✐s ✷✵✳ ❖♥❡ ♦❢ t❤❡ ♥✉♠❜❡rs ✐s ♠✉❧t✐♣❧✐❡❞ ❜② t❤❡ sq✉❛r❡ ♦❢ t❤❡ ♦t❤❡r✳

❋✐♥❞ t❤❡ ♥✉♠❜❡rs t❤❛t ♠❛❦❡ t❤✐s ♣r♦❞✉❝t ❛ ♠❛①✐♠✉♠✳

✷✳ ❆ ✇♦♦❞❡♥ ❜❧♦❝❦ ✐s ♠❛❞❡ ❛s s❤♦✇♥ ✐♥ t❤❡ ❞✐❛❣r❛♠✳ ❚❤❡ ❡♥❞s ❛r❡ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡s ❤❛✈✐♥❣ s✐❞❡s 3x✱

4x ❛♥❞ 5x✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❜❧♦❝❦ ✐s y✳ ❚❤❡ t♦t❛❧ s✉r❢❛❝❡ ❛r❡❛ ♦❢ t❤❡ ❜❧♦❝❦ ✐s 3600cm2✳

❋✐❣✉r❡ ✺✳✶✸

❛✳ ❙❤♦✇ t❤❛t y = 300−x2 ✳

x

❜✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ x ❢♦r ✇❤✐❝❤ t❤❡ ❜❧♦❝❦ ✇✐❧❧ ❤❛✈❡ ❛ ♠❛①✐♠✉♠ ✈♦❧✉♠❡✳ ✭❱♦❧✉♠❡ ❂ ❛r❡❛ ♦❢ ❜❛s❡ ×

❤❡✐❣❤t✳✮

✸✳ ❚❤❡ ❞✐❛❣r❛♠ s❤♦✇s t❤❡ ♣❧❛♥ ❢♦r ❛ ✈❡r❛♥❞❛❤ ✇❤✐❝❤ ✐s t♦ ❜❡ ❜✉✐❧t ♦♥ t❤❡ ❝♦r♥❡r ♦❢ ❛ ❝♦tt❛❣❡✳ ❆ r❛✐❧✐♥❣

ABCDE ✐s t♦ ❜❡ ❝♦♥str✉❝t❡❞ ❛r♦✉♥❞ t❤❡ ❢♦✉r ❡❞❣❡s ♦❢ t❤❡ ✈❡r❛♥❞❛❤✳

✼✹

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❋✐❣✉r❡ ✺✳✶✹

■❢ AB = DE = x ❛♥❞ BC = CD = y✱ ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r❛✐❧✐♥❣ ♠✉st ❜❡ ✸✵ ♠❡tr❡s✱ ✜♥❞ t❤❡ ✈❛❧✉❡s

♦❢ x ❛♥❞ y ❢♦r ✇❤✐❝❤ t❤❡ ✈❡r❛♥❞❛❤ ✇✐❧❧ ❤❛✈❡ ❛ ♠❛①✐♠✉♠ ❛r❡❛✳

✺✳✸✳✶✳✷ ❘❛t❡ ♦❢ ❈❤❛♥❣❡ ♣r♦❜❧❡♠s

❚✇♦ ❝♦♥❝❡♣ts ✇❡r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✿ ❆✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ❂ f(b)−f(a) ❛♥❞ ■♥st❛♥t❛♥❡♦✉s r❛t❡

b−a

♦❢ ❝❤❛♥❣❡ ❂ lim

f (x+h)−f (x)

h→0

✳ ❲❤❡♥ ✇❡ ♠❡♥t✐♦♥ r❛t❡ ♦❢ ❝❤❛♥❣❡✱ t❤❡ ❧❛tt❡r ✐s ✐♠♣❧✐❡❞✳ ■♥st❛♥t❛♥❡♦✉s r❛t❡

h

♦❢ ❝❤❛♥❣❡ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❲❤❡♥ ❆✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ✐s r❡q✉✐r❡❞✱ ✐t ✇✐❧❧ ❜❡ s♣❡❝✐✜❝❛❧❧② r❡❢❡r t♦ ❛s

❛✈❡r❛❣❡ r❛t❡ ♦❢ ❝❤❛♥❣❡✳

❱❡❧♦❝✐t② ✐s ♦♥❡ ♦❢ t❤❡ ♠♦st ❝♦♠♠♦♥ ❢♦r♠s ♦❢ r❛t❡ ♦❢ ❝❤❛♥❣❡✳ ❆❣❛✐♥✱ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ❂ ❛✈❡r❛❣❡ r❛t❡

♦❢ ❝❤❛♥❣❡ ❛♥❞ ✐♥st❛♥t❛♥❡♦✉s ✈❡❧♦❝✐t② ❂ ✐♥st❛♥t❛♥❡♦✉s r❛t❡ ♦❢ ❝❤❛♥❣❡ ❂ ❞❡r✐✈❛t✐✈❡✳ ❱❡❧♦❝✐t② r❡❢❡rs t♦

t❤❡ ✐♥❝r❡❛s❡ ♦❢ ❞✐st❛♥❝❡✭s✮ ❢♦r ❛ ❝♦rr❡s♣♦♥❞✐♥❣ ✐♥❝r❡❛❞❡ ✐♥ t✐♠❡ ✭t✮✳ ❚❤❡ ♥♦t❛t✐♦♥ ❝♦♠♠♦♥❧② ✉s❡❞ ❢♦r t❤✐s ✐s✿

v (t) = ds = s✬ (t)

dt

✇❤❡r❡ s✬ (t) ✐s t❤❡ ♣♦s✐t✐♦♥ ❢✉♥❝t✐♦♥✳ ❆❝❝❡❧❡r❛t✐♦♥ ✐s t❤❡ ❝❤❛♥❣❡ ✐♥ ✈❡❧♦❝✐t② ❢♦r ❛ ❝♦r❡rs♣♦♥❞✐♥❣ ✐♥❝r❡❛s❡

✐♥ t✐♠❡✳ ❚❤❡r❡❢♦r❡✱ ❛❝❝❡❧❡r❛t✐♦♥ ✐s t❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✈❡❧♦❝✐t② a (t) = v✬ (t) ❚❤✐s ✐♠♣❧✐❡s t❤❛t ❛❝❝❡❧❡r❛t✐♦♥ ✐s t❤❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡ ♦❢ t❤❡ ❞✐st❛♥❝❡✭s✮✳

❊①❡r❝✐s❡ ✺✳✶✺✿ ❘❛t❡ ♦❢ ❈❤❛♥❣❡

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✽✷✳✮

❚❤❡ ❤❡✐❣❤t ✭✐♥ ♠❡tr❡s✮ ♦❢ ❛ ❣♦❧❢ ❜❛❧❧ t❤❛t ✐s ❤✐t ✐♥t♦ t❤❡ ❛✐r ❛❢t❡r t s❡❝♦♥❞s✱ ✐s ❣✐✈❡♥ ❜② h (t) =

20t − 5t2✳ ❉❡t❡r♠✐♥❡

✶✳ t❤❡ ❛✈❡r❛❣❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❜❛❧❧ ❞✉r✐♥❣ t❤❡ ✜rst t✇♦ s❡❝♦♥❞s

✷✳ t❤❡ ✈❡❧♦❝✐t② ♦❢ t❤❡ ❜❛❧❧ ❛❢t❡r ✶✱✺ s❡❝♦♥❞s

✸✳ ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ③❡r♦

✹✳ t❤❡ ✈❡❧♦❝✐t② ❛t ✇❤✐❝❤ t❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞

✺✳ t❤❡ ❛❝❝❡❧❡r❛t✐♦♥ ♦❢ t❤❡ ❜❛❧❧

✺✳✸✳✷ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ ❉❡t❡r♠✐♥❡ f✬ (x) ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s ✐❢✿

❛✳ f (x) = x2 − 6x

❜✳ f (x) = 2x − x2

✷✳ ●✐✈❡♥✿ f (x) = −x2 + 3x✱ ✜♥❞ f✬ (x) ✉s✐♥❣ ✜rst ♣r✐♥❝✐♣❧❡s✳

✸✳ ❉❡t❡r♠✐♥❡ dx ✐❢✿

dy

❛✳ y = (2x)2 − 13x

√

❜✳ y = 2 x−5

√x

✹✳ ●✐✈❡♥✿ f (x) = x3 − 3x2 + 4

❛✳ ❈❛❧❝✉❧❛t❡ f (−1)✱ ❛♥❞ ❤❡♥❝❡ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥f (x) = 0

❜✳ ❉❡t❡r♠✐♥❡ f✬ (x)

✼✺

❝✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f ♥❡❛t❧② ❛♥❞ ❝❧❡❛r❧②✱ s❤♦✇✐♥❣ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛s ✇❡❧❧

❛s t❤❡ ✐♥t❡r❝❡♣ts ♦♥ ❜♦t❤ ❛①❡s✳

❞✳ ❉❡t❡r♠✐♥❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❣r❛♣❤ ♦❢ f ✇❤❡r❡ t❤❡ ❣r❛❞✐❡♥t ✐s ✾✳

✺✳ ●✐✈❡♥✿ f (x) = 2x3 − 5x2 − 4x + 3✳ ❚❤❡ x✲✐♥t❡r❝❡♣ts ♦❢ f ❛r❡✿ ✭✲✶❀✵✮ ✭1❀✵✮ ❛♥❞ ✭✸❀✵✮✳

2

❛✳ ❉❡t❡r♠✐♥❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ♦❢ f✳

❜✳ ❉r❛✇ ❛ ♥❡❛t s❦❡t❝❤ ❣r❛♣❤ ♦❢ f✳ ❈❧❡❛r❧② ✐♥❞✐❝❛t❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ ✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡ ❛①❡s✱

❛s ✇❡❧❧ ❛s t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts✳

❝✳ ❋♦r ✇❤✐❝❤ ✈❛❧✉❡s ♦❢ k ✇✐❧❧ t❤❡ ❡q✉❛t✐♦♥ f (x) = k ✱ ❤❛✈❡ ❡①❛❝t❧② t✇♦ r❡❛❧ r♦♦ts❄

❞✳ ❉❡t❡r♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❣r❛♣❤ ♦❢ f (x) = 2x3 − 5x2 − 4x + 3 ❛t t❤❡ ♣♦✐♥t

✇❤❡r❡ x = 1✳

✻✳ ❆♥s✇❡r t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❡st✐♦♥s✿

❛✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ f (x) = x3 − 9x2 + 24x − 20✱ s❤♦✇✐♥❣ ❛❧❧ ✐♥t❡r❝❡♣ts ✇✐t❤ t❤❡ ❛①❡s ❛♥❞ t✉r♥✐♥❣

♣♦✐♥ts✳

❜✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ f (x) ❛t x = 4✳

✼✳ ❈❛❧❝✉❧❛t❡✿ lim

1−x3

x→1 1−x

✽✳ ●✐✈❡♥✿ f (x) = 2x2 − x

❛✳ ❯s❡ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ t❤❡ ❞❡r✐✈❛t✐✈❡ t♦ ❝❛❧❝✉❧❛t❡ f✬ (x)✳

❜✳ ❍❡♥❝❡✱ ❝❛❧❝✉❧❛t❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥t ❛t ✇❤✐❝❤ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❣r❛♣❤

♦❢ f ✐s ✼✳

√

✾✳ ■❢ xy − 5 = x3✱ ❞❡t❡r♠✐♥❡ dx

dy

✶✵✳ ●✐✈❡♥✿ g (x) = x−2 + x2 2✳ ❈❛❧❝✉❧❛t❡ g✬ (2)✳

✶✶✳ ●✐✈❡♥✿

f (x) = 2x − 3

❛✳ ❋✐♥❞✿

f −1 (x)

❜✳ ❙♦❧✈❡✿

f −1 (x) = 3f ✬ (x)

✶✷✳ ❋✐♥❞ f✬ (x) ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣✿

√

❛✳

5

f (x) =

x3 + 10

3

❜✳

(2x2−5)(3x+2)

f (x) =

x2

✶✸✳ ❉❡t❡r♠✐♥❡ t❤❡ ♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦❢ t❤❡ s✉♠ ♦❢ ❛ ♣♦s✐t✐✈❡ ♥✉♠❜❡r ❛♥❞ ✐ts r❡❝✐♣r♦❝❛❧✳

✶✹✳ ■❢ t❤❡ ❞✐s♣❧❛❝❡♠❡♥t s ✭✐♥ ♠❡tr❡s✮ ♦❢ ❛ ♣❛rt✐❝❧❡ ❛t t✐♠❡ t ✭✐♥ s❡❝♦♥❞s✮ ✐s ❣♦✈❡r♥❡❞ ❜② t❤❡ ❡q✉❛t✐♦♥

s = 1 t3 − 2t✱ ✜♥❞ ✐ts ❛❝❝❡❧❡r❛t✐♦♥ ❛❢t❡r ✷ s❡❝♦♥❞s✳ ✭❆❝❝❡❧❡r❛t✐♦♥ ✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ✈❡❧♦❝✐t②✱ ❛♥❞

2

✈❡❧♦❝✐t② ✐s t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ ❞✐s♣❧❛❝❡♠❡♥t✳✮

✶✺✳ ❆❢t❡r ❞♦✐♥❣ s♦♠❡ r❡s❡❛r❝❤✱ ❛ tr❛♥s♣♦rt ❝♦♠♣❛♥② ❤❛s ❞❡t❡r♠✐♥❡❞ t❤❛t t❤❡ r❛t❡ ❛t ✇❤✐❝❤ ♣❡tr♦❧ ✐s

❝♦♥s✉♠❡❞ ❜② ♦♥❡ ♦❢ ✐ts ❧❛r❣❡ ❝❛rr✐❡rs✱ tr❛✈❡❧❧✐♥❣ ❛t ❛♥ ❛✈❡r❛❣❡ s♣❡❡❞ ♦❢ x ❦♠ ♣❡r ❤♦✉r✱ ✐s ❣✐✈❡♥ ❜②✿

P (x) = 55 + x

litresperkilometre

2x

200

❛✳ ❆ss✉♠❡ t❤❛t t❤❡ ♣❡tr♦❧ ❝♦sts ❘✹✱✵✵ ♣❡r ❧✐tr❡ ❛♥❞ t❤❡ ❞r✐✈❡r ❡❛r♥s ❘✶✽✱✵✵ ♣❡r ❤♦✉r ✭tr❛✈❡❧❧✐♥❣ t✐♠❡✮✳

◆♦✇ ❞❡❞✉❝❡ t❤❛t t❤❡ t♦t❛❧ ❝♦st✱ C✱ ✐♥ ❘❛♥❞s✱ ❢♦r ❛ ✷ ✵✵✵ ❦♠ tr✐♣ ✐s ❣✐✈❡♥ ❜②✿ C (x) = 256000 +40x

x

❜✳ ❍❡♥❝❡ ❞❡t❡r♠✐♥❡ t❤❡ ❛✈❡r❛❣❡ s♣❡❡❞ t♦ ❜❡ ♠❛✐♥t❛✐♥❡❞ t♦ ❡✛❡❝t ❛ ♠✐♥✐♠✉♠ ❝♦st ❢♦r ❛ ✷ ✵✵✵ ❦♠

tr✐♣✳

✶✻✳ ❉✉r✐♥❣ ❛♥ ❡①♣❡r✐♠❡♥t t❤❡ t❡♠♣❡r❛t✉r❡ T ✭✐♥ ❞❡❣r❡❡s ❈❡❧s✐✉s✮✱ ✈❛r✐❡s ✇✐t❤ t✐♠❡ t ✭✐♥ ❤♦✉rs✮✱ ❛❝❝♦r❞✐♥❣

t♦ t❤❡ ❢♦r♠✉❧❛✿ T (t) = 30 + 4t − 1t2 t ∈ [1; 10]

2

❛✳ ❉❡t❡r♠✐♥❡ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ r❛t❡ ♦❢ ❝❤❛♥❣❡ ♦❢ t❡♠♣❡r❛t✉r❡ ✇✐t❤ t✐♠❡✳

❜✳ ❉✉r✐♥❣ ✇❤✐❝❤ t✐♠❡ ✐♥t❡r✈❛❧ ✇❛s t❤❡ t❡♠♣❡r❛t✉r❡ ❞r♦♣♣✐♥❣❄

✶✼✳ ❚❤❡ ❞❡♣t❤✱ d✱ ♦❢ ✇❛t❡r ✐♥ ❛ ❦❡tt❧❡ t ♠✐♥✉t❡s ❛❢t❡r ✐t st❛rts t♦ ❜♦✐❧✱ ✐s ❣✐✈❡♥ ❜② d = 86 − 1t − 1t3✱ ✇❤❡r❡

8

4

d ✐s ♠❡❛s✉r❡❞ ✐♥ ♠✐❧❧✐♠❡tr❡s✳

❛✳ ❍♦✇ ♠❛♥② ♠✐❧❧✐♠❡tr❡s ♦❢ ✇❛t❡r ❛r❡ t❤❡r❡ ✐♥ t❤❡ ❦❡tt❧❡ ❥✉st ❜❡❢♦r❡ ✐t st❛rts t♦ ❜♦✐❧❄

✼✻

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❜✳ ❆s t❤❡ ✇❛t❡r ❜♦✐❧s✱ t❤❡ ❧❡✈❡❧ ✐♥ t❤❡ ❦❡tt❧❡ ❞r♦♣s✳ ❋✐♥❞ t❤❡ r❛t❡ ❛t ✇❤✐❝❤ t❤❡ ✇❛t❡r ❧❡✈❡❧ ✐s ❞❡❝r❡❛s✐♥❣

✇❤❡♥ t ❂ ✷ ♠✐♥✉t❡s✳

❝✳ ❍♦✇ ♠❛♥② ♠✐♥✉t❡s ❛❢t❡r t❤❡ ❦❡tt❧❡ st❛rts ❜♦✐❧✐♥❣ ✇✐❧❧ t❤❡ ✇❛t❡r ❧❡✈❡❧ ❜❡ ❞r♦♣♣✐♥❣ ❛t ❛ r❛t❡ ♦❢ 1218

♠♠✴♠✐♥✉t❡❄

✼✼

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✺

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶ ✭♣✳ ✻✺✮

❙t❡♣ ✶✳ ❚❤✐s ✐s ✇r✐tt❡♥ ❛s✿ limx→1x + 2 = 3 ✐♥ ❧✐♠✐t ♥♦t❛t✐♦♥✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✷ ✭♣✳ ✻✻✮

❙t❡♣ ✶✳ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ s✐♠♣❧✐❢②✳

❙t❡♣ ✷✳ ❚❤❡r❡ ❛r❡ ♥♦ t❡r♠s t♦ ❝❛♥❝❡❧✳

❙t❡♣ ✸✳ limx→110 = 10

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✸ ✭♣✳ ✻✻✮

❙t❡♣ ✶✳ ❚❤❡r❡ ✐s ♥♦t❤✐♥❣ t♦ s✐♠♣❧✐❢②✳

❙t❡♣ ✷✳ ❚❤❡r❡ ❛r❡ ♥♦ t❡r♠s t♦ ❝❛♥❝❡❧✳

❙t❡♣ ✸✳ limx→2x = 2

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✹ ✭♣✳ ✻✻✮

❙t❡♣ ✶✳ ❚❤❡ ♥✉♠❡r❛t♦r ❝❛♥ ❜❡ ❢❛❝t♦r✐s❡❞✳ x2−100 = (x+10)(x−10)

x−10

x−10

❙t❡♣ ✷✳ x − 10 ❝❛♥ ❜❡ ❝❛♥❝❡❧❧❡❞ ❢r♦♠ t❤❡ ♥✉♠❡r❛t♦r ❛♥❞ ❞❡♥♦♠✐♥❛t♦r✳

(x+10)(x−10) = x + 10

x−10

❙t❡♣ ✸✳ lim

x2−100

x→10

= 20

x−10

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✺ ✭♣✳ ✻✼✮

❙t❡♣ ✶✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❣r❛❞✐❡♥t ❛t ❛ ♣♦✐♥t x ✐s ❣✐✈❡♥ ❜②✿ lim

f (x+h)−f (x)

h→0

■♥ ♦✉r ❝❛s❡ x = 2✳ ■t ✐s s✐♠♣❧❡r

h

t♦ s✉❜st✐t✉t❡ x = 2 ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝❛❧❝✉❧❛t✐♦♥✳

❙t❡♣ ✷✳

f (x + h)

=

2(x + h)2 − 5 (x + h)

=

2 x2 + 2xh + h2 − 5x − 5h

✭✺✳✶✵✮

=

2x2 + 4xh + 2h2 − 5x − 5h

❙t❡♣ ✸✳

2x2+4xh+2h2−5x−5h−(2x2−5x)

lim f(x+h)−f(x)

=

;

h = 0

h→0

h

h

=

lim 2x2+4xh+2h2−5x−5h−2x2+5x

h→0

h

=

lim 4xh+2h2−5h

h→0

h

✭✺✳✶✶✮

=

lim h(4x+2h−5)

h→0

h

=

lim4x + 2h − 5

h→0

=

4x − 5

❙t❡♣ ✹✳ 4x − 5 = 4 (2) − 5 = 3

❙t❡♣ ✺✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ f (x) = 2x2 − 5x ❛t x = 2 ✐s ✸✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✻ ✭♣✳ ✻✼✮

❙t❡♣ ✶✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❣r❛❞✐❡♥t ❛t ❛ ♣♦✐♥t x ✐s ❣✐✈❡♥ ❜②✿ lim

f (x+h)−f (x)

h→0

■♥ ♦✉r ❝❛s❡ x = a✳ ■t ✐s s✐♠♣❧❡r

h

t♦ s✉❜st✐t✉t❡ x = a ❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❝❛❧❝✉❧❛t✐♦♥✳

✼✽

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❙t❡♣ ✷✳

f (x + h)

=

5(x + h)2 − 4 (x + h) + 1

=

5 x2 + 2xh + h2 − 4x − 4h + 1

✭✺✳✶✷✮

=

5x2 + 10xh + 5h2 − 4x − 4h + 1

❙t❡♣ ✸✳

5x2+10xh+5h2−4x−4h+1−(5x2−4x+1)

lim f(x+h)−f(x)

=

h→0

h

h

=

lim 5x2+10xh+5h2−4x−4h+1−5x2+4x−1

h→0

h

=

lim 10xh+5h2−4h

h→0

h

✭✺✳✶✸✮

=

lim h(10x+5h−4)

h→0

h

=

lim10x + 5h − 4

h→0

=

10x − 4

❙t❡♣ ✹✳ 10x − 4 = 10a − 5

❙t❡♣ ✺✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ f (x) = 5x2 − 4x + 1 ❛t x = 1 ✐s 10a − 5✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✼ ✭♣✳ ✻✽✮

❙t❡♣ ✶✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❣r❛❞✐❡♥t ❛t ❛ ♣♦✐♥t x ✐s ❣✐✈❡♥ ❜②✿ g✬ (x) = lim

g(x+h)−g(x)

h→0

h

❙t❡♣ ✷✳ g (x + h) = x + h − 1

❙t❡♣ ✸✳

g✬ (x)

=

lim g(x+h)−g(x)

h→0

h

=

lim x+h−1−(x−1)

h→0

h

=

lim x+h−1−x+1

h→0

h

✭✺✳✶✹✮

=

lim h

h→0 h

=

lim1

h→0

=

1

❙t❡♣ ✹✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ g✬ (x) ♦❢ g (x) = x − 1 ✐s ✶✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✽ ✭♣✳ ✻✾✮

❙t❡♣ ✶✳ ❲❡ ✇✐❧❧ ❛♣♣❧② t✇♦ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥✿ d (xn) = nxn−1 ❛♥❞ d [f (x) − g (x)] = d [f (x)]− d [g (x)]

dx

dx

dx

dx

❙t❡♣ ✷✳ ■♥ ♦✉r ❝❛s❡ f (x) = x ❛♥❞ g (x) = 1✳ f✬ (x) = 1 ❛♥❞ g✬ (x) = 0

❙t❡♣ ✸✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ x − 1 ✐s ✶ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ r❡s✉❧t ❛s ✇❛s ♦❜t❛✐♥❡❞ ❡❛r❧✐❡r✱ ❢r♦♠ ✜rst ♣r✐♥❝✐♣❧❡s✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✾ ✭♣✳ ✼✵✮

❙t❡♣ ✶✳ ❲❡ ❛r❡ r❡q✉✐r❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ❞❡✜♥❡❞ ❜② y = x2 ❛t t❤❡ ♣♦✐♥t

✭✶✱✶✮✳ ❚❤❡ t❛♥❣❡♥t ✐s ❛ str❛✐❣❤t ❧✐♥❡ ❛♥❞ ✇❡ ❝❛♥ ✜♥❞ t❤❡ ❡q✉❛t✐♦♥ ❜② ✉s✐♥❣ ❞❡r✐✈❛t✐✈❡s t♦ ✜♥❞ t❤❡

❣r❛❞✐❡♥t ♦❢ t❤❡ str❛✐❣❤t ❧✐♥❡✳ ❚❤❡♥ ✇❡ ✇✐❧❧ ❤❛✈❡ t❤❡ ❣r❛❞✐❡♥t ❛♥❞ ♦♥❡ ♣♦✐♥t ♦♥ t❤❡ ❧✐♥❡✱ s♦ ✇❡ ❝❛♥ ✜♥❞

t❤❡ ❡q✉❛t✐♦♥ ✉s✐♥❣✿ y − y1 = m (x − x1) ❢r♦♠ ❣r❛❞❡ ✶✶ ❈♦♦r❞✐♥❛t❡ ●❡♦♠❡tr②✳

❙t❡♣ ✷✳ ❯s✐♥❣ ♦✉r r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇❡ ❣❡t✿ y✬ = 2x

❙t❡♣ ✸✳ ■♥ ♦r❞❡r t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❣r❛❞✐❡♥t ❛t t❤❡ ♣♦✐♥t ✭✶✱✶✮✱ ✇❡ s✉❜st✐t✉t❡ t❤❡ x✲✈❛❧✉❡ ✐♥t♦ t❤❡ ❡q✉❛t✐♦♥ ❢♦r

t❤❡ ❞❡r✐✈❛t✐✈❡✳ ❙♦✱ y✬ ❛t x = 1 ✐s✿ m = 2 (1) = 2

✼✾

❙t❡♣ ✹✳

y − y1

=

m (x − x1)

y − 1

=

(2) (x − 1)

✭✺✳✶✺✮

y

=

2x − 2 + 1

y

=

2x − 1

❙t❡♣ ✺✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ ❞❡✜♥❡❞ ❜② y = x2 ❛t t❤❡ ♣♦✐♥t ✭✶✱✶✮ ✐s y = 2x − 1✳

❙t❡♣ ✻✳

❋✐❣✉r❡ ✺✳✶✺

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶✵ ✭♣✳ ✼✵✮

❙t❡♣ ✶✳ ❯s✐♥❣ t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇❡ ❣❡t✿ f✬ (x) = 6x2 − 18x + 12

❙t❡♣ ✷✳

6x2 − 18x + 12

=

0

x2 − 3x + 2

=

0

✭✺✳✶✻✮

(x − 2) (x − 1)

=

0

❚❤❡r❡❢♦r❡✱ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛r❡ ❛t x = 2 ❛♥❞ x = 1✳

❙t❡♣ ✸✳

f (2)

=

2(2)3 − 9(2)2 + 12 (2) − 15

=

16 − 36 + 24 − 15

✭✺✳✶✼✮

=

−11

f (1)

=

2(1)3 − 9(1)2 + 12 (1) − 15

=

2 − 9 + 12 − 15

✭✺✳✶✽✮

=

−10

❙t❡♣ ✹✳ ❚❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ♦❢ t❤❡ ❣r❛♣❤ ♦❢ f (x) = 2x3 − 9x2 + 12x − 15 ❛r❡ ✭✷✱✲✶✶✮ ❛♥❞ ✭✶✱✲✶✵✮✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶✶ ✭♣✳ ✼✶✮

❙t❡♣ ✶✳ ❚❤❡ y✲✐♥t❡r❝❡♣t ✐s ♦❜t❛✐♥❡❞ ❜② s❡tt✐♥❣ x = 0✳ g (0) = (0)2 − 0 + 2 = 2 ❚❤❡ t✉r♥✐♥❣ ♣♦✐♥t ✐s ❛t ✭✵✱✷✮✳

❙t❡♣ ✷✳ ❚❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡ ❢♦✉♥❞ ❜② s❡tt✐♥❣ g (x) = 0✳

g (x)

=

x2 − x + 2

✭✺✳✶✾✮

0

=

x2 − x + 2

❯s✐♥❣ t❤❡ q✉❛❞r❛t✐❝ ❢♦r♠✉❧❛ ❛♥❞ ❧♦♦❦✐♥❣ ❛t b2 − 4ac ✇❡ ❝❛♥ s❡❡ t❤❛t t❤✐s ✇♦✉❧❞ ❜❡ ♥❡❣❛t✐✈❡ ❛♥❞ s♦

t❤✐s ❢✉♥❝t✐♦♥ ❞♦❡s ♥♦t ❤❛✈❡ r❡❛❧ r♦♦ts✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ❣r❛♣❤ ♦❢ g (x) ❞♦❡s ♥♦t ❤❛✈❡ ❛♥② x✲✐♥t❡r❝❡♣ts✳

✽✵

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❙t❡♣ ✸✳ ❲♦r❦ ♦✉t t❤❡ ❞❡r✐✈❛t✐✈❡ dg ❛♥❞ s❡t ✐t t♦ ③❡r♦ t♦ ❢♦r t❤❡ x ❝♦♦r❞✐♥❛t❡ ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥t✳ dg = 2x − 1

dx

dx

dg

=

0

dx

2x − 1

=

0

✭✺✳✷✵✮

2x

=

1

x

=

1

2

❙t❡♣ ✹✳ y ❝♦♦r❞✐♥❛t❡ ♦❢ t✉r♥✐♥❣ ♣♦✐♥t ✐s ❣✐✈❡♥ ❜② ❝❛❧❝✉❧❛t✐♥❣ g 1 ✳

2

2

g 1

=

1

− 1 + 2

2

2

2

=

1 − 1 + 2

✭✺✳✷✶✮

4

2

=

7

4

❚❤❡ t✉r♥✐♥❣ ♣♦✐♥t ✐s ❛t 1, 7

2

4

❙t❡♣ ✺✳

❋✐❣✉r❡ ✺✳✶✻

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶✷ ✭♣✳ ✼✶✮

❙t❡♣ ✶✳ ❋✐♥❞ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❜② s❡tt✐♥❣ g✬ (x) = 0✳

■❢ ✇❡ ✉s❡ t❤❡ r✉❧❡s ♦❢ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇❡ ❣❡t g✬ (x) = −3x2 + 12x − 9

g✬ (x)

=

0

−3x2 + 12x − 9

=

0

✭✺✳✷✷✮

x2 − 4x + 3

=

0

(x − 3) (x − 1)

=

0

❚❤❡ x✲❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛r❡✿ x = 1 ❛♥❞ x = 3✳

❚❤❡ y✲❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛r❡ ❝❛❧❝✉❧❛t❡❞ ❛s✿

g (x)

=

−x3 + 6x2 − 9x + 4

g (1)

=

−(1)3 + 6(1)2 − 9 (1) + 4

✭✺✳✷✸✮

=

−1 + 6 − 9 + 4

=

0

g (x)

=

−x3 + 6x2 − 9x + 4

g (3)

=

−(3)3 + 6(3)2 − 9 (3) + 4

✭✺✳✷✹✮

=

−27 + 54 − 27 + 4

=

4

❚❤❡r❡❢♦r❡ t❤❡ t✉r♥✐♥❣ ♣♦✐♥ts ❛r❡✿ (1, 0) ❛♥❞ (3, 4)✳

✽✶

❙t❡♣ ✷✳ ❲❡ ✜♥❞ t❤❡ y✲✐♥t❡r❝❡♣ts ❜② ✜♥❞✐♥❣ t❤❡ ✈❛❧✉❡ ❢♦r g (0)✳

g (x)

=

−x3 + 6x2 − 9x + 4

y

✭✺✳✷✺✮

int = g (0)

=

−(0)3 + 6(0)2 − 9 (0) + 4

=

4

❙t❡♣ ✸✳ ❲❡ ✜♥❞ t❤❡ x✲✐♥t❡r❝❡♣ts ❜② ✜♥❞✐♥❣ t❤❡ ♣♦✐♥ts ❢♦r ✇❤✐❝❤ t❤❡ ❢✉♥❝t✐♦♥ g (x) = 0✳

g (x) = −x3 + 6x2 − 9x + 4

❯s❡ t❤❡ ❢❛❝t♦r t❤❡♦r❡♠ t♦ ❝♦♥✜r♠ t❤❛t (x − 1) ✐s ❛ ❢❛❝t♦r✳ ■❢ g (1) = 0✱ t❤❡♥ (x − 1) ✐s ❛ ❢❛❝t♦r✳

g (x)

=

−x3 + 6x2 − 9x + 4

g (1)

=

−(1)3 + 6(1)2 − 9 (1) + 4

✭✺✳✷✻✮

=

−1 + 6 − 9 + 4

=

0

❚❤❡r❡❢♦r❡✱ (x − 1) ✐s ❛ ❢❛❝t♦r✳

■❢ ✇❡ ❞✐✈✐❞❡ g (x) ❜② (x − 1) ✇❡ ❛r❡ ❧❡❢t ✇✐t❤✿ −x2 + 5x − 4 ❚❤✐s ❤❛s ❢❛❝t♦rs − (x − 4) (x − 1)

❚❤❡r❡❢♦r❡✿ g (x) = − (x − 1) (x − 1) (x − 4)

❚❤❡ x✲✐♥t❡r❝❡♣ts ❛r❡✿ xint = 1, 4

❙t❡♣ ✹✳

❋✐❣✉r❡ ✺✳✶✼

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶✸ ✭♣✳ ✼✸✮

❙t❡♣ ✶✳ ▲❡t t❤❡ t✇♦ ♥✉♠❜❡rs ❜❡ a ❛♥❞ b✳ ❚❤❡♥ ✇❡ ❤❛✈❡✿

a + b = 10

❲❡ ❛r❡ r❡q✉✐r❡❞ t♦ ♠✐♥✐♠✐s❡ t❤❡ ♣r♦❞✉❝t ♦❢ a ❛♥❞ b✳ ❈❛❧❧ t❤❡ ♣r♦❞✉❝t P ✳ ❚❤❡♥✿

P = a · b

❲❡ ❝❛♥ s♦❧✈❡ ❢♦r b ❢r♦♠ t♦ ❣❡t✿

b = 10 − a

❙✉❜st✐t✉t❡ t❤✐s ✐♥t♦ t♦ ✇r✐t❡ P ✐♥ t❡r♠s ♦❢ a ♦♥❧②✳

P = a (10 − a) = 10a − a2

❙t❡♣ ✷✳ ❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ✐s✿ P ✬ (a) = 10 − 2a

❙t❡♣ ✸✳ ❙❡t P ✬ (a) = 0 t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ♦❢ a ✇❤✐❝❤ ♠❛❦❡s P ❛ ♠❛①✐♠✉♠✳

P ✬ (a)

=

10 − 2a

0

=

10 − 2a

2a

=

10

✭✺✳✷✼✮

a

=

10

2

a

=

5

✽✷

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❙✉❜st✐t✉t❡ ✐♥t♦ t♦ s♦❧✈❡ ❢♦r t❤❡ ✇✐❞t❤✳

b

=

10 − a

=

10 − 5

✭✺✳✷✽✮

=

5

❙t❡♣ ✹✳ ❚❤❡ ♣r♦❞✉❝t ✐s ♠❛①✐♠✐s❡❞ ✐❢ a ❛♥❞ b ❛r❡ ❜♦t❤ ❡q✉❛❧ t♦ ✺✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶✹ ✭♣✳ ✼✸✮

❙t❡♣ ✶✳ ❚❤❡ ✐♠♣♦rt❛♥t ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ❣✐✈❡♥ ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ❛r❡❛ ❛♥❞ ♠♦❞✐✜❡❞ ♣❡r✐♠❡t❡r ♦❢ t❤❡ ❣❛r❞❡♥✳

❲❡ ❦♥♦✇ t❤❛t t❤❡ ❛r❡❛ ♦❢ t❤❡ ❣❛r❞❡♥ ✐s✿ A = w · l ❲❡ ❛r❡ ❛❧s♦ t♦❧❞ t❤❛t t❤❡ ❢❡♥❝❡ ❝♦✈❡rs ♦♥❧② ✸ s✐❞❡s

❛♥❞ t❤❡ t❤r❡❡ s✐❞❡s s❤♦✉❧❞ ❛❞❞ ✉♣ t♦ ✶✻✵ ♠✳ ❚❤✐s ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s✿ 160 = w + l + l

❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ✉s❡ t♦ ✇r✐t❡ w ✐♥ t❡r♠s ♦❢ l✿ w = 160 − 2l ❙✉❜st✐t✉t❡ ✐♥t♦ t♦ ❣❡t✿ A = (160 − 2l) l =

160l − 2l2

❙t❡♣ ✷✳ ❙✐♥❝❡ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ♠❛①✐♠✐s✐♥❣ t❤❡ ❛r❡❛✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t♦ ❣❡t✿ A✬ (l) = 160 − 4l

❙t❡♣ ✸✳ ❚♦ ✜♥❞ t❤❡ st❛t✐♦♥❛r② ♣♦✐♥t✱ ✇❡ s❡t A✬ (l) = 0 ❛♥❞ s♦❧✈❡ ❢♦r t❤❡ ✈❛❧✉❡ ♦❢ l t❤❛t ♠❛①✐♠✐s❡s t❤❡ ❛r❡❛✳

A✬ (l)

=

160 − 4l

0

=

160 − 4l

∴ 4l =

160

✭✺✳✷✾✮

l

=

160

4

l

=

40m

❙✉❜st✐t✉t❡ ✐♥t♦ t♦ s♦❧✈❡ ❢♦r t❤❡ ✇✐❞t❤✳

w

=

160 − 2l

=

160 − 2 (40)

✭✺✳✸✵✮

=

160 − 80

=

80m

❙t❡♣ ✹✳ ❆ ✇✐❞t❤ ♦❢ ✽✵ ♠ ❛♥❞ ❛ ❧❡♥❣t❤ ♦❢ ✹✵ ♠ ✇✐❧❧ ②✐❡❧❞ t❤❡ ♠❛①✐♠❛❧ ❛r❡❛ ❢❡♥❝❡❞ ♦✛✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✺✳✶✺ ✭♣✳ ✼✹✮

❙t❡♣ ✶✳

vave

=

h(2)−h(0)

2−0

[20(2)−5(2)2]−[20(0)−5(0)2]

=

2

✭✺✳✸✶✮

=

40−20

2

=

10 ms−1

❙t❡♣ ✷✳

v (t)

=

dh

dt

✭✺✳✸✷✮

=

20 − 10t

❱❡❧♦❝✐t② ❛❢t❡r ✶✱✺ s❡❝♦♥❞s✿

v (1, 5)

=

20 − 10 (1, 5)

✭✺✳✸✸✮

=

5 ms−1

✽✸

❙t❡♣ ✸✳

v (t)

=

0

20 − 10t

=

0

✭✺✳✸✹✮

10t

=

20

t

=

2

❚❤❡r❡❢♦r❡ t❤❡ ✈❡❧♦❝✐t② ✐s ③❡r♦ ❛❢t❡r ✷ s❡❝♦♥❞s

❙t❡♣ ✹✳ ❚❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞ ✇❤❡♥ h (t) = 0

20t − 5t2

=

0

5t (4 − t)

=

0

✭✺✳✸✺✮

t = 0

or

t = 4

❚❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣r♦✉♥❞ ❛❢t❡r ✹ s❡❝♦♥❞s✳ ❚❤❡ ✈❡❧♦❝✐t② ❛❢t❡r ✹ s❡❝♦♥❞s ✇✐❧❧ ❜❡✿

v (4)

=

h✬ (4)

=

20 − 10 (4)

✭✺✳✸✻✮

=

−20 ms−1

❚❤❡ ❜❛❧❧ ❤✐ts t❤❡ ❣♦✉♥❞ ❛t ❛ s♣❡❡❞ ♦❢ 20ms−1✳ ◆♦t✐❝❡ t❤❛t t❤❡ s✐❣♥ ♦❢ t❤❡ ✈❡❧♦❝✐t② ✐s ♥❡❣❛t✐✈❡ ✇❤✐❝❤

♠❡❛♥s t❤❛t t❤❡ ❜❛❧❧ ✐s ♠♦✈✐♥❣ ❞♦✇♥✇❛r❞ ✭t❤❡ r❡✈❡rs❡ ♦❢ ✉♣✇❛r❞✱ ✇❤✐❝❤ ✐s ✇❤❡♥ t❤❡ ✈❡❧♦❝✐t② ✐s ♣♦s✐t✐✈❡✮✳

❙t❡♣ ✺✳

a

=

v✬ (t)

✭✺✳✸✼✮

=

−10 ms−1

❏✉st ❜❡❝❛✉s❡ ❣r❛✈✐t② ✐s ❝♦♥st❛♥t ❞♦❡s ♥♦t ♠❡❛♥ ✇❡ s❤♦✉❧❞ t❤✐♥❦ ♦❢ ❛❝❝❡❧❡r❛t✐♦♥ ❛s ❛ ❝♦♥st❛♥t✳ ❲❡

s❤♦✉❧❞ st✐❧❧ ❝♦♥s✐❞❡r ✐t ❛ ❢✉♥❝t✐♦♥✳

✽✹

❈❍❆P❚❊❘ ✺✳ ❉■❋❋❊❘❊◆❚■❆▲ ❈❆▲❈❯▲❯❙

❈❤❛♣t❡r ✻

▲✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣

✻✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❚❡r♠✐♥♦❧♦❣②✱ ❚❤❡ ❋❡❛s✐❜❧❡ ❘❡❣✐♦♥✶

✻✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

■♥ ●r❛❞❡ ✶✶ ②♦✉ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ t♦ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ❛♥❞ s♦❧✈❡❞ ♣r♦❜❧❡♠s ❜② ❧♦♦❦✐♥❣ ❛t ♣♦✐♥ts ♦♥ t❤❡

❡❞❣❡s ♦❢ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ■♥ ●r❛❞❡ ✶✷ ②♦✉ ✇✐❧❧ ❧♦♦❦ ❛t ❤♦✇ t♦ s♦❧✈❡ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠s ✐♥ ❛

♠♦r❡ ❣❡♥❡r❛❧ ♠❛♥♥❡r✳

✻✳✶✳✷ ❚❡r♠✐♥♦❧♦❣②

❍❡r❡ ✐s ❛ r❡❝❛♣ ♦❢ s♦♠❡ ♦❢ t❤❡ ✐♠♣♦rt❛♥t ❝♦♥❝❡♣ts ✐♥ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✳

✻✳✶✳✷✳✶ ❋❡❛s✐❜❧❡ ❘❡❣✐♦♥ ❛♥❞ P♦✐♥ts

❈♦♥str❛✐♥ts ♠❡❛♥ t❤❛t ✇❡ ❝❛♥♥♦t ❥✉st t❛❦❡ ❛♥② x ❛♥❞ y ✇❤❡♥ ❧♦♦❦✐♥❣ ❢♦r t❤❡ x ❛♥❞ y t❤❛t ♦♣t✐♠✐s❡ ♦✉r

♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳ ■❢ ✇❡ t❤✐♥❦ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s x ❛♥❞ y ❛s ❛ ♣♦✐♥t (x, y) ✐♥ t❤❡ xy✲♣❧❛♥❡ t❤❡♥ ✇❡ ❝❛❧❧ t❤❡

s❡t ♦❢ ❛❧❧ ♣♦✐♥ts ✐♥ t❤❡ xy✲♣❧❛♥❡ t❤❛t s❛t✐s❢② ♦✉r ❝♦♥str❛✐♥ts t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❆♥② ♣♦✐♥t ✐♥ t❤❡ ❢❡❛s✐❜❧❡

r❡❣✐♦♥ ✐s ❝❛❧❧❡❞ ❛ ❢❡❛s✐❜❧❡ ♣♦✐♥t✳

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❝♦♥str❛✐♥ts

x ≥ 0

✭✻✳✶✮

y ≥ 0

♠❡❛♥ t❤❛t ❡✈❡r② (x, y) ✇❡ ❝❛♥ ❝♦♥s✐❞❡r ♠✉st ❧✐❡ ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t ♦❢ t❤❡ xy ♣❧❛♥❡✳ ❚❤❡ ❝♦♥str❛✐♥t

x ≥ y

✭✻✳✷✮

♠❡❛♥s t❤❛t ❡✈❡r② (x, y) ♠✉st ❧✐❡ ♦♥ ♦r ❜❡❧♦✇ t❤❡ ❧✐♥❡ y = x ❛♥❞ t❤❡ ❝♦♥str❛✐♥t

x ≤ 20

✭✻✳✸✮

♠❡❛♥s t❤❛t x ♠✉st ❧✐❡ ♦♥ ♦r t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ❧✐♥❡ x = 20✳

❲❡ ❝❛♥ ✉s❡ t❤❡s❡ ❝♦♥str❛✐♥ts t♦ ❞r❛✇ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ❛s s❤♦✇♥ ❜② t❤❡ s❤❛❞❡❞ r❡❣✐♦♥ ✐♥ ❋✐❣✉r❡ ✻✳✶✳

t✐♣✿ ❚❤❡ ❝♦♥str❛✐♥ts ❛r❡ ✉s❡❞ t♦ ❝r❡❛t❡ ❜♦✉♥❞s ♦❢ t❤❡ s♦❧✉t✐♦♥✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✾✸✴✶✳✶✴❃✳

✽✺

✽✻

❈❍❆P❚❊❘ ✻✳ ▲■◆❊❆❘ P❘❖●❘❆▼▼■◆●

❋✐❣✉r❡ ✻✳✶✿ ❚❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝♦♥str❛✐♥ts x ≥ 0✱ y ≥ 0✱ x ≥ y ❛♥❞ x ≤ 20✳

t✐♣✿

ax + by = c

■❢ b = 0✱ ❢❡❛s✐❜❧❡ ♣♦✐♥ts ♠✉st ❧✐❡ ♦♥ t❤❡ ❧✐♥❡ y = −ax + c

b

b

■❢ b = 0✱ ❢❡❛s✐❜❧❡ ♣♦✐♥ts ♠✉st ❧✐❡ ♦♥ t❤❡ ❧✐♥❡ x = c/a

ax + by ≤ c

■❢ b = 0✱ ❢❡❛s✐❜❧❡ ♣♦✐♥ts ♠✉st ❧✐❡ ♦♥ ♦r ❜❡❧♦✇ t❤❡ ❧✐♥❡ y = −ax + c✳

b

b

■❢ b = 0✱ ❢❡❛s✐❜❧❡ ♣♦✐♥ts ♠✉st ❧✐❡ ♦♥ ♦r t♦ t❤❡ ❧❡❢t ♦❢ t❤❡ ❧✐♥❡ x = c/a✳

❚❛❜❧❡ ✻✳✶

❲❤❡♥ ❛ ❝♦♥str❛✐♥t ✐s ❧✐♥❡❛r✱ ✐t ♠❡❛♥s t❤❛t ✐t r❡q✉✐r❡s t❤❛t ❛♥② ❢❡❛s✐❜❧❡ ♣♦✐♥t (x, y) ❧✐❡s ♦♥ ♦♥❡ s✐❞❡ ♦❢ ♦r ♦♥ ❛

❧✐♥❡✳ ■♥t❡r♣r❡t✐♥❣ ❝♦♥str❛✐♥ts ❛s ❣r❛♣❤s ✐♥ t❤❡ xy ♣❧❛♥❡ ✐s ✈❡r② ✐♠♣♦rt❛♥t s✐♥❝❡ ✐t ❛❧❧♦✇s ✉s t♦ ❝♦♥str✉❝t t❤❡

❢❡❛s✐❜❧❡ r❡❣✐♦♥ s✉❝❤ ❛s ✐♥ ❋✐❣✉r❡ ✻✳✶✳

✻✳✶✳✸ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣ ❛♥❞ t❤❡ ❋❡❛s✐❜❧❡ ❘❡❣✐♦♥

■❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛♥❞ ❛❧❧ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❛r❡ ❧✐♥❡❛r t❤❡♥ ✇❡ ❝❛❧❧ t❤❡ ♣r♦❜❧❡♠ ♦❢ ♦♣t✐♠✐s✐♥❣ t❤❡

♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ s✉❜❥❡❝t t♦ t❤❡s❡ ❝♦♥str❛✐♥ts ❛ ❧✐♥❡❛r ♣r♦❣r❛♠✳ ❆❧❧ ♦♣t✐♠✐s❛t✐♦♥ ♣r♦❜❧❡♠s ✇❡ ✇✐❧❧ ❧♦♦❦

❛t ✇✐❧❧ ❜❡ ❧✐♥❡❛r ♣r♦❣r❛♠s✳

❚❤❡ ♠❛❥♦r ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❜❡✐♥❣ ❧✐♥❡❛r ✐s t❤❛t t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ✐s ❛❧✇❛②s ❛ ♣♦❧②❣♦♥✳ ❚❤✐s

✐s ❡✈✐❞❡♥t s✐♥❝❡ t❤❡ ❝♦♥str❛✐♥ts t❤❛t ❞❡✜♥❡ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ❛❧❧ ❝♦♥tr✐❜✉t❡ ❛ ❧✐♥❡ s❡❣♠❡♥t t♦ ✐ts ❜♦✉♥❞❛r②

✭s❡❡ ❋✐❣✉r❡ ✻✳✶✮✳ ■t ✐s ❛❧s♦ ❛❧✇❛②s tr✉❡ t❤❛t t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ✐s ❛ ❝♦♥✈❡① ♣♦❧②❣♦♥✳

❚❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❜❡✐♥❣ ❧✐♥❡❛r ♠❡❛♥s t❤❛t t❤❡ ❢❡❛s✐❜❧❡ ♣♦✐♥t✭s✮ t❤❛t ❣✐✈❡s t❤❡ s♦❧✉t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r

♣r♦❣r❛♠ ❛❧✇❛②s ❧✐❡s ♦♥ ♦♥❡ ♦❢ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❚❤✐s ✐s ✈❡r② ✐♠♣♦rt❛♥t s✐♥❝❡✱ ❛s ✇❡ ✇✐❧❧

s♦♦♥ s❡❡✱ ✐t ❣✐✈❡s ✉s ❛ ✇❛② ♦❢ s♦❧✈✐♥❣ ❧✐♥❡❛r ♣r♦❣r❛♠s✳

❲❡ ✇✐❧❧ ♥♦✇ s❡❡ ✇❤② t❤❡ s♦❧✉t✐♦♥s ♦❢ ❛ ❧✐♥❡❛r ♣r♦❣r❛♠ ❛❧✇❛②s ❧✐❡ ♦♥ t❤❡ ✈❡rt✐❝❡s ♦❢ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳

❋✐rst❧②✱ ♥♦t❡ t❤❛t ✐❢ ✇❡ t❤✐♥❦ ♦❢ f (x, y) ❛s ❧②✐♥❣ ♦♥ t❤❡ z ❛①✐s✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥ f (x, y) = ax + by ✭✇❤❡r❡ a

❛♥❞ b ❛r❡ r❡❛❧ ♥✉♠❜❡rs✮ ✐s t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ♣❧❛♥❡✳ ■❢ ✇❡ s♦❧✈❡ ❢♦r y ✐♥ t❤❡ ❡q✉❛t✐♦♥ ❞❡✜♥✐♥❣ t❤❡ ♦❜❥❡❝t✐✈❡

❢✉♥❝t✐♦♥ t❤❡♥

f (x, y) = ax + by

✭✻✳✹✮

∴

y = −a x + f(x,y)

b

b

❲❤❛t t❤✐s ♠❡❛♥s ✐s t❤❛t ✐❢ ✇❡ ✜♥❞ ❛❧❧ t❤❡ ♣♦✐♥ts ✇❤❡r❡ f (x, y) = c ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡r c ✭✐✳❡✳ f (x, y) ✐s

❝♦♥st❛♥t ✇✐t❤ ❛ ✈❛❧✉❡ ♦❢ c✮✱ t❤❡♥ ✇❡ ❤❛✈❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❧✐♥❡✳ ❚❤✐s ❧✐♥❡ ✇❡ ❝❛❧❧ ❛ ❧❡✈❡❧ ❧✐♥❡ ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡

❢✉♥❝t✐♦♥✳

❈♦♥s✐❞❡r ❛❣❛✐♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ❞❡s❝r✐❜❡❞ ✐♥ ❋✐❣✉r❡ ✻✳✶✳ ▲❡t✬s s❛② t❤❛t ✇❡ ❤❛✈❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥

f (x, y) = x − 2y ✇✐t❤ t❤✐s ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ■❢ ✇❡ ❝♦♥s✐❞❡r ❊q✉❛t✐♦♥ ✭✻✳✹✮ ❝♦rr❡s♣♦♥❞✐♥❣ t♦

f (x, y) = −20

✭✻✳✺✮

✽✼

t❤❡♥ ✇❡ ❣❡t t❤❡ ❧❡✈❡❧ ❧✐♥❡

1

y =

x + 10

✭✻✳✻✮

2

✇❤✐❝❤ ❤❛s ❜❡❡♥ ❞r❛✇♥ ✐♥ ✳ ▲❡✈❡❧ ❧✐♥❡s ❝♦rr❡s♣♦♥❞✐♥❣ t♦

f (x, y) = −10

♦r y = x + 5

2

f (x, y) = 0

♦r

y = x2

✭✻✳✼✮

f (x, y) = 10

♦r y = x − 5

2

f (x, y) = 20

♦r y = x − 10

2

❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❞r❛✇♥ ✐♥✳ ■t ✐s ✈❡r② ✐♠♣♦rt❛♥t t♦ r❡❛❧✐s❡ t❤❛t t❤❡s❡ ❛r❡ ♥♦t t❤❡ ♦♥❧② ❧❡✈❡❧ ❧✐♥❡s❀ ✐♥ ❢❛❝t✱ t❤❡r❡

❛r❡ ✐♥✜♥✐t❡❧② ♠❛♥② ♦❢ t❤❡♠ ❛♥❞ t❤❡② ❛r❡ ❛❧❧ ♣❛r❛❧❧❡❧ t♦ ❡❛❝❤ ♦t❤❡r✳ ❘❡♠❡♠❜❡r t❤❛t ✐❢ ✇❡ ❧♦♦❦ ❛t ❛♥② ♦♥❡

❧❡✈❡❧ ❧✐♥❡ f (x, y) ❤❛s t❤❡ s❛♠❡ ✈❛❧✉❡ ❢♦r ❡✈❡r② ♣♦✐♥t (x, y) t❤❛t ❧✐❡s ♦♥ t❤❛t ❧✐♥❡✳ ❆❧s♦✱ f (x, y) ✇✐❧❧ ❛❧✇❛②s

❤❛✈❡ ❞✐✛❡r❡♥t ✈❛❧✉❡s ♦♥ ❞✐✛❡r❡♥t ❧❡✈❡❧ ❧✐♥❡s✳

❋✐❣✉r❡ ✻✳✷✿ ❚❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ ❝♦♥str❛✐♥ts x ≥ 0✱ y ≥ 0✱ x ≥ y ❛♥❞ x ≤ 20 ✇✐t❤

♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ f (x, y) = x − 2y✳ ❚❤❡ ❞❛s❤❡❞ ❧✐♥❡s r❡♣r❡s❡♥t ✈❛r✐♦✉s ❧❡✈❡❧ ❧✐♥❡s ♦❢ f (x, y)✳

■❢ ❛ r✉❧❡r ✐s ♣❧❛❝❡❞ ♦♥ t❤❡ ❧❡✈❡❧ ❧✐♥❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ f (x, y) = −20 ✐♥ ❋✐❣✉r❡ ✻✳✷ ❛♥❞ ♠♦✈❡❞ ❞♦✇♥ t❤❡

♣❛❣❡ ♣❛r❛❧❧❡❧ t♦ t❤✐s ❧✐♥❡ t❤❡♥ ✐t ✐s ❝❧❡❛r t❤❛t t❤❡ r✉❧❡r ✇✐❧❧ ❜❡ ♠♦✈✐♥❣ ♦✈❡r ❧❡✈❡❧ ❧✐♥❡s ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞

t♦ ❧❛r❣❡r ✈❛❧✉❡s ♦❢ f (x, y)✳ ❙♦ ✐❢ ✇❡ ✇❛♥t❡❞ t♦ ♠❛①✐♠✐s❡ f (x, y) t❤❡♥ ✇❡ s✐♠♣❧② ♠♦✈❡ t❤❡ r✉❧❡r ❞♦✇♥ t❤❡

♣❛❣❡ ✉♥t✐❧ ✇❡ r❡❛❝❤ t❤❡ ✏❧♦✇❡st✑ ♣♦✐♥t ✐♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❚❤✐s ♣♦✐♥t ✇✐❧❧ t❤❡♥ ❜❡ t❤❡ ❢❡❛s✐❜❧❡ ♣♦✐♥t t❤❛t

♠❛①✐♠✐s❡s f (x, y)✳ ❙✐♠✐❧❛r❧②✱ ✐❢ ✇❡ ✇❛♥t❡❞ t♦ ♠✐♥✐♠✐s❡ f (x, y) t❤❡♥ t❤❡ ✏❤✐❣❤❡st✑ ❢❡❛s✐❜❧❡ ♣♦✐♥t ✇✐❧❧ ❣✐✈❡ t❤❡

♠✐♥✐♠✉♠ ✈❛❧✉❡ ♦❢ f (x, y)✳

❙✐♥❝❡ ♦✉r ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ✐s ❛ ♣♦❧②❣♦♥✱ t❤❡s❡ ♣♦✐♥ts ✇✐❧❧ ❛❧✇❛②s ❧✐❡ ♦♥ ✈❡rt✐❝❡s ✐♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❚❤❡

❢❛❝t t❤❛t t❤❡ ✈❛❧✉❡ ♦❢ ♦✉r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❛❧♦♥❣ t❤❡ ❧✐♥❡ ♦❢ t❤❡ r✉❧❡r ✐♥❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ ✐t ❞♦✇♥ ❛♥❞

❞❡❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ ✐t ✉♣ ❞❡♣❡♥❞s ♦♥ t❤✐s ♣❛rt✐❝✉❧❛r ❡①❛♠♣❧❡✳ ❙♦♠❡ ♦t❤❡r ❡①❛♠♣❧❡s ♠✐❣❤t ❤❛✈❡ t❤❛t t❤❡

❢✉♥❝t✐♦♥ ✐♥❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ✉♣ ❛♥❞ ❞❡❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ ✐t ❞♦✇♥✳

■t ✐s ❛ ❣❡♥❡r❛❧ ♣r♦♣❡rt②✱ t❤♦✉❣❤✱ ♦❢ ❧✐♥❡❛r ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s t❤❛t t❤❡② ✇✐❧❧ ❝♦♥s✐st❡♥t❧② ✐♥❝r❡❛s❡ ♦r ❞❡❝r❡❛s❡

❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ✉♣ ♦r ❞♦✇♥✳ ❑♥♦✇✐♥❣ ✇❤✐❝❤ ❞✐r❡❝t✐♦♥ t♦ ♠♦✈❡ t❤❡ r✉❧❡r ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐s❡✴♠✐♥✐♠✐s❡

f (x, y) = ax+by ✐s ❛s s✐♠♣❧❡ ❛s ❧♦♦❦✐♥❣ ❛t t❤❡ s✐❣♥ ♦❢ b ✭✐✳❡✳ ✏✐s b ♥❡❣❛t✐✈❡✱ ♣♦s✐t✐✈❡ ♦r ③❡r♦❄✧✮✳ ■❢ b ✐s ♣♦s✐t✐✈❡✱

t❤❡♥ f (x, y)✐♥❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ✉♣ ❛♥❞ f (x, y)❞❡❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ❞♦✇♥✳ ❚❤❡ ♦♣♣♦s✐t❡

❤❛♣♣❡♥s ❢♦r t❤❡ ❝❛s❡ ✇❤❡♥ b ✐s ♥❡❣❛t✐✈❡✿ f (x, y)❞❡❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ✉♣ ❛♥❞ f (x, y)✐♥❝r❡❛s❡s ❛s

✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ❞♦✇♥✳ ■❢ b = 0 t❤❡♥ ✇❡ ♥❡❡❞ t♦ ❧♦♦❦ ❛t t❤❡ s✐❣♥ ♦❢ a✳

■❢ a ✐s ♣♦s✐t✐✈❡ t❤❡♥ f (x, y) ✐♥❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r t♦ t❤❡ r✐❣❤t ❛♥❞ ❞❡❝r❡❛s❡s ✐❢ ✇❡ ♠♦✈❡ t❤❡

r✉❧❡r t♦ t❤❡ ❧❡❢t✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ♦♣♣♦s✐t❡ ❤❛♣♣❡♥s ❢♦r a ♥❡❣❛t✐✈❡✳ ■❢ ✇❡ ❧♦♦❦ ❛❣❛✐♥ ❛t t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥

♠❡♥t✐♦♥❡❞ ❡❛r❧✐❡r✱

f (x, y) = x − 2y

✭✻✳✽✮

✇✐t❤ a = 1 ❛♥❞ b = −2✱ t❤❡♥ ✇❡ s❤♦✉❧❞ ✜♥❞ t❤❛t f (x, y) ✐♥❝r❡❛s❡s ❛s ✇❡ ♠♦✈❡ t❤❡ r✉❧❡r ❞♦✇♥ t❤❡ ♣❛❣❡ s✐♥❝❡

b = −2 < 0✳ ❚❤✐s ✐s ❡①❛❝t❧② ✇❤❛t ✇❡ ❢♦✉♥❞ ❤❛♣♣❡♥✐♥❣ ✐♥ ❋✐❣✉r❡ ✻✳✷✳

❚❤❡ ♠❛✐♥ ♣♦✐♥ts ❛❜♦✉t ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ✇❡ ❤❛✈❡ ❡♥❝♦✉♥t❡r❡❞ s♦ ❢❛r ❛r❡

✽✽

❈❍❆P❚❊❘ ✻✳ ▲■◆❊❆❘ P❘❖●❘❆▼▼■◆●

• ❚❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ✐s ❛❧✇❛②s ❛ ♣♦❧②❣♦♥✳

• ❙♦❧✉t✐♦♥s ♦❝❝✉r ❛t ✈❡rt✐❝❡s ♦❢ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳

• ▼♦✈✐♥❣ ❛ r✉❧❡r ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧❡✈❡❧ ❧✐♥❡s ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✉♣✴❞♦✇♥ t♦ t❤❡ t♦♣✴❜♦tt♦♠ ♦❢ t❤❡

❢❡❛s✐❜❧❡ r❡❣✐♦♥ s❤♦✇s ✉s ✇❤✐❝❤ ♦❢ t❤❡ ✈❡rt✐❝❡s ✐s t❤❡ s♦❧✉t✐♦♥✳

• ❚❤❡ ❞✐r❡❝t✐♦♥ ✐♥ ✇❤✐❝❤ t♦ ♠♦✈❡ t❤❡ r✉❧❡r ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ s✐❣♥ ♦❢ b ❛♥❞ ❛❧s♦ ♣♦ss✐❜❧② ❜② t❤❡ s✐❣♥

♦❢ a✳

❚❤❡s❡ ♣♦✐♥ts ❛r❡ s✉✣❝✐❡♥t t♦ ❞❡t❡r♠✐♥❡ ❛ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❛♥② ❧✐♥❡❛r ♣r♦❣r❛♠✳

✻✳✷ ▼❡t❤♦❞✱ ❊①❡r❝✐s❡s✷

✻✳✷✳✶ ▼❡t❤♦❞✿ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣

■❢ ✇❡ ✇✐s❤ t♦ ♠❛①✐♠✐s❡ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ f (x, y) t❤❡♥✿

✶✳ ❋✐♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧❡✈❡❧ ❧✐♥❡s ♦❢ f (x, y) ✭t❤✐s ✐s ❛❧✇❛②s ❣♦✐♥❣ t♦ ❜❡ −a ❛s ✇❡ s❛✇ ✐♥ ❊q✉❛t✐♦♥ ✮

b

✷✳ P❧❛❝❡ ②♦✉r r✉❧❡r ♦♥ t❤❡ xy ♣❧❛♥❡✱ ♠❛❦✐♥❣ ❛ ❧✐♥❡ ✇✐t❤ ❣r❛❞✐❡♥t −a ✭✐✳❡✳ b ✉♥✐ts ♦♥ t❤❡ x✲❛①✐s ❛♥❞ −a

b

✉♥✐ts ♦♥ t❤❡ y✲❛①✐s✮

✸✳ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r ♣r♦❣r❛♠ ✐s ❣✐✈❡♥ ❜② ❛♣♣r♦♣r✐❛t❡❧② ♠♦✈✐♥❣ t❤❡ r✉❧❡r✳ ❋✐rst❧② ✇❡ ♥❡❡❞ t♦ ❝❤❡❝❦

✇❤❡t❤❡r b ✐s ♥❡❣❛t✐✈❡✱ ♣♦s✐t✐✈❡ ♦r ③❡r♦✳

❛✳ ■❢ b > 0✱ ♠♦✈❡ t❤❡ r✉❧❡r ✉♣ t❤❡ ♣❛❣❡✱ ❦❡❡♣✐♥❣ t❤❡ r✉❧❡r ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧❡✈❡❧ ❧✐♥❡s ❛❧❧ t❤❡ t✐♠❡✱ ✉♥t✐❧

✐t t♦✉❝❤❡s t❤❡ ✏❤✐❣❤❡st✑ ♣♦✐♥t ✐♥ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳ ❚❤✐s ♣♦✐♥t ✐s t❤❡♥ t❤❡ s♦❧✉t✐♦♥✳

❜✳ ■❢ b < 0✱ ♠♦✈❡ t❤❡ r✉❧❡r ✐♥ t❤❡ ♦♣♣♦s✐t❡ ❞✐r❡❝t✐♦♥ t♦ ❣❡t t❤❡ s♦❧✉t✐♦♥ ❛t t❤❡ ✏❧♦✇❡st✑ ♣♦✐♥t ✐♥ t❤❡

❢❡❛s✐❜❧❡ r❡❣✐♦♥✳

❝✳ ■❢ b = 0✱ ❝❤❡❝❦ t❤❡ s✐❣♥ ♦❢ a

✶✳ ■❢ a < 0 ♠♦✈❡ t❤❡ r✉❧❡r t♦ t❤❡ ✏❧❡❢t♠♦st✑ ❢❡❛s✐❜❧❡ ♣♦✐♥t✳ ❚❤✐s ♣♦✐♥t ✐s t❤❡♥ t❤❡ s♦❧✉t✐♦♥✳

✷✳ ■❢ a > 0 ♠♦✈❡ t❤❡ r✉❧❡r t♦ t❤❡ ✏r✐❣❤t♠♦st✑ ❢❡❛s✐❜❧❡ ♣♦✐♥t✳ ❚❤✐s ♣♦✐♥t ✐s t❤❡♥ t❤❡ s♦❧✉t✐♦♥✳

❊①❡r❝✐s❡ ✻✳✶✿ Pr✐③❡s✦

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✾✶✳✮

❆s ♣❛rt ♦❢ t❤❡✐r ♦♣❡♥✐♥❣ s♣❡❝✐❛❧s✱ ❛ ❢✉r♥✐t✉r❡ st♦r❡ ❤❛s ♣r♦♠✐s❡❞ t♦ ❣✐✈❡ ❛✇❛② ❛t ❧❡❛st ✹✵ ♣r✐③❡s

✇✐t❤ ❛ t♦t❛❧ ✈❛❧✉❡ ♦❢ ❛t ❧❡❛st ❘✷ ✵✵✵✳ ❚❤❡ ♣r✐③❡s ❛r❡ ❦❡tt❧❡s ❛♥❞ t♦❛st❡rs✳

✶✳ ■❢ t❤❡ ❝♦♠♣❛♥② ❞❡❝✐❞❡s t❤❛t t❤❡r❡ ✇✐❧❧ ❜❡ ❛t ❧❡❛st ✶✵ ♦❢ ❡❛❝❤ ♣r✐③❡✱ ✇r✐t❡ ❞♦✇♥ t✇♦ ♠♦r❡

✐♥❡q✉❛❧✐t✐❡s ❢r♦♠ t❤❡s❡ ❝♦♥str❛✐♥ts✳

✷✳ ■❢ t❤❡ ❝♦st ♦❢ ♠❛♥✉❢❛❝t✉r✐♥❣ ❛ ❦❡tt❧❡ ✐s ❘✻✵ ❛♥❞ ❛ t♦❛st❡r ✐s ❘✺✵✱ ✇r✐t❡ ❞♦✇♥ ❛♥ ♦❜❥❡❝t✐✈❡

❢✉♥❝t✐♦♥ C ✇❤✐❝❤ ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❝♦st t♦ t❤❡ ❝♦♠♣❛♥② ♦❢ ❜♦t❤ ❦❡tt❧❡s ❛♥❞

t♦❛st❡rs✳

✸✳ ❙❦❡t❝❤ t❤❡ ❣r❛♣❤ ♦❢ t❤❡ ❢❡❛s✐❜✐❧✐t② r❡❣✐♦♥ t❤❛t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ ❛❧❧ t❤❡ ♣♦ss✐❜❧❡

❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❦❡tt❧❡s ❛♥❞ t♦❛st❡rs t❤❛t ❤♦♥♦✉r t❤❡ ♣r♦♠✐s❡s ♦❢ t❤❡ ❝♦♠♣❛♥②✳

✹✳ ❍♦✇ ♠❛♥② ♦❢ ❡❛❝❤ ♣r✐③❡ ✇✐❧❧ r❡♣r❡s❡♥t t❤❡ ❝❤❡❛♣❡st ♦♣t✐♦♥ ❢♦r t❤❡ ❝♦♠♣❛♥②❄

✺✳ ❍♦✇ ♠✉❝❤ ✇✐❧❧ t❤✐s ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❦❡tt❧❡s ❛♥❞ t♦❛st❡rs ❝♦st❄

❊①❡r❝✐s❡ ✻✳✷✿ ❙❡❛r❝❤ ▲✐♥❡ ▼❡t❤♦❞

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✾✷✳✮

❆s ❛ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥♥❡r ❛t ❛ ❢❛❝t♦r② ♠❛♥✉❢❛❝t✉r✐♥❣ ❧❛✇♥ ❝✉tt❡rs ②♦✉r ❥♦❜ ✇✐❧❧ ❜❡ t♦ ❛❞✈✐s❡ t❤❡

♠❛♥❛❣❡♠❡♥t ♦♥ ❤♦✇ ♠❛♥② ♦❢ ❡❛❝❤ ♠♦❞❡❧ s❤♦✉❧❞ ❜❡ ♣r♦❞✉❝❡❞ ♣❡r ✇❡❡❦ ✐♥ ♦r❞❡r t♦ ♠❛①✐♠✐s❡ t❤❡

♣r♦✜t ♦♥ t❤❡ ❧♦❝❛❧ ♣r♦❞✉❝t✐♦♥✳ ❚❤❡ ❢❛❝t♦r② ✐s ♣r♦❞✉❝✐♥❣ t✇♦ t②♣❡s ♦❢ ❧❛✇♥ ❝✉tt❡rs✿ ◗✉❛❞r❛♥t ❛♥❞

P❡♥t❛❣♦♥✳ ❚✇♦ ♦❢ t❤❡ ♣r♦❞✉❝t✐♦♥ ♣r♦❝❡ss❡s t❤❛t t❤❡ ❧❛✇♥ ❝✉tt❡rs ♠✉st ❣♦ t❤r♦✉❣❤ ❛r❡✿ ❜♦❞②✇♦r❦

❛♥❞ ❡♥❣✐♥❡ ✇♦r❦✳

• ❚❤❡ ❢❛❝t♦r② ❝❛♥♥♦t ♦♣❡r❛t❡ ❢♦r ❧❡ss t❤❛♥ ✸✻✵ ❤♦✉rs ♦♥ ❡♥❣✐♥❡ ✇♦r❦ ❢♦r t❤❡ ❧❛✇♥ ❝✉tt❡rs✳

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✾✼✴✶✳✶✴❃✳

✽✾

• ❚❤❡ ❢❛❝t♦r② ❤❛s ❛ ♠❛①✐♠✉♠ ❝❛♣❛❝✐t② ♦❢ ✹✽✵ ❤♦✉rs ❢♦r ❜♦❞②✇♦r❦ ❢♦r t❤❡ ❧❛✇♥ ❝✉tt❡rs✳

• ❍❛❧❢ ❛♥ ❤♦✉r ♦❢ ❡♥❣✐♥❡ ✇♦r❦ ❛♥❞ ❤❛❧❢ ❛♥ ❤♦✉r ♦❢ ❜♦❞②✇♦r❦ ✐s r❡q✉✐r❡❞ t♦ ♣r♦❞✉❝❡ ♦♥❡ ◗✉❛❞r❛♥t✳

• ❚❤❡ r❛t✐♦ ♦❢ P❡♥t❛❣♦♥ ❧❛✇♥ ❝✉tt❡rs t♦ ◗✉❛❞r❛♥t ❧❛✇♥ ❝✉tt❡rs ♣r♦❞✉❝❡❞ ♣❡r ✇❡❡❦ ♠✉st ❜❡ ❛t

❧❡❛st ✸✿✷✳

• ❆ ♠✐♥✐♠✉♠ ♦❢ ✷✵✵ ◗✉❛❞r❛♥t ❧❛✇♥ ❝✉tt❡rs ♠✉st ❜❡ ♣r♦❞✉❝❡❞ ♣❡r ✇❡❡❦✳

▲❡t t❤❡ ♥✉♠❜❡r ♦❢ ◗✉❛❞r❛♥t ❧❛✇♥ ❝✉tt❡rs ♠❛♥✉❢❛❝t✉r❡❞ ✐♥ ❛ ✇❡❡❦ ❜❡ x✳

▲❡t t❤❡ ♥✉♠❜❡r ♦❢ P❡♥t❛❣♦♥ ❧❛✇♥ ❝✉tt❡rs ♠❛♥✉❢❛❝t✉r❡❞ ✐♥ ❛ ✇❡❡❦ ❜❡ y✳

❚✇♦ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❛r❡✿

x ≥ 200

✭✻✳✾✮

3x + 2y ≥ 2 160

✭✻✳✶✵✮

✶✳ ❲r✐t❡ ❞♦✇♥ t❤❡ r❡♠❛✐♥✐♥❣ ❝♦♥str❛✐♥ts ✐♥ t❡r♠s ♦❢ x ❛♥❞ y t♦ r❡♣r❡s❡♥t t❤❡ ❛❜♦✈❡♠❡♥t✐♦♥❡❞

✐♥❢♦r♠❛t✐♦♥✳

✷✳ ❯s❡ ❣r❛♣❤ ♣❛♣❡r t♦ r❡♣r❡s❡♥t t❤❡ ❝♦♥str❛✐♥ts ❣r❛♣❤✐❝❛❧❧②✳

✸✳ ❈❧❡❛r❧② ✐♥❞✐❝❛t❡ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥ ❜② s❤❛❞✐♥❣ ✐t✳

✹✳ ■❢ t❤❡ ♣r♦✜t ♦♥ ♦♥❡ ◗✉❛❞r❛♥t ❧❛✇♥ ❝✉tt❡r ✐s ❘✶ ✷✵✵ ❛♥❞ t❤❡ ♣r♦✜t ♦♥ ♦♥❡ P❡♥t❛❣♦♥ ❧❛✇♥

❝✉tt❡r ✐s ❘✹✵✵✱ ✇r✐t❡ ❞♦✇♥ ❛♥ ❡q✉❛t✐♦♥ t❤❛t ✇✐❧❧ r❡♣r❡s❡♥t t❤❡ ♣r♦✜t ♦♥ t❤❡ ❧❛✇♥ ❝✉tt❡rs✳

✺✳ ❯s✐♥❣ ❛ s❡❛r❝❤ ❧✐♥❡ ❛♥❞ ②♦✉r ❣r❛♣❤✱ ❞❡t❡r♠✐♥❡ t❤❡ ♥✉♠❜❡r ♦❢ ◗✉❛❞r❛♥t ❛♥❞ P❡♥t❛❣♦♥ ❧❛✇♥

❝✉tt❡rs t❤❛t ✇✐❧❧ ②✐❡❧❞ ❛ ♠❛①✐♠✉♠ ♣r♦✜t✳

✻✳ ❉❡t❡r♠✐♥❡ t❤❡ ♠❛①✐♠✉♠ ♣r♦✜t ♣❡r ✇❡❡❦✳

✻✳✷✳✷ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

✶✳ P♦❧❦❛❞♦ts ✐s ❛ s♠❛❧❧ ❝♦♠♣❛♥② t❤❛t ♠❛❦❡s t✇♦ t②♣❡s ♦❢ ❝❛r❞s✱ t②♣❡ ❳ ❛♥❞ t②♣❡ ❨✳ ❲✐t❤ t❤❡ ❛✈❛✐❧❛❜❧❡

❧❛❜♦✉r ❛♥❞ ♠❛t❡r✐❛❧✱ t❤❡ ❝♦♠♣❛♥② ❝❛♥ ♠❛❦❡ ♥♦t ♠♦r❡ t❤❛♥ ✶✺✵ ❝❛r❞s ♦❢ t②♣❡ ❳ ❛♥❞ ♥♦t ♠♦r❡ t❤❛♥

✶✷✵ ❝❛r❞s ♦❢ t②♣❡ ❨ ♣❡r ✇❡❡❦✳ ❆❧t♦❣❡t❤❡r t❤❡② ❝❛♥♥♦t ♠❛❦❡ ♠♦r❡ t❤❛♥ ✷✵✵ ❝❛r❞s ♣❡r ✇❡❡❦✳ ❚❤❡r❡ ✐s

❛♥ ♦r❞❡r ❢♦r ❛t ❧❡❛st ✹✵ t②♣❡ ❳ ❝❛r❞s ❛♥❞ ✶✵ t②♣❡ ❨ ❝❛r❞s ♣❡r ✇❡❡❦✳ P♦❧❦❛❞♦ts ♠❛❦❡s ❛ ♣r♦✜t ♦❢ ❘✺

❢♦r ❡❛❝❤ t②♣❡ ❳ ❝❛r❞ s♦❧❞ ❛♥❞ ❘✶✵ ❢♦r ❡❛❝❤ t②♣❡ ❨ ❝❛r❞✳ ▲❡t t❤❡ ♥✉♠❜❡r ♦❢ t②♣❡ ❳ ❝❛r❞s ❜❡ ① ❛♥❞ t❤❡

♥✉♠❜❡r ♦❢ t②♣❡ ❨ ❝❛r❞s ❜❡ ②✱ ♠❛♥✉❢❛❝t✉r❡❞ ♣❡r ✇❡❡❦✳

❛✳ ❖♥❡ ♦❢ t❤❡ ❝♦♥str❛✐♥t ✐♥❡q✉❛❧✐t✐❡s ✇❤✐❝❤ r❡♣r❡s❡♥ts t❤❡ r❡str✐❝t✐♦♥s ❛❜♦✈❡ ✐s x ≤ 150✳ ❲r✐t❡ t❤❡

♦t❤❡r ❝♦♥str❛✐♥t ✐♥❡q✉❛❧✐t✐❡s✳

❜✳ ❘❡♣r❡s❡♥t t❤❡ ❝♦♥str❛✐♥ts ❣r❛♣❤✐❝❛❧❧② ❛♥❞ s❤❛❞❡ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳

❝✳ ❲r✐t❡ t❤❡ ❡q✉❛t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤❡ ♣r♦✜t P ✭t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✮✱ ✐♥ t❡r♠s ♦❢ x ❛♥❞ y✳

❞✳ ❖♥ ②♦✉r ❣r❛♣❤✱ ❞r❛✇ ❛ str❛✐❣❤t ❧✐♥❡ ✇❤✐❝❤ ✇✐❧❧ ❤❡❧♣ ②♦✉ t♦ ❞❡t❡r♠✐♥❡ ❤♦✇ ♠❛♥② ♦❢ ❡❛❝❤ t②♣❡ ♠✉st

❜❡ ♠❛❞❡ ✇❡❡❦❧② t♦ ♣r♦❞✉❝❡ t❤❡ ♠❛①✐♠✉♠ P

❡✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♠❛①✐♠✉♠ ✇❡❡❦❧② ♣r♦✜t✳

✷✳ ❆ ❜r✐❝❦✇♦r❦s ♣r♦❞✉❝❡s ✏❢❛❝❡ ❜r✐❝❦s✧ ❛♥❞ ✏❝❧✐♥❦❡rs✧✳ ❇♦t❤ t②♣❡s ♦❢ ❜r✐❝❦s ❛r❡ ♣r♦❞✉❝❡❞ ❛♥❞ s♦❧❞ ✐♥

❜❛t❝❤❡s ♦❢ ❛ t❤♦✉s❛♥❞✳ ❋❛❝❡ ❜r✐❝❦s ❛r❡ s♦❧❞ ❛t ❘✶✺✵ ♣❡r t❤♦✉s❛♥❞✱ ❛♥❞ ❝❧✐♥❦❡rs ❛t ❘✶✵✵ ♣❡r t❤♦✉s❛♥❞✱

✇❤❡r❡ ❛♥ ✐♥❝♦♠❡ ♦❢ ❛t ❧❡❛st ❘✾✱✵✵✵ ♣❡r ♠♦♥t❤ ✐s r❡q✉✐r❡❞ t♦ ❝♦✈❡r ❝♦sts✳ ❚❤❡ ❜r✐❝❦✇♦r❦s ✐s ❛❜❧❡ t♦

♣r♦❞✉❝❡ ❛t ♠♦st ✹✵✱✵✵✵ ❢❛❝❡ ❜r✐❝❦s ❛♥❞ ✾✵✱✵✵✵ ❝❧✐♥❦❡rs ♣❡r ♠♦♥t❤✱ ❛♥❞ ❤❛s tr❛♥s♣♦rt ❢❛❝✐❧✐t✐❡s t♦ ❞❡❧✐✈❡r

❛t ♠♦st ✶✵✵✱✵✵✵ ❜r✐❝❦s ♣❡r ♠♦♥t❤✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❝❧✐♥❦❡rs ♣r♦❞✉❝❡❞ ♠✉st ❜❡ ❛t ❧❡❛st t❤❡ s❛♠❡ ♥✉♠❜❡r

♦❢ ❢❛❝❡ ❜r✐❝❦s ♣r♦❞✉❝❡❞✳ ▲❡t t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝❡ ❜r✐❝❦s ✐♥ t❤♦✉s❛♥❞s ❜❡ x✱ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ ❝❧✐♥❦❡rs

✐♥ t❤♦✉s❛♥❞s ❜❡ y✳

❛✳ ▲✐st ❛❧❧ t❤❡ ❝♦♥str❛✐♥ts✳

✾✵

❈❍❆P❚❊❘ ✻✳ ▲■◆❊❆❘ P❘❖●❘❆▼▼■◆●

❜✳ ●r❛♣❤ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳

❝✳ ■❢ t❤❡ s❛❧❡ ♦❢ ❢❛❝❡ ❜r✐❝❦s ②✐❡❧❞s ❛ ♣r♦✜t ♦❢ ❘✷✺ ♣❡r t❤♦✉s❛♥❞ ❛♥❞ ❝❧✐♥❦❡rs ❘✹✺ ♣❡r t❤♦✉s❛♥❞✱ ✉s❡

②♦✉r ❣r❛♣❤ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ♠❛①✐♠✉♠ ♣r♦✜t✳

❞✳ ■❢ t❤❡ ♣r♦✜t ♠❛r❣✐♥s ♦♥ ❢❛❝❡ ❜r✐❝❦s ❛♥❞ ❝❧✐♥❦❡rs ❛r❡ ✐♥t❡r❝❤❛♥❣❡❞✱ ✉s❡ ②♦✉r ❣r❛♣❤ t♦ ❞❡t❡r♠✐♥❡

t❤❡ ♠❛①✐♠✉♠ ♣r♦✜t✳

✸✳ ❆ s♠❛❧❧ ❝❡❧❧ ♣❤♦♥❡ ❝♦♠♣❛♥② ♠❛❦❡s t✇♦ t②♣❡s ♦❢ ❝❡❧❧ ♣❤♦♥❡s✿ ❊❛s②❤❡❛r ❛♥❞ ▲♦♥❣t❛❧❦✳ Pr♦❞✉❝t✐♦♥

✜❣✉r❡s ❛r❡ ❝❤❡❝❦❡❞ ✇❡❡❦❧②✳ ❆t ♠♦st✱ ✹✷ ❊❛s②❤❡❛r ❛♥❞ ✻✵ ▲♦♥❣t❛❧❦ ♣❤♦♥❡s ❝❛♥ ❜❡ ♠❛♥✉❢❛❝t✉r❡❞ ❡❛❝❤

✇❡❡❦✳ ❆t ❧❡❛st ✸✵ ❝❡❧❧ ♣❤♦♥❡s ♠✉st ❜❡ ♣r♦❞✉❝❡❞ ❡❛❝❤ ✇❡❡❦ t♦ ❝♦✈❡r ❝♦sts✳ ■♥ ♦r❞❡r ♥♦t t♦ ✢♦♦❞ t❤❡

♠❛r❦❡t✱ t❤❡ ♥✉♠❜❡r ♦❢ ❊❛s②❤❡❛r ♣❤♦♥❡s ❝❛♥♥♦t ❜❡ ♠♦r❡ t❤❛♥ t✇✐❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ▲♦♥❣t❛❧❦ ♣❤♦♥❡s✳

■t t❛❦❡s 2 ❤♦✉r t♦ ❛ss❡♠❜❧❡ ❛♥ ❊❛s②❤❡❛r ♣❤♦♥❡ ❛♥❞ 1 ❤♦✉r t♦ ♣✉t t♦❣❡t❤❡r ❛ ▲♦♥❣t❛❧❦ ♣❤♦♥❡✳ ❚❤❡

3

2

tr❛❞❡ ✉♥✐♦♥s ♦♥❧② ❛❧❧♦✇ ❢♦r ❛ ✺✵✲❤♦✉r ✇❡❡❦✳ ▲❡t x ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❊❛s②❤❡❛r ♣❤♦♥❡s ❛♥❞ y ❜❡ t❤❡

♥✉♠❜❡r ♦❢ ▲♦♥❣t❛❧❦ ♣❤♦♥❡s ♠❛♥✉❢❛❝t✉r❡❞ ❡❛❝❤ ✇❡❡❦✳

❛✳ ❚✇♦ ♦❢ t❤❡ ❝♦♥str❛✐♥ts ❛r❡✿

0 ≤ x ≤ 42

and

0 ≤ y ≤ 60

✭✻✳✶✶✮

❲r✐t❡ ❞♦✇♥ t❤❡ ♦t❤❡r t❤r❡❡ ❝♦♥str❛✐♥ts✳

❜✳ ❉r❛✇ ❛ ❣r❛♣❤ t♦ r❡♣r❡s❡♥t t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥

❝✳ ■❢ t❤❡ ♣r♦✜t ♦♥ ❛♥ ❊❛s②❤❡❛r ♣❤♦♥❡ ✐s ❘✷✷✺ ❛♥❞ t❤❡ ♣r♦✜t ♦♥ ❛ ▲♦♥❣t❛❧❦ ✐s ❘✼✺✱ ❞❡t❡r♠✐♥❡ t❤❡

♠❛①✐♠✉♠ ♣r♦✜t ♣❡r ✇❡❡❦✳

✹✳ ❍❛✐r ❢♦r ❆❢r✐❝❛ ✐s ❛ ✜r♠ t❤❛t s♣❡❝✐❛❧✐s❡s ✐♥ ♠❛❦✐♥❣ t✇♦ ❦✐♥❞s ♦❢ ✉♣✲♠❛r❦❡t s❤❛♠♣♦♦✱ ●❧♦✇❤❛✐r ❛♥❞

▲♦♥❣❝✉r❧s✳ ❚❤❡② ♠✉st ♣r♦❞✉❝❡ ❛t ❧❡❛st t✇♦ ❝❛s❡s ♦❢ ●❧♦✇❤❛✐r ❛♥❞ ♦♥❡ ❝❛s❡ ♦❢ ▲♦♥❣❝✉r❧s ♣❡r ❞❛② t♦

st❛② ✐♥ t❤❡ ♠❛r❦❡t✳ ❉✉❡ t♦ ❛ ❧✐♠✐t❡❞ s✉♣♣❧② ♦❢ ❝❤❡♠✐❝❛❧s✱ t❤❡② ❝❛♥♥♦t ♣r♦❞✉❝❡ ♠♦r❡ t❤❛♥ ✽ ❝❛s❡s ♦❢

●❧♦✇❤❛✐r ❛♥❞ ✻ ❝❛s❡s ♦❢ ▲♦♥❣❝✉r❧s ♣❡r ❞❛②✳ ■t t❛❦❡s ❤❛❧❢✲❛♥✲❤♦✉r t♦ ♣r♦❞✉❝❡ ♦♥❡ ❝❛s❡ ♦❢ ●❧♦✇❤❛✐r ❛♥❞

♦♥❡ ❤♦✉r t♦ ♣r♦❞✉❝❡ ❛ ❝❛s❡ ♦❢ ▲♦♥❣❝✉r❧s✱ ❛♥❞ ❞✉❡ t♦ r❡str✐❝t✐♦♥s ❜② t❤❡ ✉♥✐♦♥s✱ t❤❡ ♣❧❛♥t ♠❛② ♦♣❡r❛t❡

❢♦r ❛t ♠♦st ✼ ❤♦✉rs ♣❡r ❞❛②✳ ❚❤❡ ✇♦r❦❢♦r❝❡ ❛t ❍❛✐r ❢♦r ❆❢r✐❝❛✱ ✇❤✐❝❤ ✐s st✐❧❧ ✐♥ tr❛✐♥✐♥❣✱ ❝❛♥ ♦♥❧②

♣r♦❞✉❝❡ ❛ ♠❛①✐♠✉♠ ♦❢ ✶✵ ❝❛s❡s ♦❢ s❤❛♠♣♦♦ ♣❡r ❞❛②✳ ▲❡t x ❜❡ t❤❡ ♥✉♠❜❡r ♦❢ ❝❛s❡s ♦❢ ●❧♦✇❤❛✐r ❛♥❞ y

t❤❡ ♥✉♠❜❡r ♦❢ ❝❛s❡s ♦❢ ▲♦♥❣❝✉r❧s ♣r♦❞✉❝❡❞ ♣❡r ❞❛②✳

❛✳ ❲r✐t❡ ❞♦✇♥ t❤❡ ✐♥❡q✉❛❧✐t✐❡s t❤❛t r❡♣r❡s❡♥t ❛❧❧ t❤❡ ❝♦♥str❛✐♥ts✳

❜✳ ❙❦❡t❝❤ t❤❡ ❢❡❛s✐❜❧❡ r❡❣✐♦♥✳

❝✳ ■❢ t❤❡ ♣r♦✜t ♦♥ ❛ ❝❛s❡ ♦❢ ●❧♦✇❤❛✐r ✐s ❘✹✵✵ ❛♥❞ t❤❡ ♣r♦✜t ♦♥ ❛ ❝❛s❡ ♦❢ ▲♦♥❣❝✉r❧s ✐s ❘✸✵✵✱ ❞❡t❡r♠✐♥❡

t❤❡ ♠❛①✐♠✉♠ ♣r♦✜t t❤❛t ❍❛✐r ❢♦r ❆❢r✐❝❛ ❝❛♥ ♠❛❦❡ ♣❡r ❞❛②✳

✺✳ ❆ tr❛♥s♣♦rt ❝♦♥tr❛❝t❡r ❤❛s ✻ ✺✲t♦♥ tr✉❝❦s ❛♥❞ ✽ ✸✲t♦♥ tr✉❝❦s✳ ❍❡ ♠✉st ❞❡❧✐✈❡r ❛t ❧❡❛st ✶✷✵ t♦♥s ♦❢ s❛♥❞

♣❡r ❞❛② t♦ ❛ ❝♦♥str✉❝t✐♦♥ s✐t❡✱ ❜✉t ❤❡ ♠❛② ♥♦t ❞❡❧✐✈❡r ♠♦r❡ t❤❛♥ ✶✽✵ t♦♥s ♣❡r ❞❛②✳ ❚❤❡ ✺✲t♦♥ tr✉❝❦s

❝❛♥ ❡❛❝❤ ♠❛❦❡ t❤r❡❡ tr✐♣s ♣❡r ❞❛② ❛t ❛ ❝♦st ♦❢ ❘✸✵ ♣❡r tr✐♣✱ ❛♥❞ t❤❡ ✸✲t♦♥ tr✉❝❦s ❝❛♥ ❡❛❝❤ ♠❛❦❡ ❢♦✉r

tr✐♣s ♣❡r ❞❛② ❛t ❛ ❝♦st ♦❢ ❘✶✷✵ ♣❡r tr✐♣✳ ❍♦✇ ♠✉st t❤❡ ❝♦♥tr❛❝t❡r ✉t✐❧✐s❡ ❤✐s tr✉❝❦s s♦ t❤❛t ❤❡ ❤❛s

♠✐♥✐♠✉♠ ❡①♣❡♥s❡❄

✾✶

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✻

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✻✳✶ ✭♣✳ ✽✽✮

❙t❡♣ ✶✳ ▲❡t t❤❡ ♥✉♠❜❡r ♦❢ ❦❡tt❧❡s ❜❡ xk ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ t♦❛st❡rs ❜❡ yt ❛♥❞ ✇r✐t❡ ❞♦✇♥ t✇♦ ❝♦♥str❛✐♥ts ❛♣❛rt

❢r♦♠ xk ≥ 0 ❛♥❞ yt ≥ 0 t❤❛t ♠✉st ❜❡ ❛❞❤❡r❡❞ t♦✳

❙t❡♣ ✷✳ ❙✐♥❝❡ t❤❡r❡ ✇✐❧❧ ❜❡ ❛t ❧❡❛st ✶✵ ♦❢ ❡❛❝❤ ♣r✐③❡ ✇❡ ❝❛♥ ✇r✐t❡✿

xk ≥ 10

✭✻✳✶✷✮

❛♥❞

yt ≥ 10

✭✻✳✶✸✮

❆❧s♦ t❤❡ st♦r❡ ❤❛s ♣r♦♠✐s❡❞ t♦ ❣✐✈❡ ❛✇❛② ❛t ❧❡❛st ✹✵ ♣r✐③❡s ✐♥ t♦t❛❧✳ ❚❤❡r❡❢♦r❡✿

xk + yt ≥ 40

✭✻✳✶✹✮

❙t❡♣ ✸✳ ❚❤❡ ❝♦st ♦❢ ♠❛♥✉❢❛❝t✉r✐♥❣ ❛ ❦❡tt❧❡ ✐s ❘✻✵ ❛♥❞ ❛ t♦❛st❡r ✐s ❘✺✵✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝♦st t❤❡ t♦t❛❧ ❝♦st C ✐s✿

C = 60xk + 50yt

✭✻✳✶✺✮

❙t❡♣ ✹✳

❋✐❣✉r❡ ✻✳✸

❙t❡♣ ✺✳ ❋r♦♠ t❤❡ ❣r❛♣❤✱ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✈❡rt❡① ❆ ❛r❡ ✭✸✵✱✶✵✮ ❛♥❞ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ ✈❡rt❡① ❇ ❛r❡ ✭✶✵✱✸✵✮✳

❙t❡♣ ✻✳ ❚❤❡ s❡❛❝❤ ❧✐♥❡ ✐s t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥✳ ❚❤❛t ✐s✱ ✐❢ t❤❡ ❡q✉❛t✐♦♥ C = 60x + 50y ✐s ♥♦✇

✇r✐tt❡♥ ✐♥ t❤❡ st❛♥❞❛r❞ ❢♦r♠ y = ...✱ t❤❡♥ t❤❡ ❣r❛❞✐❡♥t ✐s✿

6

m = − ,

✭✻✳✶✻✮

5

✇❤✐❝❤ ✐s s❤♦✇♥ ✇✐t❤ t❤❡ ❜r♦❦❡♥ ❧✐♥❡ ♦♥ t❤❡ ❣r❛♣❤✳

❋✐❣✉r❡ ✻✳✹

❙t❡♣ ✼✳ ❆t ✈❡rt❡① ❆✱ t❤❡ ❝♦st ✐s✿

C

=

60xk + 50yt

=

60 (30) + 50 (10)

✭✻✳✶✼✮

=

1800 + 500

=

2300

✾✷

❈❍❆P❚❊❘ ✻✳ ▲■◆❊❆❘ P❘❖●❘❆▼▼■◆●

❆t ✈❡rt❡① ❇✱ t❤❡ ❝♦st ✐s✿

C

=

60xk + 50yt

=

60 (10) + 50 (30)

✭✻✳✶✽✮

=

600 + 1500

=

2100

❙t❡♣ ✽✳ ❚❤❡ ❝❤❡❛♣❡st ❝♦♠❜✐♥❛t✐♦♥ ♦❢ ♣r✐③❡s ✐s ✶✵ ❦❡tt❧❡s ❛♥❞ ✸✵ t♦❛st❡rs✱ ❝♦st✐♥❣ t❤❡ ❝♦♠♣❛♥② ❘✷ ✶✵✵✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✻✳✷ ✭♣✳ ✽✽✮

❙t❡♣ ✶✳

1

1

x +

y ≤ 480

✭✻✳✶✾✮

2

5

y

3

≥

✭✻✳✷✵✮

x

2

❙t❡♣ ✷✳

❋✐❣✉r❡ ✻✳✺

❙t❡♣ ✸✳

P = 1 200x + 400y

✭✻✳✷✶✮

❙t❡♣ ✹✳ ❇② ♠♦✈✐♥❣ t❤❡ s❡❛r❝❤ ❧✐♥❡ ✉♣✇❛r❞s✱ ✇❡ s❡❡ t❤❛t t❤❡ ♣♦✐♥t ♦❢ ♠❛①✐♠✉♠ ♣r♦✜t ✐s ❛t ✭✻✵✵✱✾✵✵✮✳ ❚❤❡r❡❢♦r❡

P = 1 200 (600) + 400 (900)

✭✻✳✷✷✮

P = R1 080 000

✭✻✳✷✸✮

❈❤❛♣t❡r ✼

●❡♦♠❡tr②

✼✳✶ ■♥tr♦❞✉❝t✐♦♥✱ ❈✐r❝❧❡ ●❡♦♠❡tr②✶

✼✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

✼✳✶✳✶✳✶ ❉✐s❝✉ss✐♦♥ ✿ ❉✐s❝✉ss t❤❡s❡ ❘❡s❡❛r❝❤ ❚♦♣✐❝s

❘❡s❡❛r❝❤ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣❡♦♠❡tr✐❝❛❧ ✐❞❡❛s ❛♥❞ ❞❡s❝r✐❜❡ ✐t t♦ ②♦✉r ❣r♦✉♣✿

✶✳ t❛①✐❝❛❜ ❣❡♦♠❡tr②✱

✷✳ s♣❤❡r✐❝❛❧ ❣❡♦♠❡tr②✱

✸✳ ❢r❛❝t❛❧s✱

✹✳ t❤❡ ❑♦❝❤ s♥♦✇✢❛❦❡✳

✼✳✶✳✷ ❈✐r❝❧❡ ●❡♦♠❡tr②

✼✳✶✳✷✳✶ ❚❡r♠✐♥♦❧♦❣②

❚❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ r❡❝❛♣ ♦❢ t❡r♠s t❤❛t ❛r❡ r❡❣✉❧❛r❧② ✉s❡❞ ✇❤❡♥ r❡❢❡rr✐♥❣ t♦ ❝✐r❝❧❡s✳

❛r❝✿ ❆♥ ❛r❝ ✐s ❛ ♣❛rt ♦❢ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡✳

❝❤♦r❞✿ ❆ ❝❤♦r❞ ✐s ❞❡✜♥❡❞ ❛s ❛ str❛✐❣❤t ❧✐♥❡ ❥♦✐♥✐♥❣ t❤❡ ❡♥❞s ♦❢ ❛♥ ❛r❝✳

r❛❞✐✉s✿ ❚❤❡ r❛❞✐✉s✱ r✱ ✐s t❤❡ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ t♦ ❛♥② ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✳

❞✐❛♠❡t❡r✿ ❚❤❡ ❞✐❛♠❡t❡r ✐s ❛ s♣❡❝✐❛❧ ❝❤♦r❞ t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡✳ ❚❤❡ ❞✐❛♠❡t❡r ✐s

t❤❡ str❛✐❣❤t ❧✐♥❡ ❢r♦♠ ❛ ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ t♦ ❛♥♦t❤❡r ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✱ t❤❛t ♣❛ss❡s

t❤r♦✉❣❤ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡✳

s❡❣♠❡♥t✿ ❆ s❡❣♠❡♥t ✐s t❤❡ ♣❛rt ♦❢ t❤❡ ❝✐r❝❧❡ t❤❛t ✐s ❝✉t ♦✛ ❜② ❛ ❝❤♦r❞✳ ❆ ❝❤♦r❞ ❞✐✈✐❞❡s ❛ ❝✐r❝❧❡ ✐♥t♦ t✇♦

s❡❣♠❡♥ts✳

t❛♥❣❡♥t✿ ❆ t❛♥❣❡♥t ✐s ❛ ❧✐♥❡ t❤❛t ♠❛❦❡s ❝♦♥t❛❝t ✇✐t❤ ❛ ❝✐r❝❧❡ ❛t ♦♥❡ ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✳ ✭AB ✐s ❛

t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ❛t ♣♦✐♥t P ✮✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✷✼✴✶✳✶✴❃✳

✾✸

✾✹

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❋✐❣✉r❡ ✼✳✶✿ P❛rts ♦❢ ❛ ❈✐r❝❧❡

✼✳✶✳✷✳✷ ❆①✐♦♠s

❆♥ ❛①✐♦♠ ✐s ❛♥ ❡st❛❜❧✐s❤❡❞ ♦r ❛❝❝❡♣t❡❞ ♣r✐♥❝✐♣❧❡✳ ❋♦r t❤✐s s❡❝t✐♦♥✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛r❡ ❛❝❝❡♣t❡❞ ❛s ❛①✐♦♠s✳

✶✳ ❚❤❡ ❚❤❡♦r❡♠ ♦❢ P②t❤❛❣♦r❛s✱ ✇❤✐❝❤ st❛t❡s t❤❛t t❤❡ sq✉❛r❡ ♦♥ t❤❡ ❤②♣♦t❡♥✉s❡ ♦❢ ❛ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡

✐s ❡q✉❛❧ t♦ t❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡s ♦♥ t❤❡ ♦t❤❡r t✇♦ s✐❞❡s✳ ■♥ [U+25B5]ABC✱ t❤✐s ♠❡❛♥s t❤❛t (AB)2 +

(BC)2 = (AC)2

❋✐❣✉r❡ ✼✳✷✿ ❆ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡

✷✳ ❆ t❛♥❣❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ r❛❞✐✉s✱ ❞r❛✇♥ ❛t t❤❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t ✇✐t❤ t❤❡ ❝✐r❝❧❡✳

✼✳✶✳✷✳✸ ❚❤❡♦r❡♠s ♦❢ t❤❡ ●❡♦♠❡tr② ♦❢ ❈✐r❝❧❡s

❆ t❤❡♦r❡♠ ✐s ❛ ❣❡♥❡r❛❧ ♣r♦♣♦s✐t✐♦♥ t❤❛t ✐s ♥♦t s❡❧❢✲❡✈✐❞❡♥t ❜✉t ✐s ♣r♦✈❡❞ ❜② r❡❛s♦♥✐♥❣ ✭t❤❡s❡ ♣r♦♦❢s ♥❡❡❞ ♥♦t

❜❡ ❧❡❛r♥❡❞ ❢♦r ❡①❛♠✐♥❛t✐♦♥ ♣✉r♣♦s❡s✮✳

❚❤❡♦r❡♠ ✶ ❚❤❡ ❧✐♥❡ ❞r❛✇♥ ❢r♦♠ t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡✱ ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❛ ❝❤♦r❞✱ ❜✐s❡❝ts t❤❡ ❝❤♦r❞✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✸

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O✳ ❉r❛✇ ❛ ❝❤♦r❞ AB ❛♥❞ ❞r❛✇ ❛ ♣❡r♣❡♥❞✐❝✉❧❛r ❧✐♥❡ ❢r♦♠ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡

❝✐r❝❧❡ t♦ ✐♥t❡rs❡❝t t❤❡ ❝❤♦r❞ ❛t ♣♦✐♥t P ✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t AP ❂ BP

✶✳ [U+25B5]OAP ❛♥❞ [U+25B5]OBP ❛r❡ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡s✳

✷✳ OA = OB ❛s ❜♦t❤ ♦❢ t❤❡s❡ ❛r❡ r❛❞✐✐ ❛♥❞ OP ✐s ❝♦♠♠♦♥ t♦ ❜♦t❤ tr✐❛♥❣❧❡s✳

❆♣♣❧② t❤❡ ❚❤❡♦r❡♠ ♦❢ P②t❤❛❣♦r❛s t♦ ❡❛❝❤ tr✐❛♥❣❧❡✱ t♦ ❣❡t✿

OA2

=

OP 2 + AP 2

✭✼✳✶✮

OB2

=

OP 2 + BP 2

✾✺

❍♦✇❡✈❡r✱ OA = OB✳ ❙♦✱

OP 2 + AP 2

=

OP 2 + BP 2

∴ AP 2

=

BP 2

✭✼✳✷✮

and AP

=

BP

❚❤✐s ♠❡❛♥s t❤❛t OP ❜✐s❡❝ts AB✳

❚❤❡♦r❡♠ ✷ ❚❤❡ ❧✐♥❡ ❞r❛✇♥ ❢r♦♠ t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡✱ t❤❛t ❜✐s❡❝ts ❛ ❝❤♦r❞✱ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡

❝❤♦r❞✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✹

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O✳ ❉r❛✇ ❛ ❝❤♦r❞ AB ❛♥❞ ❞r❛✇ ❛ ❧✐♥❡ ❢r♦♠ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ t♦ ❜✐s❡❝t

t❤❡ ❝❤♦r❞ ❛t ♣♦✐♥t P ✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t OP ⊥ AB ■♥ [U+25B5]OAP ❛♥❞ [U+25B5]OBP ✱

✶✳ AP = P B ✭❣✐✈❡♥✮

✷✳ OA = OB ✭r❛❞✐✐✮

✸✳ OP ✐s ❝♦♠♠♦♥ t♦ ❜♦t❤ tr✐❛♥❣❧❡s✳

∴ [U+25B5]OAP ≡ [U+25B5]OBP ✭❙❙❙✮✳

❫

❫

OP A

=

OP B

❫

❫

OP A + OP B

=

180◦

(APB is a str. line)

✭✼✳✸✮

❫

❫

∴OP A =

OP B= 90◦

∴ OP

⊥

AB

❚❤❡♦r❡♠ ✸ ❚❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ ❛ ❝❤♦r❞ ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✺

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✳ ❉r❛✇ ❛ ❝❤♦r❞ AB✳ ❉r❛✇ ❛ ❧✐♥❡ P Q ♣❡r♣❡♥❞✐❝✉❧❛r t♦ AB s✉❝❤ t❤❛t P Q ❜✐s❡❝ts AB ❛t

♣♦✐♥t P ✳ ❉r❛✇ ❧✐♥❡s AQ ❛♥❞ BQ✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t Q ✐s t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡✱ ❜② s❤♦✇✐♥❣

t❤❛t AQ = BQ✳ ■♥ [U+25B5]OAP ❛♥❞ [U+25B5]OBP ✱

✶✳ AP = P B ✭❣✐✈❡♥✮

✷✳ ∠QP A = ∠QP B ✭QP ⊥ AB✮

✸✳ QP ✐s ❝♦♠♠♦♥ t♦ ❜♦t❤ tr✐❛♥❣❧❡s✳

✾✻

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

∴ [U+25B5]QAP ≡ [U+25B5]QBP ✭❙❆❙✮✳ ❋r♦♠ t❤✐s✱ QA = QB✳ ❙✐♥❝❡ t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡ ✐s t❤❡ ♦♥❧②

♣♦✐♥t ✐♥s✐❞❡ ❛ ❝✐r❝❧❡ t❤❛t ❤❛s ♣♦✐♥ts ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ❛t ❛♥ ❡q✉❛❧ ❞✐st❛♥❝❡ ❢r♦♠ ✐t✱ Q ♠✉st ❜❡ t❤❡ ❝❡♥tr❡

♦❢ t❤❡ ❝✐r❝❧❡✳

✼✳✶✳✷✳✸✳✶ ❈✐r❝❧❡s ■

✶✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ x✿

❋✐❣✉r❡ ✼✳✻

❋✐❣✉r❡ ✼✳✼

❚❤❡♦r❡♠ ✹ ❚❤❡ ❛♥❣❧❡ s✉❜t❡♥❞❡❞ ❜② ❛♥ ❛r❝ ❛t t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡ ✐s ❞♦✉❜❧❡ t❤❡ s✐③❡ ♦❢ t❤❡ ❛♥❣❧❡ s✉❜t❡♥❞❡❞

❜② t❤❡ s❛♠❡ ❛r❝ ❛t t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ t❤❡ ❝✐r❝❧❡✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✽

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O ❛♥❞ ✇✐t❤ A ❛♥❞ B ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✳ ❉r❛✇ ❛ ❝❤♦r❞ AB✳ ❉r❛✇ r❛❞✐✐

OA ❛♥❞ OB✳ ❙❡❧❡❝t ❛♥② ♣♦✐♥t P ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ t❤❡ ❝✐r❝❧❡✳ ❉r❛✇ ❧✐♥❡s P A ❛♥❞ P B✳ ❉r❛✇ P O ❛♥❞

❫

❫

❫

❫

❫

❡①t❡♥❞ t♦ R✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t AOB= 2· AP B✳AOR=P AO + AP O ✭❡①t❡r✐♦r ❛♥❣❧❡ ❂ s✉♠ ♦❢

❫

❫

❫

❫

✐♥t❡r✐♦r ♦♣♣✳ ❛♥❣❧❡s✮ ❇✉t✱ P AO=AP O ✭[U+25B5]AOP ✐s ❛♥ ✐s♦s❝❡❧❡s [U+25B5]✮ ∴AOR= 2 AP O ❙✐♠✐❧❛r❧②✱

❫

❫

BOR= 2 BP O✳ ❙♦✱

❫

❫

❫

AOB

=

AOR + BOR

❫

❫

=

2 AP O +2 BP O

❫

❫

✭✼✳✹✮

=

2

AP O + BP O

❫

=

2

AP B

✾✼

✼✳✶✳✷✳✸✳✷ ❈✐r❝❧❡s ■■

✶✳ ❋✐♥❞ t❤❡ ❛♥❣❧❡s ✭a t♦ f✮ ✐♥❞✐❝❛t❡❞ ✐♥ ❡❛❝❤ ❞✐❛❣r❛♠✿

❋✐❣✉r❡ ✼✳✾

❚❤❡♦r❡♠ ✺ ❚❤❡ ❛♥❣❧❡s s✉❜t❡♥❞❡❞ ❜② ❛ ❝❤♦r❞ ❛t t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡ ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢ t❤❡ ❝❤♦r❞

❛r❡ ❡q✉❛❧✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✶✵

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O✳ ❉r❛✇ ❛ ❝❤♦r❞ AB✳ ❙❡❧❡❝t ❛♥② ♣♦✐♥ts P ❛♥❞ Q ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢

t❤❡ ❝✐r❝❧❡✱ s✉❝❤ t❤❛t ❜♦t❤ P ❛♥❞ Q ❛r❡ ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢ t❤❡ ❝❤♦r❞✳ ❉r❛✇ ❧✐♥❡s P A✱ P B✱ QA ❛♥❞ QB✳

❫

❫

❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t AQB=AP B✳

❫

❫

AOB

=

2 AQB ∠ at centre = twice ∠at circumference

❫

❫

and AOB

=

2 AP B ∠ at centre = twice ∠ at circumference

❫

❫

✭✼✳✺✮

∴ 2 AQB

=

2 AP B

❫

❫

∴AQB

=

AP B

❚❤❡♦r❡♠ ✻ ✭❈♦♥✈❡rs❡ ♦❢ ❚❤❡♦r❡♠ ✮ ■❢ ❛ ❧✐♥❡ s❡❣♠❡♥t s✉❜t❡♥❞s ❡q✉❛❧ ❛♥❣❧❡s ❛t t✇♦ ♦t❤❡r ♣♦✐♥ts ♦♥ t❤❡

s❛♠❡ s✐❞❡ ♦❢ t❤❡ ❧✐♥❡✱ t❤❡♥ t❤❡s❡ ❢♦✉r ♣♦✐♥ts ❧✐❡ ♦♥ ❛ ❝✐r❝❧❡✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✶✶

❈♦♥s✐❞❡r ❛ ❧✐♥❡ s❡❣♠❡♥t AB✱ t❤❛t s✉❜t❡♥❞s ❡q✉❛❧ ❛♥❣❧❡s ❛t ♣♦✐♥ts P ❛♥❞ Q ♦♥ t❤❡ s❛♠❡ s✐❞❡ ♦❢ AB✳ ❚❤❡

❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t ♣♦✐♥ts A✱ B✱ P ❛♥❞ Q ❧✐❡ ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡✳ ❇② ❝♦♥tr❛❞✐❝t✐♦♥✳

❆ss✉♠❡ t❤❛t ♣♦✐♥t P ❞♦❡s ♥♦t ❧✐❡ ♦♥ ❛ ❝✐r❝❧❡ ❞r❛✇♥ t❤r♦✉❣❤ ♣♦✐♥ts A✱ B ❛♥❞ Q✳ ▲❡t t❤❡ ❝✐r❝❧❡ ❝✉t AP ✭♦r

✾✽

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

AP ❡①t❡♥❞❡❞✮ ❛t ♣♦✐♥t R✳

❫

❫

AQB

=

ARB ∠ ✬s on same side of chord

❫

❫

but AQB

=

AP B (given)

❫

❫

✭✼✳✻✮

∴ARB

=

AP B

❫

❫

❫

but this cannot be true since ARB

=

AP B + RBP (ext. ∠ of [U+25B5])

∴ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❝✐r❝❧❡ ❞♦❡s ♥♦t ♣❛ss t❤r♦✉❣❤ P ✱ ♠✉st ❜❡ ❢❛❧s❡✱ ❛♥❞ A✱ B✱ P ❛♥❞ Q ❧✐❡ ♦♥ t❤❡

❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡✳

✼✳✶✳✷✳✸✳✸ ❈✐r❝❧❡s ■■■

✶✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❧❡tt❡rs✳

❋✐❣✉r❡ ✼✳✶✷

✼✳✶✳✷✳✸✳✹ ❈②❝❧✐❝ ◗✉❛❞r✐❧❛t❡r❛❧s

❈②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧s ❛r❡ q✉❛❞r✐❧❛t❡r❛❧s ✇✐t❤ ❛❧❧ ❢♦✉r ✈❡rt✐❝❡s ❧②✐♥❣ ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡✳ ❚❤❡

✈❡rt✐❝❡s ♦❢ ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ❛r❡ s❛✐❞ t♦ ❜❡ ❝♦♥❝②❝❧✐❝✳

❚❤❡♦r❡♠ ✼ ❚❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ♦❢ ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ❛r❡ s✉♣♣❧❡♠❡♥t❛r②✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✶✸

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O✳ ❉r❛✇ ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ABP Q✳ ❉r❛✇ AO ❛♥❞ P O✳ ❚❤❡ ❛✐♠ ✐s

❫

❫

❫

❫

t♦ ♣r♦✈❡ t❤❛t ABP + AQP = 180◦ ❛♥❞ QAB + QP B= 180◦✳

❫

❫

O1

=

2 ABP ∠ ✬s at centre

❫

❫

O2

=

2 AQP ∠ ✬s at centre

❫

❫

But, O1 + O2

=

360◦

❫

❫

✭✼✳✼✮

∴ 2 ABP +2 AQP

=

360◦

❫

❫

∴ABP + AQP

=

180◦

❫

❫

Similarly, QAB + QPB

=

180◦

✾✾

❚❤❡♦r❡♠ ✽ ✭❈♦♥✈❡rs❡ ♦❢ ❚❤❡♦r❡♠ ✮ ■❢ t❤❡ ♦♣♣♦s✐t❡ ❛♥❣❧❡s ♦❢ ❛ q✉❛❞r✐❧❛t❡r❛❧ ❛r❡ s✉♣♣❧❡♠❡♥t❛r②✱ t❤❡♥ t❤❡

q✉❛❞r✐❧❛t❡r❛❧ ✐s ❝②❝❧✐❝✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✶✹

❫

❫

❫

❫

❈♦♥s✐❞❡r ❛ q✉❛❞r✐❧❛t❡r❛❧ ABP Q✱ s✉❝❤ t❤❛t ABP + AQP = 180◦ ❛♥❞ QAB + QP B= 180◦✳ ❚❤❡ ❛✐♠

✐s t♦ ♣r♦✈❡ t❤❛t ♣♦✐♥ts A✱ B✱ P ❛♥❞ Q ❧✐❡ ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡✳ ❇② ❝♦♥tr❛❞✐❝t✐♦♥✳

❆ss✉♠❡ t❤❛t ♣♦✐♥t P ❞♦❡s ♥♦t ❧✐❡ ♦♥ ❛ ❝✐r❝❧❡ ❞r❛✇♥ t❤r♦✉❣❤ ♣♦✐♥ts A✱ B ❛♥❞ Q✳ ▲❡t t❤❡ ❝✐r❝❧❡ ❝✉t AP ✭♦r

AP ❡①t❡♥❞❡❞✮ ❛t ♣♦✐♥t R✳ ❉r❛✇ BR✳

❫

❫

QAB + QRB

=

180◦ opp. ∠ ✬s of cyclic quad

❫

❫

but QAB + QPB

=

180◦ (given)

❫

❫

✭✼✳✽✮

∴QRB

=

QP B

❫

❫

❫

but this cannot be true since QRB

=

QP B + RBP (ext.∠of[U+25B5])

∴ t❤❡ ❛ss✉♠♣t✐♦♥ t❤❛t t❤❡ ❝✐r❝❧❡ ❞♦❡s ♥♦t ♣❛ss t❤r♦✉❣❤ P ✱ ♠✉st ❜❡ ❢❛❧s❡✱ ❛♥❞ A✱ B✱ P ❛♥❞ Q ❧✐❡ ♦♥ t❤❡

❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡ ❛♥❞ ABP Q ✐s ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧✳

✼✳✶✳✷✳✸✳✹✳✶ ❈✐r❝❧❡s ■❱

✶✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❧❡tt❡rs✳

❋✐❣✉r❡ ✼✳✶✺

❚❤❡♦r❡♠ ✾ ❚✇♦ t❛♥❣❡♥ts ❞r❛✇♥ t♦ ❛ ❝✐r❝❧❡ ❢r♦♠ t❤❡ s❛♠❡ ♣♦✐♥t ♦✉ts✐❞❡ t❤❡ ❝✐r❝❧❡ ❛r❡ ❡q✉❛❧ ✐♥ ❧❡♥❣t❤✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✶✻

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O✳ ❈❤♦♦s❡ ❛ ♣♦✐♥t P ♦✉ts✐❞❡ t❤❡ ❝✐r❝❧❡✳ ❉r❛✇ t✇♦ t❛♥❣❡♥ts t♦ t❤❡ ❝✐r❝❧❡

❢r♦♠ ♣♦✐♥t P ✱ t❤❛t ♠❡❡t t❤❡ ❝✐r❝❧❡ ❛t A ❛♥❞ B✳ ❉r❛✇ ❧✐♥❡s OA✱ OB ❛♥❞ OP ✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t

AP = BP ✳ ■♥ [U+25B5]OAP ❛♥❞ [U+25B5]OBP ✱

✶✵✵

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

✶✳ OA = OB ✭r❛❞✐✐✮

✷✳ ∠OAP = ∠OP B = 90◦ ✭OA ⊥ AP ❛♥❞ OB ⊥ BP ✮

✸✳ OP ✐s ❝♦♠♠♦♥ t♦ ❜♦t❤ tr✐❛♥❣❧❡s✳

[U+25B5]OAP ≡ [U+25B5]OBP ✭r✐❣❤t ❛♥❣❧❡✱ ❤②♣♦t❡♥✉s❡✱ s✐❞❡✮ ∴ AP = BP

✼✳✶✳✷✳✸✳✹✳✷ ❈✐r❝❧❡s ❱

✶✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❧❡♥❣t❤s✳

❋✐❣✉r❡ ✼✳✶✼

❚❤❡♦r❡♠ ✶✵ ❚❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ❛ t❛♥❣❡♥t ❛♥❞ ❛ ❝❤♦r❞✱ ❞r❛✇♥ ❛t t❤❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t ♦❢ t❤❡ ❝❤♦r❞✱ ✐s ❡q✉❛❧

t♦ t❤❡ ❛♥❣❧❡ ✇❤✐❝❤ t❤❡ ❝❤♦r❞ s✉❜t❡♥❞s ✐♥ t❤❡ ❛❧t❡r♥❛t❡ s❡❣♠❡♥t✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✶✽

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O✳ ❉r❛✇ ❛ ❝❤♦r❞ AB ❛♥❞ ❛ t❛♥❣❡♥t SR t♦ t❤❡ ❝✐r❝❧❡ ❛t ♣♦✐♥t B✳ ❈❤♦r❞

AB s✉❜t❡♥❞s ❛♥❣❧❡s ❛t ♣♦✐♥ts P ❛♥❞ Q ♦♥ t❤❡ ♠✐♥♦r ❛♥❞ ♠❛❥♦r ❛r❝s✱ r❡s♣❡❝t✐✈❡❧②✳ ❉r❛✇ ❛ ❞✐❛♠❡t❡r BT ❛♥❞

❫

❫

❫

❫

❫

❫

❥♦✐♥ A t♦ T ✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t AP B=ABR ❛♥❞ AQB=ABS✳ ❋✐rst ♣r♦✈❡ t❤❛t AQB=ABS ❛s

✶✵✶

❫

❫

t❤✐s r❡s✉❧t ✐s ♥❡❡❞❡❞ t♦ ♣r♦✈❡ t❤❛t AP B=ABR✳

❫

❫

ABS + ABT

=

90◦ (TB ⊥ SR)

❫

BAT

=

90◦ (∠ ✬s at centre)

❫

❫

∴ABT + AT B

=

90◦ (sum of angles in [U+25B5]BAT)

❫

❫

∴ABS

=

ABT

❫

❫

However, AQB

=

AT B (angles subtended by same chord AB)

❫

❫

∴AQB

=

ABS

✭✼✳✾✮

❫

❫

SBQ + QBR

=

180◦

(SBT is a str. line)

❫

❫

AP B + AQB

=

180◦

(ABPQ is a cyclic quad)

❫

❫

❫

❫

∴SBQ + QBR =

AP B + AQB

❫

❫

AQB

=

ABS

❫

❫

∴AP B

=

ABR

✼✳✶✳✷✳✸✳✹✳✸ ❈✐r❝❧❡s ❱■

✶✳ ❋✐♥❞ t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ✉♥❦♥♦✇♥ ❧❡tt❡rs✳

❋✐❣✉r❡ ✼✳✶✾

❚❤❡♦r❡♠ ✶✶ ✭❈♦♥✈❡rs❡ ♦❢ ✮ ■❢ t❤❡ ❛♥❣❧❡ ❢♦r♠❡❞ ❜❡t✇❡❡♥ ❛ ❧✐♥❡✱ t❤❛t ✐s ❞r❛✇♥ t❤r♦✉❣❤ t❤❡ ❡♥❞ ♣♦✐♥t ♦❢ ❛

❝❤♦r❞✱ ❛♥❞ t❤❡ ❝❤♦r❞✱ ✐s ❡q✉❛❧ t♦ t❤❡ ❛♥❣❧❡ s✉❜t❡♥❞❡❞ ❜② t❤❡ ❝❤♦r❞ ✐♥ t❤❡ ❛❧t❡r♥❛t❡ s❡❣♠❡♥t✱ t❤❡♥ t❤❡ ❧✐♥❡ ✐s

❛ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡✳

Pr♦♦❢✿

❋✐❣✉r❡ ✼✳✷✵

❈♦♥s✐❞❡r ❛ ❝✐r❝❧❡✱ ✇✐t❤ ❝❡♥tr❡ O ❛♥❞ ❝❤♦r❞ AB✳ ▲❡t ❧✐♥❡ SR ♣❛ss t❤r♦✉❣❤ ♣♦✐♥t B✳ ❈❤♦r❞ AB s✉❜t❡♥❞s

❫

❫

❛♥ ❛♥❣❧❡ ❛t ♣♦✐♥t Q s✉❝❤ t❤❛t ABS=AQB✳ ❚❤❡ ❛✐♠ ✐s t♦ ♣r♦✈❡ t❤❛t SBR ✐s ❛ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡✳

✶✵✷

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❇② ❝♦♥tr❛❞✐❝t✐♦♥✳ ❆ss✉♠❡ t❤❛t SBR ✐s ♥♦t ❛ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ❛♥❞ ❞r❛✇ XBY s✉❝❤ t❤❛t XBY ✐s ❛

t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡✳

❫

❫

ABX

=

AQB

(tan − chord theorem)

❫

❫

However,

ABS

=

AQB

(given)

❫

❫

∴ABX

=

ABS

✭✼✳✶✵✮

❫

❫

❫

But,

ABX

=

ABS + XBS

❫

can only be true if,

XBS

=

0

❫

■❢ XBS ✐s ③❡r♦✱ t❤❡♥ ❜♦t❤ XBY ❛♥❞ SBR ❝♦✐♥❝✐❞❡ ❛♥❞ SBR ✐s ❛ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡✳

✼✳✶✳✷✳✸✳✹✳✹ ❆♣♣❧②✐♥❣ ❚❤❡♦r❡♠

✶✳ ❙❤♦✇ t❤❛t ❚❤❡♦r❡♠ ❛❧s♦ ❛♣♣❧✐❡s t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝❛s❡s✿

❋✐❣✉r❡ ✼✳✷✶

❊①❡r❝✐s❡ ✼✳✶✿ ❈✐r❝❧❡ ●❡♦♠❡tr② ■

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✶✶✳✮

❋✐❣✉r❡ ✼✳✷✷

BD ✐s ❛ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ O✳ BO ⊥ AD✳ Pr♦✈❡ t❤❛t✿

✶✳ CF OE ✐s ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧

✷✳ F B = BC

✸✳ [U+25B5]COE///[U+25B5]CBF

✹✳ CD2 = ED.AD

✺✳ OE = CD

BC

CO

❊①❡r❝✐s❡ ✼✳✷✿ ❈✐r❝❧❡ ●❡♦♠❡tr② ■■

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✶✷✳✮

❋✐❣✉r❡ ✼✳✷✸

✶✵✸

F D ✐s ❞r❛✇♥ ♣❛r❛❧❧❡❧ t♦ t❤❡ t❛♥❣❡♥t CB Pr♦✈❡ t❤❛t✿

✶✳ F ADE ✐s ❝②❝❧✐❝

✷✳ [U+25B5]AF E|||[U+25B5]CBD

✸✳ FC.AG = DC.FE

GH

BD

✼✳✷ ❈♦♦r❞✐♥❛t❡ ●❡♦♠❡tr②✱ ❊q✉❛t✐♦♥ ♦❢ ❚❛♥❣❡♥t✱ ❚r❛♥s❢♦r♠❛t✐♦♥s✷

✼✳✷✳✶ ❈♦✲♦r❞✐♥❛t❡ ●❡♦♠❡tr②

✼✳✷✳✶✳✶ ❊q✉❛t✐♦♥ ♦❢ ❛ ❈✐r❝❧❡

❲❡ ❦♥♦✇ t❤❛t ❡✈❡r② ♣♦✐♥t ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡ ✐s t❤❡ s❛♠❡ ❞✐st❛♥❝❡ ❛✇❛② ❢r♦♠ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡

❝✐r❝❧❡✳ ❈♦♥s✐❞❡r ❛ ♣♦✐♥t (x1, y1) ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ ❛ ❝✐r❝❧❡ ♦❢ r❛❞✐✉s r ✇✐t❤ ❝❡♥tr❡ ❛t (x0, y0)✳

❋✐❣✉r❡ ✼✳✷✹

■♥ ❋✐❣✉r❡ ✼✳✷✹✱ [U+25B5]OP Q ✐s ❛ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡✳ ❚❤❡r❡❢♦r❡✱ ❢r♦♠ t❤❡ ❚❤❡♦r❡♠ ♦❢ P②t❤❛❣♦r❛s✱

✇❡ ❦♥♦✇ t❤❛t✿

OP 2 = P Q2 + OQ2

✭✼✳✶✶✮

❇✉t✱

P Q

=

y1 − y0

OQ

=

x1 − x0

✭✼✳✶✷✮

OP

=

r

∴

r2

=

(y1 − y0)2 + (x1 − x0)2

❇✉t✱ t❤✐s s❛♠❡ r❡❧❛t✐♦♥ ❤♦❧❞s ❢♦r ❛♥② ♣♦✐♥t P ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✳ ■♥ ❢❛❝t✱ t❤❡ r❡❧❛t✐♦♥ ❤♦❧❞s ❢♦r ❛❧❧ ♣♦✐♥ts

P ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡✳ ❚❤❡r❡❢♦r❡✱ ✇❡ ❝❛♥ ✇r✐t❡✿

(x − x0)2 + (y − y0)2 = r2

✭✼✳✶✸✮

❢♦r ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ ❛t (x0, y0) ❛♥❞ r❛❞✐✉s r✳

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ (0, 0) ❛♥❞ r❛❞✐✉s ✹ ✐s✿

(y − y0)2 + (x − x0)2

=

r2

(y − 0)2 + (x − 0)2

=

42

✭✼✳✶✹✮

y2 + x2

=

16

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✽✽✴✶✳✶✴❃✳

✶✵✹

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ❝✐r❝❧❡s ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴✻r✶●◗❈①②▼❑■✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✼✳✷✺

❊①❡r❝✐s❡ ✼✳✸✿ ❊q✉❛t✐♦♥ ♦❢ ❛ ❈✐r❝❧❡ ■

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✶✸✳✮

❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ ❝✐r❝❧❡ ✭❝❡♥tr❡ O✮ ✇✐t❤ ❛ ❞✐❛♠❡t❡r ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts✱ P ❛t (−5, 5) ❛♥❞ Q

❛t (5, −5)✳

❊①❡r❝✐s❡ ✼✳✹✿ ❊q✉❛t✐♦♥ ♦❢ ❛ ❈✐r❝❧❡ ■■

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✶✹✳✮

❋✐♥❞ t❤❡ ❝❡♥t❡r ❛♥❞ r❛❞✐✉s ♦❢ t❤❡ ❝✐r❝❧❡

x2 − 14x + y2 + 4y = −28✳

✼✳✷✳✶✳✷ ❊q✉❛t✐♦♥ ♦❢ ❛ ❚❛♥❣❡♥t t♦ ❛ ❈✐r❝❧❡ ❛t ❛ P♦✐♥t ♦♥ t❤❡ ❈✐r❝❧❡

❲❡ ❛r❡ ❣✐✈❡♥ t❤❛t ❛ t❛♥❣❡♥t t♦ ❛ ❝✐r❝❧❡ ✐s ❞r❛✇♥ t❤r♦✉❣❤ ❛ ♣♦✐♥t P ✇✐t❤ ❝♦✲♦r❞✐♥❛t❡s (x1, y1)✳ ■♥ t❤✐s s❡❝t✐♦♥✱

✇❡ ✜♥❞ ♦✉t ❤♦✇ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❛t t❛♥❣❡♥t✳

❋✐❣✉r❡ ✼✳✷✻

❲❡ st❛rt ❜② ♠❛❦✐♥❣ ❛ ❧✐st ♦❢ ✇❤❛t ✇❡ ❦♥♦✇✿

✶✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ (x0, y0) ❛♥❞ r❛❞✐✉s r ✐s (x − x0)2 + (y − y0)2 = r2✳

✷✳ ❲❡ ❦♥♦✇ t❤❛t ❛ t❛♥❣❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ r❛❞✐✉s✱ ❞r❛✇♥ ❛t t❤❡ ♣♦✐♥t ♦❢ ❝♦♥t❛❝t ✇✐t❤ t❤❡ ❝✐r❝❧❡✳

❆s ✇❡ ❤❛✈❡ s❡❡♥ ✐♥ ❡❛r❧✐❡r ❣r❛❞❡s✱ t❤❡r❡ ❛r❡ t✇♦ st❡♣s t♦ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ❡q✉❛t✐♦♥ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡✿

❙t❡♣ ✶✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐♥❡✱ m✳

❙t❡♣ ✷✳ ❈❛❧❝✉❧❛t❡ t❤❡ y✲✐♥t❡r❝❡♣t ♦❢ t❤❡ ❧✐♥❡✱ c✳

❚❤❡ s❛♠❡ ♠❡t❤♦❞ ✐s ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t✳ ❋✐rst ✇❡ ♥❡❡❞ t♦ ✜♥❞ t❤❡ ❣r❛❞✐❡♥t ♦❢

t❤❡ t❛♥❣❡♥t✳ ❲❡ ❞♦ t❤✐s ❜② ✜♥❞✐♥❣ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐♥❡ t❤❛t ♣❛ss❡s t❤r♦✉❣❤ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ❛♥❞

♣♦✐♥t P ✭❧✐♥❡ f ✐♥ ✮✱ ❜❡❝❛✉s❡ t❤✐s ❧✐♥❡ ✐s ❛ r❛❞✐✉s ❧✐♥❡ ❛♥❞ t❤❡ t❛♥❣❡♥t ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ✐t✳

y1 − y0

mf =

✭✼✳✶✺✮

x1 − x0

❚❤❡ t❛♥❣❡♥t ✭❧✐♥❡ g✮ ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤✐s ❧✐♥❡✳ ❚❤❡r❡❢♦r❡✱

mf × mg = −1

✭✼✳✶✻✮

✶✵✺

❙♦✱

1

mg = −

✭✼✳✶✼✮

mf

◆♦✇✱ ✇❡ ❦♥♦✇ t❤❛t t❤❡ t❛♥❣❡♥t ♣❛ss❡s t❤r♦✉❣❤ (x1, y1) s♦ t❤❡ ❡q✉❛t✐♦♥ ✐s ❣✐✈❡♥ ❜②✿

y − y1

=

m (x − x1)

y − y1

=

− 1 (x − x

m

1)

f

✭✼✳✶✽✮

y − y1

=

−

1

(x − x

y1−y0

1)

x1−x0

y − y1

=

− x1−x0 (x − x

y

1)

1 −y0

❋♦r ❡①❛♠♣❧❡✱ ✜♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡ ❛t ♣♦✐♥t (1, 1)✳ ❚❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛t

(0, 0)✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s x2 + y2 = 2✳

❯s❡

x1 − x0

y − y1 = −

(x − x1)

✭✼✳✶✾✮

y1 − y0

✇✐t❤ (x0, y0) = (0, 0) ❛♥❞ (x1, y1) = (1, 1)✳

y − y1

=

− x1−x0 (x − x

y

1)

1 −y0

y − 1

=

− 1−0 (x − 1)

1−0

y − 1

=

− 1 (x − 1)

1

✭✼✳✷✵✮

y

=

− (x − 1) + 1

y

=

−x + 1 + 1

y

=

−x + 2

✼✳✷✳✶✳✷✳✶ ❈♦✲♦r❞✐♥❛t❡ ●❡♦♠❡tr②

✶✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐❝❧❡✿

❛✳ ✇✐t❤ ❝❡♥tr❡ (0; 5) ❛♥❞ r❛❞✐✉s ✺

❜✳ ✇✐t❤ ❝❡♥tr❡ (2; 0) ❛♥❞ r❛❞✐✉s ✹

❝✳ ✇✐t❤ ❝❡♥tr❡ (5; 7) ❛♥❞ r❛❞✐✉s ✶✽

❞✳ ✇✐t❤ ❝❡♥tr❡ (−2; 0) ❛♥❞ r❛❞✐✉s ✻ √

❡✳ ✇✐t❤ ❝❡♥tr❡ (−5; −3) ❛♥❞ r❛❞✐✉s 3

✷✳

❛✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥tr❡ (2; 1) ✇❤✐❝❤ ♣❛ss❡s t❤r♦✉❣❤ (4; 1)✳

❜✳ ❲❤❡r❡ ❞♦❡s ✐t ❝✉t t❤❡ ❧✐♥❡ y = x + 1❄

❝✳ ❉r❛✇ ❛ s❦❡t❝❤ t♦ ✐❧❧✉str❛t❡ ②♦✉r ❛♥s✇❡rs✳

✸✳

❛✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥t❡r (−3; −2) ✇❤✐❝❤ ♣❛ss❡s t❤r♦✉❣❤ (1; −4)✳

❜✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❝✐r❝❧❡ ✇✐t❤ ❝❡♥t❡r (3; 1) ✇❤✐❝❤ ♣❛ss❡s t❤r♦✉❣❤ (2; 5)✳

❝✳ ❋✐♥❞ t❤❡ ♣♦✐♥t ✇❤❡r❡ t❤❡s❡ t✇♦ ❝✐r❝❧❡s ❝✉t ❡❛❝❤ ♦t❤❡r✳

✹✳ ❋✐♥❞ t❤❡ ❝❡♥t❡r ❛♥❞ r❛❞✐✉s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝✐r❝❧❡s✿

❛✳ (x − 9)2 + (y − 6)2 = 36

❜✳ (x − 2)2 + (y − 9)2 = 1

❝✳ (x + 5)2 + (y + 7)2 = 12

❞✳ (x + 4)2 + (y + 4)2 = 23

❡✳ 3(x − 2)2 + 3(y + 3)2 = 12

✶✵✻

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❢✳ x2 − 3x + 9 = y2 + 5y + 25 = 17

✺✳ ❋✐♥❞ t❤❡ x− ❛♥❞ y− ✐♥t❡r❝❡♣ts ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s ❛♥❞ ❞r❛✇ ❛ s❦❡t❝❤ t♦ ✐❧❧✉str❛t❡ ②♦✉r ❛♥s✇❡r✿

❛✳ (x + 7)2 + (y − 2)2 = 8

❜✳ x2 + (y − 6)2 = 100

❝✳ (x + 4)2 + y2 = 16

❞✳ (x − 5)2 + (y + 1)2 = 25

✻✳ ❋✐♥❞ t❤❡ ❝❡♥t❡r ❛♥❞ r❛❞✐✉s ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝✐r❝❧❡s✿

❛✳ x2 + 6x + y2 − 12y = −20

❜✳ x2 + 4x + y2 − 8y = 0

❝✳ x2 + y2 + 8y = 7

❞✳ x2 − 6x + y2 = 16

❡✳ x2 − 5x + y2 + 3y = −34

❢✳ x2 − 6nx + y2 + 10ny = 9n2

✼✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥s t♦ t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✐r❝❧❡✿

❛✳ x2 + y2 = 17 ❛t t❤❡ ♣♦✐♥t (1; 4)

❜✳ x2 + y2 = 25 ❛t t❤❡ ♣♦✐♥t (3; 4)

❝✳ (x + 1)2 + (y − 2)2 = 25 ❛t t❤❡ ♣♦✐♥t (3; 5)

❞✳ (x − 2)2 + (y − 1)2 = 13 ❛t t❤❡ ♣♦✐♥t (5; 3)

✼✳✷✳✷ ❚r❛♥s❢♦r♠❛t✐♦♥s

✼✳✷✳✷✳✶ ❘♦t❛t✐♦♥ ♦❢ ❛ P♦✐♥t ❛❜♦✉t ❛♥ ❛♥❣❧❡ θ

❋✐rst ✇❡ ✇✐❧❧ ✜♥❞ ❛ ❢♦r♠✉❧❛ ❢♦r t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ P ❛❢t❡r ❛ r♦t❛t✐♦♥ ♦❢ θ✳

❲❡ ♥❡❡❞ t♦ ❦♥♦✇ s♦♠❡t❤✐♥❣ ❛❜♦✉t ♣♦❧❛r ❝♦✲♦r❞✐♥❛t❡s ❛♥❞ ❝♦♠♣♦✉♥❞ ❛♥❣❧❡s ❜❡❢♦r❡ ✇❡ st❛rt✳

✼✳✷✳✷✳✶✳✶ P♦❧❛r ❝♦✲♦r❞✐♥❛t❡s

❋✐❣✉r❡ ✼✳✷✼

◆♦t✐❝❡ t❤❛t ✿ sinα = y ∴ y = rsinα

r

❛♥❞

cosα = x ∴ x = rcosα

r

❚❛❜❧❡ ✼✳✶

s♦ P ❝❛♥ ❜❡ ❡①♣r❡ss❡❞ ✐♥ t✇♦ ✇❛②s✿

✶✳ P (x; y) r❡❝t❛♥❣✉❧❛r ❝♦✲♦r❞✐♥❛t❡s

✷✳ P (rcosα; rsinα) ♣♦❧❛r ❝♦✲♦r❞✐♥❛t❡s✳

✶✵✼

✼✳✷✳✷✳✶✳✷ ❈♦♠♣♦✉♥❞ ❛♥❣❧❡s

✭❙❡❡ ❞❡r✐✈❛t✐♦♥ ♦❢ ❢♦r♠✉❧❛❡ ✐♥ ❈❤✳ ✶✷✮

sin (α + β)

=

sinαcosβ + sinβcosα

✭✼✳✷✶✮

cos (α + β)

=

cosαcosβ − sinαsinβ

✼✳✷✳✷✳✶✳✸ ◆♦✇ ❝♦♥s✐❞❡r P ✬ ❛❢t❡r ❛ r♦t❛t✐♦♥ ♦❢ θ

P (x; y) = P (rcosα; rsinα)

✭✼✳✷✷✮

P ✬ (rcos (α + θ) ; rsin (α + θ))

❊①♣❛♥❞ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ P ✬

x − co − ordinate ofP ✬

=

rcos (α + θ)

=

r [cosαcosθ − sinαsinθ]

✭✼✳✷✸✮

=

rcosαcosθ − rsinαsinθ

=

xcosθ − ysinθ

y − co − ordinate of P✬ =

rsin (α + θ)

=

r [sinαcosθ + sinθcosα]

✭✼✳✷✹✮

=

rsinαcosθ + rcosαsinθ

=

ycosθ + xsinθ

❋✐❣✉r❡ ✼✳✷✽

✇❤✐❝❤ ❣✐✈❡s t❤❡ ❢♦r♠✉❧❛ P ✬ = [(xcosθ − ysinθ; ycosθ + xsinθ)]✳

√

❙♦ t♦ ✜♥❞ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ P 1; 3 ❛❢t❡r ❛ r♦t❛t✐♦♥ ♦❢ ✹✺◦✱ ✇❡ ❛rr✐✈❡ ❛t✿

P ✬

=

[(xcosθ − ysinθ) ; (ycosθ + xsinθ)]

√

√

=

1cos45◦ −

3sin45◦ ;

3cos45◦ + 1sin45◦

√

√

✭✼✳✷✺✮

=

1

√

−

3

√

;

3

√

+ 1

√

2

2

2

2

√

√

=

1−

3

√

;

3+1

√

2

2

✼✳✷✳✷✳✶✳✸✳✶ ❘♦t❛t✐♦♥s

❆♥② ❧✐♥❡ OP ✐s ❞r❛✇♥ ✭♥♦t ♥❡❝❡ss❛r✐❧② ✐♥ t❤❡ ✜rst q✉❛❞r❛♥t✮✱ ♠❛❦✐♥❣ ❛♥ ❛♥❣❧❡ ♦❢ θ ❞❡❣r❡❡s ✇✐t❤ t❤❡ x✲❛①✐s✳

❯s✐♥❣ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ P ❛♥❞ t❤❡ ❛♥❣❧❡ α✱ ❝❛❧❝✉❧❛t❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ P ✬✱ ✐❢ t❤❡ ❧✐♥❡ OP ✐s r♦t❛t❡❞ ❛❜♦✉t t❤❡ ♦r✐❣✐♥ t❤r♦✉❣❤ α ❞❡❣r❡❡s✳

✶✵✽

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

P

α

✶✳ ✭✷✱ ✻✮

✻✵◦

✷✳ ✭✹✱ ✷✮

✸✵◦

✸✳ ✭✺✱ ✲✶✮

✹✺◦

✹✳ ✭✲✸✱ ✷✮

✶✷✵◦

✺✳ ✭✲✹✱ ✲✶✮ ✷✷✺◦

✻✳ ✭✷✱ ✺✮

✲✶✺✵◦

❚❛❜❧❡ ✼✳✷

❋✐❣✉r❡ ✼✳✷✾

✼✳✷✳✷✳✷ ❈❤❛r❛❝t❡r✐st✐❝s ♦❢ ❚r❛♥s❢♦r♠❛t✐♦♥s

❘✐❣✐❞ tr❛♥s❢♦r♠❛t✐♦♥s ❧✐❦❡ tr❛♥s❧❛t✐♦♥s✱ r❡✢❡❝t✐♦♥s✱ r♦t❛t✐♦♥s ❛♥❞ ❣❧✐❞❡ r❡✢❡❝t✐♦♥s ♣r❡s❡r✈❡ s❤❛♣❡ ❛♥❞ s✐③❡✱

❛♥❞ t❤❛t ❡♥❧❛r❣❡♠❡♥t ♣r❡s❡r✈❡s s❤❛♣❡ ❜✉t ♥♦t s✐③❡✳

✼✳✷✳✷✳✷✳✶ ●❡♦♠❡tr✐❝ ❚r❛♥s❢♦r♠❛t✐♦♥s✿

❉r❛✇ ❛ ❧❛r❣❡ ✶✺×✶✺ ❣r✐❞ ❛♥❞ ♣❧♦t [U+25B5]ABC ✇✐t❤ A (2; 6)✱ B (5; 6) ❛♥❞ C (5; 1)✳ ❋✐❧❧ ✐♥ t❤❡ ❧✐♥❡s

y = x ❛♥❞ y = −x✳ ❈♦♠♣❧❡t❡ t❤❡ t❛❜❧❡ ❜❡❧♦✇ ✱ ❜② ❞r❛✇✐♥❣ t❤❡ ✐♠❛❣❡s ♦❢ [U+25B5]ABC ✉♥❞❡r t❤❡ ❣✐✈❡♥

tr❛♥s❢♦r♠❛t✐♦♥s✳ ❚❤❡ ✜rst ♦♥❡ ❤❛s ❜❡❡♥ ❞♦♥❡ ❢♦r ②♦✉✳

❋✐❣✉r❡ ✼✳✸✵

✶✵✾

❉❡s❝r✐♣t✐♦♥

❚r❛♥s❢♦r♠❛t✐♦♥

✭tr❛♥s❧❛t✐♦♥✱ r❡✢❡❝t✐♦♥✱

❈♦✲♦r❞✐♥❛t❡s ▲❡♥❣t❤s

❆♥❣❧❡s

r♦t❛t✐♦♥✱ ❡♥❧❛r❣❡♠❡♥t✮

❫✬

A✬ (2; −6)

A✬B✬ = 3

B = 90◦

❫

(x; y) → (x; −y)

r❡✢❡❝t✐♦♥ ❛❜♦✉t t❤❡ x✲❛①✐s B✬ (5; −6)

B✬C✬ = 4

tan A= 4/3

❫

❫

C✬ (5; −2)

A✬C✬ = 5

∴A= 53◦, C= 37◦

(x; y) → (x + 1; y − 2)

(x; y) → (−x; y)

(x; y) → (−y; x)

(x; y) → (−x; −y)

(x; y) → (2x; 2y)

(x; y) → (y; x)

(x; y) → (y; x + 1)

❚❛❜❧❡ ✼✳✸

❆ tr❛♥s❢♦r♠❛t✐♦♥ t❤❛t ❧❡❛✈❡s ❧❡♥❣t❤s ❛♥❞ ❛♥❣❧❡s ✉♥❝❤❛♥❣❡❞ ✐s ❝❛❧❧❡❞ ❛ r✐❣✐❞ tr❛♥s❢♦r♠❛t✐♦♥✳ ❲❤✐❝❤ ♦❢

t❤❡ ❛❜♦✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ❛r❡ r✐❣✐❞❄

✼✳✷✳✸ ❊①❡r❝✐s❡s

✶✳ ∆ABC ✉♥❞❡r❣♦❡s s❡✈❡r❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ❢♦r♠✐♥❣ ∆A✬B✬C✬✳ ❉❡s❝r✐❜❡ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡

❛♥❣❧❡s ❛♥❞ s✐❞❡s ♦❢ ∆ABC ❛♥❞ ∆A✬B✬C✬ ✭❡✳❣✳✱ t❤❡② ❛r❡ t✇✐❝❡ ❛s ❧❛r❣❡✱ t❤❡ s❛♠❡✱ ❡t❝✳✮

✶✶✵

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❚r❛♥s❢♦r♠❛t✐♦♥

❙✐❞❡s ❆♥❣❧❡s ❆r❡❛

❘❡✢❡❝t

❘❡❞✉❝❡ ❜② ❛ s❝❛❧❡ ❢❛❝t♦r ♦❢ ✸

❘♦t❛t❡ ❜② ✾✵◦

❚r❛♥s❧❛t❡ ✹ ✉♥✐ts r✐❣❤t

❊♥❧❛r❣❡ ❜② ❛ s❝❛❧❡ ❢❛❝t♦r ♦❢ ✷

❚❛❜❧❡ ✼✳✹

❫

✷✳ ∆DEF ❤❛s E= 30◦✱ DE = 4 cm✱ EF = 5 cm✳ ∆DEF ✐s ❡♥❧❛r❣❡❞ ❜② ❛ s❝❛❧❡ ❢❛❝t♦r ♦❢ ✻ t♦ ❢♦r♠

∆D✬E✬F ✬✳

❛✳ ❙♦❧✈❡ ∆DEF

❜✳ ❍❡♥❝❡✱ s♦❧✈❡ ∆D✬E✬F ✬

✸✳ ∆XY Z ❤❛s ❛♥ ❛r❡❛ ♦❢ 6 cm2✳ ❋✐♥❞ t❤❡ ❛r❡❛ ♦❢ ∆X✬Y ✬Z✬ ✐❢ t❤❡ ♣♦✐♥ts ❤❛✈❡ ❜❡❡♥ tr❛♥s❢♦r♠❡❞ ❛s ❢♦❧❧♦✇s✿

❛✳ (x, y) → (x + 2; y + 3)

❜✳ (x, y) → (y; x)

❝✳ (x, y) → (4x; y)

❞✳ (x, y) → (3x; y + 2)

❡✳ (x, y) → (−x; −y)

❢✳ (x, y) → (x; −y + 3)

❣✳ (x, y) → (4x; 4y)

❤✳ (x, y) → (−3x; 4y)

✶✶✶

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✼

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✼✳✶ ✭♣✳ ✶✵✷✮

❙t❡♣ ✶✳

❫

F OE

=

90◦ (BO ⊥ OD)

❫

✭✼✳✷✻✮

F CE

=

90◦ (∠ subtended by diameter AE)

∴

CF OE is a cyclic quad (opposite ∠ ✬s supplementary)

❫

❙t❡♣ ✷✳ ▲❡t OEC= x✳

❫

∴

F CB= x (∠ between tangentBD and chord CE)

❫

✭✼✳✷✼✮

∴

BF C= x (exterior ∠ to cyclic quad CFOE)

∴ BF = BC (sides opposite equal ∠✬s in isosceles [U+25B5]BFC)

❙t❡♣ ✸✳

❫

CBF

=

180◦ − 2x (sum of ∠ ✬s in [U+25B5]BFC)

OC

=

OE (radii of circle O)

❫

✭✼✳✷✽✮

∴

ECO= x (isosceles [U+25B5]COE)

❫

∴

COE= 180◦ − 2x (sum of ∠ ✬s in [U+25B5]COE)

❫

❫

• COE=CBF

❫

❫

• ECO=F CB

❫

❫

• OEC=CF B

∴ [U+25B5]COE|||[U+25B5]CBF (3 ∠ ✬s equal)

✭✼✳✷✾✮

❙t❡♣ ✹✳

❙t❡♣ ❛✳ ■♥ [U+25B5]EDC

❫

CED

=

180◦ − x (∠ ✬s on a str. lineAD)

❫

✭✼✳✸✵✮

ECD

=

90◦ − x (complementary∠ ✬s)

❙t❡♣ ❜✳ ■♥ [U+25B5]ADC

❫

❫

❫

ACE

=

180◦ − x

sum of ∠ ✬s ACE and ECO

✭✼✳✸✶✮

❫

CAD

=

90◦ − x (sum of ∠ ✬s in [U+25B5]CAE)

❫

❫

❙t❡♣ ❝✳ ▲❛st❧②✱ ADC=EDC s✐♥❝❡ t❤❡② ❛r❡ t❤❡ s❛♠❡ ∠✳

✶✶✷

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❙t❡♣ ❞✳

∴ [U+25B5]ADC|||[U+25B5]CDE (3 ∠ ✬s equal)

∴

ED = CD

✭✼✳✸✷✮

CD

AD

∴

CD2 = ED.AD

❙t❡♣ ✺✳

❙t❡♣ ❛✳

OE

=

CD ([U+25B5]OEC is isosceles)

✭✼✳✸✸✮

❙t❡♣ ❜✳ ■♥ [U+25B5]BCO

❫

OCB

=

90◦ (radius OC on tangent BD)

❫

✭✼✳✸✹✮

CBO

=

180◦ − 2x (sum of ∠ ✬s in [U+25B5]BFC)

❙t❡♣ ❝✳ ■♥ [U+25B5]OCD

❫

OCD

=

90◦ (radius OC on tangent BD)

❫

✭✼✳✸✺✮

COD

=

180◦ − 2x (sum of ∠ ✬s in [U+25B5]OCE)

❙t❡♣ ❞✳ ▲❛st❧②✱ OC ✐s ❛ ❝♦♠♠♦♥ s✐❞❡ t♦ ❜♦t❤ [U+25B5]✬s✳

❙t❡♣ ❡✳

∴ [U+25B5]BOC|||[U+25B5]ODC (common side and 2 equal angles)

∴

CO = CD

✭✼✳✸✻✮

BC

CO

∴

OE = CD (OE = CD isosceles [U+25B5]OEC)

BC

CO

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✼✳✷ ✭♣✳ ✶✵✷✮

❙t❡♣ ✶✳ ▲❡t ∠BCD = x

∴

∠CAH = x (∠ between tangent BC and chord CE)

∴

∠F DC = x (alternate∠, FD

CB)

✭✼✳✸✼✮

∴ FADE is a cyclic quad (chord FE subtends equal ∠✬s)

❙t❡♣ ✷✳

❙t❡♣ ❛✳ ▲❡t ∠F EA = y

∴ ∠F DA = y (∠ ✬s subtended by same chordAF in cyclic quad FADE)

∴

∠CBD = y (corresponding ∠ ✬s, FD

CB)

✭✼✳✸✽✮

∴

∠F EA = ∠CBD

❙t❡♣ ❜✳

∠BCD = ∠F AE (above)

✭✼✳✸✾✮

❙t❡♣ ❝✳

∠AF E

=

180◦ − x − y (∠ ✬s in [U+25B5]AFE)

∠CBD =

180◦ − x − y (∠ ✬s in [U+25B5]CBD)

✭✼✳✹✵✮

∴

[U+25B5]AF E||| [U+25B5]CBD (3 ∠ ✬s equal)

✶✶✸

❙t❡♣ ✸✳

❙t❡♣ ❛✳

DC

=

F A

BD

F E

✭✼✳✹✶✮

∴

DC.F E = F A

BD

❙t❡♣ ❜✳

AG

=

F A (FG CH splits up lines AH and AC proportionally)

GH

F C

✭✼✳✹✷✮

∴

F A = F C.AG

GH

❙t❡♣ ❝✳

∴

F C.AG = DC.F E

✭✼✳✹✸✮

GH

BD

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✼✳✸ ✭♣✳ ✶✵✹✮

❙t❡♣ ✶✳ ❉r❛✇ ❛ ♣✐❝t✉r❡ ♦❢ t❤❡ s✐t✉❛t✐♦♥ t♦ ❤❡❧♣ ②♦✉ ✜❣✉r❡ ♦✉t ✇❤❛t ♥❡❡❞s t♦ ❜❡ ❞♦♥❡✳

❋✐❣✉r❡ ✼✳✸✶

❙t❡♣ ✷✳ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ❝❡♥tr❡ ♦❢ ❛ ❝✐r❝❧❡ ❧✐❡s ♦♥ t❤❡ ♠✐❞♣♦✐♥t ♦❢ ❛ ❞✐❛♠❡t❡r✳ ❚❤❡r❡❢♦r❡ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡

❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❢♦✉♥❞ ❜② ✜♥❞✐♥❣ t❤❡ ♠✐❞♣♦✐♥t ♦❢ t❤❡ ❧✐♥❡ ❜❡t✇❡❡♥ P ❛♥❞ Q✳ ▲❡t t❤❡ ❝♦✲♦r❞✐♥❛t❡s

♦❢ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ❜❡ (x0, y0)✱ ❧❡t t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ P ❜❡ (x1, y1) ❛♥❞ ❧❡t t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢

Q ❜❡ (x2, y2)✳ ❚❤❡♥✱ t❤❡ ❝♦✲♦r❞✐♥❛t❡s ♦❢ t❤❡ ♠✐❞♣♦✐♥t ❛r❡✿

x0

=

x1+x2

2

=

−5+5

2

=

0

✭✼✳✹✹✮

y0

=

y1+y2

2

=

5+(−5)

2

=

0

❚❤❡ ❝❡♥tr❡ ♣♦✐♥t ♦❢ ❧✐♥❡ P Q ❛♥❞ t❤❡r❡❢♦r❡ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❝✐r❝❧❡ ✐s ❛t (0, 0)✳

❙t❡♣ ✸✳ ■❢ P ❛♥❞ Q ❛r❡ t✇♦ ♣♦✐♥ts ♦♥ ❛ ❞✐❛♠❡t❡r✱ t❤❡♥ t❤❡ r❛❞✐✉s ✐s ❤❛❧❢ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡♠✳

❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ t✇♦ ♣♦✐♥ts ✐s✿

r = 1 P Q

=

1

(x

2

2

2 − x1)2 + (y2 − y1)2

=

1

(5 − (−5))2 + (−5 − 5)2

2

=

1

(10)2 + (−10)2

2

√

✭✼✳✹✺✮

=

1

100 + 100

2

=

200

4

√

=

50

❙t❡♣ ✹✳

x2 + y2 = 50

✭✼✳✹✻✮

✶✶✹

❈❍❆P❚❊❘ ✼✳ ●❊❖▼❊❚❘❨

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✼✳✹ ✭♣✳ ✶✵✹✮

❙t❡♣ ✶✳ ❲❡ ♥❡❡❞ t♦ r❡✇r✐t❡ t❤❡ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❢♦r♠ (x − x0) + (y − y0) = r2

❚♦ ❞♦ t❤✐s ✇❡ ♥❡❡❞ t♦ ❝♦♠♣❧❡t❡ t❤❡ sq✉❛r❡

✐✳❡✳ ❛❞❞ ❛♥❞ s✉❜tr❛❝t 1 ❝♦♦❡✣❝✐❡♥t ♦❢ x2 ❛♥❞ 1 ❝♦♦❡✣❝✐❡♥t ♦❢ y2

2

2

❙t❡♣ ✷✳

x2 − 14x + y2 + 4y

=

−28

✭✼✳✹✼✮

∴ x2 − 14x + (7)2 − (7)2 + y2 + 4y + (2)2 − (2)2

=

−28

❙t❡♣ ✸✳

∴ (x − 7)2 − (7)2 + (y + 2)2 − (2)2 = −28

✭✼✳✹✽✮

❙t❡♣ ✹✳

∴ (x − 7)2 − 49 + (y + 2)2 − 4 =

−28

∴ (x − 7)2 + (y + 2)2

=

−28 + 49 + 4

✭✼✳✹✾✮

∴ (x − 7)2 + (y + 2)2

=

25

❙t❡♣ ✺✳

∴ center is (7; −2) and the radius is 5 units

✭✼✳✺✵✮

❈❤❛♣t❡r ✽

❚r✐❣♦♥♦♠❡tr②

✽✳✶ ❈♦♠♣♦✉♥❞ ■❞❡♥t✐t✐❡s✱ Pr♦❜❧❡♠ ❙♦❧✈✐♥❣ ❙tr❛t❡❣✐❡s✶

✽✳✶✳✶ ❈♦♠♣♦✉♥❞ ❆♥❣❧❡ ■❞❡♥t✐t✐❡s

✽✳✶✳✶✳✶ ❉❡r✐✈❛t✐♦♥ ♦❢ sin (α + β)

❲❡ ❤❛✈❡✱ ❢♦r ❛♥② ❛♥❣❧❡s α ❛♥❞ β✱ t❤❛t

sin (α + β) = sinαcosβ + sinβcosα

✭✽✳✶✮

❍♦✇ ❞♦ ✇❡ ❞❡r✐✈❡ t❤✐s ✐❞❡♥t✐t②❄ ■t ✐s tr✐❝❦②✱ s♦ ❢♦❧❧♦✇ ❝❧♦s❡❧②✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ t❤❡ ✉♥✐t ❝✐r❝❧❡ s❤♦✇♥ ❜❡❧♦✇✳ ❚❤❡ t✇♦ ♣♦✐♥ts L (a, b) ❛♥❞ K (x, y) ❛r❡ ♦♥ t❤❡ ❝✐r❝❧❡✳

❋✐❣✉r❡ ✽✳✶

❲❡ ❝❛♥ ❣❡t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ L ❛♥❞ K ✐♥ t❡r♠s ♦❢ t❤❡ ❛♥❣❧❡s α ❛♥❞ β✳ ❋♦r t❤❡ tr✐❛♥❣❧❡ LOK✱ ✇❡ ❤❛✈❡

t❤❛t

sinβ = b

⇒

b = sinβ

1

✭✽✳✷✮

cosβ = a

⇒

a = cosβ

1

❚❤✉s t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ L ❛r❡ (cosβ; sinβ)✳ ■♥ t❤❡ s❛♠❡ ✇❛② ❛s ❛❜♦✈❡✱ ✇❡ ❝❛♥ s❡❡ t❤❛t t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢

K ❛r❡ (cosα; sinα)✳ ❚❤❡ ✐❞❡♥t✐t② ❢♦r cos (α − β) ✐s ♥♦✇ ❞❡t❡r♠✐♥❡❞ ❜② ❝❛❧❝✉❧❛t✐♥❣ KL2 ✐♥ t✇♦ ✇❛②s✳ ❯s✐♥❣

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✶✾✴✶✳✶✴❃✳

✶✶✺

✶✶✻

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

t❤❡ ❞✐st❛♥❝❡ ❢♦r♠✉❧❛ ✭✐✳❡✳ d = (x2 − x1)2 + (y2 − y1)2 ♦r d2 = (x2 − x1)2 + (y2 − y1)2✮✱ ✇❡ ❝❛♥ ✜♥❞ KL2✿

KL2

=

(cosα − cosβ)2 + (sinα − sinβ)2

=

cos2α − 2cosαcosβ + cos2β + sin2α − 2sinαsinβ + sin2β

=

cos2α + sin2α + cos2β + sin2β − 2cosαcosβ − 2sinαsinβ

✭✽✳✸✮

=

1 + 1 − 2 (cosαcosβ + sinαsinβ)

=

2 − 2 (cosαcosβ + sinαsinβ)

❚❤❡ s❡❝♦♥❞ ✇❛② ✇❡ ❝❛♥ ❞❡t❡r♠✐♥❡ KL2 ✐s ❜② ✉s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡ ❢♦r [U+25B5]KOL✿

KL2

=

KO2 + LO2 − 2 · KO · LO · cos (α − β)

=

12 + 12 − 2 (1) (1) cos (α − β)

✭✽✳✹✮

=

2 − 2 · cos (α − β)

❊q✉❛t✐♥❣ ♦✉r t✇♦ ✈❛❧✉❡s ❢♦r KL2✱ ✇❡ ❤❛✈❡

2 − 2 · cos (α − β)

=

2 − 2 (cosαcosβ + sinα · sinβ)

✭✽✳✺✮

⇒

cos (α − β)

=

cosα · cosβ + sinα · sinβ

◆♦✇ ❧❡t α → 90◦ − α✳ ❚❤❡♥

cos (90◦ − α − β)

=

cos (90◦ − α) cosβ + sin (90◦ − α) sinβ

✭✽✳✻✮

=

sinα · cosβ + cosα · sinβ

❇✉t cos (90◦ − (α + β)) = sin (α + β)✳ ❚❤✉s

sin (α + β) = sinα · cosβ + cosα · sinβ

✭✽✳✼✮

✽✳✶✳✶✳✷ ❉❡r✐✈❛t✐♦♥ ♦❢ sin (α − β)

❲❡ ❝❛♥ ✉s❡

sin (α + β) = sinαcosβ + cosαsinβ

✭✽✳✽✮

t♦ s❤♦✇ t❤❛t

sin (α − β) = sinαcosβ − cosαsinβ

✭✽✳✾✮

❲❡ ❦♥♦✇ t❤❛t

sin (−θ) = −sin (θ)

✭✽✳✶✵✮

❛♥❞

cos (−θ) = cosθ

✭✽✳✶✶✮

❚❤❡r❡❢♦r❡✱

sin (α − β)

=

sin (α + (−β))

=

sinαcos (−β) + cosαsin (−β)

✭✽✳✶✷✮

=

sinαcosβ − cosαsinβ

✶✶✼

✽✳✶✳✶✳✸ ❉❡r✐✈❛t✐♦♥ ♦❢ cos (α + β)

❲❡ ❝❛♥ ✉s❡

sin (α − β) = sinαcosβ − sinβcosα

✭✽✳✶✸✮

t♦ s❤♦✇ t❤❛t

cos (α + β) = cosαcosβ − sinαsinβ

✭✽✳✶✹✮

❲❡ ❦♥♦✇ t❤❛t

sin (θ) = cos (90 − θ) .

✭✽✳✶✺✮

❚❤❡r❡❢♦r❡✱

cos (α + β)

=

sin (90 − (α + β))

=

sin ((90 − α) − β)

✭✽✳✶✻✮

=

sin (90 − α) cosβ − sinβcos (90 − α)

=

cosαcosβ − sinβsinα

✽✳✶✳✶✳✹ ❉❡r✐✈❛t✐♦♥ ♦❢ cos (α − β)

❲❡ ❢♦✉♥❞ t❤✐s ✐❞❡♥t✐t② ✐♥ ♦✉r ❞❡r✐✈❛t✐♦♥ ♦❢ t❤❡ sin (α + β) ✐❞❡♥t✐t②✳ ❲❡ ❝❛♥ ❛❧s♦ ✉s❡ t❤❡ ❢❛❝t t❤❛t

sin (α + β) = sinαcosβ + cosαsinβ

✭✽✳✶✼✮

t♦ ❞❡r✐✈❡ t❤❛t

cos (α − β) = cosαcosβ + sinαsinβ

✭✽✳✶✽✮

❆s

cos (θ) = sin (90 − θ) ,

✭✽✳✶✾✮

✇❡ ❤❛✈❡ t❤❛t

cos (α − β)

=

sin (90 − (α − β))

=

sin ((90 − α) + β)

✭✽✳✷✵✮

=

sin (90 − α) cosβ + cos (90 − α) sinβ

=

cosαcosβ + sinαsinβ

✽✳✶✳✶✳✺ ❉❡r✐✈❛t✐♦♥ ♦❢ sin2α

❲❡ ❦♥♦✇ t❤❛t

sin (α + β) = sinαcosβ + cosαsinβ

✭✽✳✷✶✮

✶✶✽

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

❲❤❡♥ α = β✱ ✇❡ ❤❛✈❡ t❤❛t

sin (2α) = sin (α + α)

=

sinαcosα + cosαsinα

=

2sinαcosα

✭✽✳✷✷✮

=

sin (2α)

✽✳✶✳✶✳✻ ❉❡r✐✈❛t✐♦♥ ♦❢ cos2α

❲❡ ❦♥♦✇ t❤❛t

cos (α + β) = cosαcosβ − sinαsinβ

✭✽✳✷✸✮

❲❤❡♥ α = β✱ ✇❡ ❤❛✈❡ t❤❛t

cos (2α) = cos (α + α)

=

cosαcosα − sinαsinα

=

cos2α − sin2α

✭✽✳✷✹✮

=

cos (2α)

❍♦✇❡✈❡r✱ ✇❡ ❝❛♥ ❛❧s♦ ✇r✐t❡

cos2α = 2cos2α − 1

✭✽✳✷✺✮

❛♥❞

cos2α = 1 − 2sin2α

✭✽✳✷✻✮

❜② ✉s✐♥❣

sin2α + cos2α = 1.

✭✽✳✷✼✮

✽✳✶✳✶✳✻✳✶ ❚❤❡ cos2α ■❞❡♥t✐t②

❯s❡

sin2α + cos2α = 1

✭✽✳✷✽✮

t♦ s❤♦✇ t❤❛t✿

2cos2α − 1

cos2α

=

{

✭✽✳✷✾✮

1 − 2sin2α

✽✳✶✳✶✳✼ Pr♦❜❧❡♠✲s♦❧✈✐♥❣ ❙tr❛t❡❣② ❢♦r ■❞❡♥t✐t✐❡s

❚❤❡ ♠♦st ✐♠♣♦rt❛♥t t❤✐♥❣ t♦ r❡♠❡♠❜❡r ✇❤❡♥ ❛s❦❡❞ t♦ ♣r♦✈❡ ✐❞❡♥t✐t✐❡s ✐s✿

t✐♣✿ ❚r✐❣♦♥♦♠❡tr✐❝ ■❞❡♥t✐t✐❡s

✶✶✾

❲❤❡♥ ♣r♦✈✐♥❣ tr✐❣♦♥♦♠❡tr✐❝ ✐❞❡♥t✐t✐❡s✱ ♥❡✈❡r ❛ss✉♠❡ t❤❛t t❤❡ ❧❡❢t ❤❛♥❞ s✐❞❡ ✐s ❡q✉❛❧ t♦ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✳

❨♦✉ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ❜♦t❤ s✐❞❡s ❛r❡ ❡q✉❛❧✳

❆ s✉❣❣❡st✐♦♥ ❢♦r ♣r♦✈✐♥❣ ✐❞❡♥t✐t✐❡s✿ ■t ✐s ✉s✉❛❧❧② ♠✉❝❤ ❡❛s✐❡r s✐♠♣❧✐❢②✐♥❣ t❤❡ ♠♦r❡ ❝♦♠♣❧❡① s✐❞❡ ♦❢ ❛♥

✐❞❡♥t✐t② t♦ ❣❡t t❤❡ s✐♠♣❧❡r s✐❞❡ t❤❛♥ t❤❡ ♦t❤❡r ✇❛② r♦✉♥❞✳

❊①❡r❝✐s❡ ✽✳✶✿ ❚r✐❣♦♥♦♠❡tr✐❝ ■❞❡♥t✐t✐❡s ✶

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✷✺✳✮

√

√

Pr♦✈❡ t❤❛t

2( 3+1)

sin75◦ =

✇✐t❤♦✉t ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r✳

4

❊①❡r❝✐s❡ ✽✳✷✿ ❚r✐❣♦♥♦♠❡tr✐❝ ■❞❡♥t✐t✐❡s ✷

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✷✺✳✮

❉❡❞✉❝❡ ❛ ❢♦r♠✉❧❛ ❢♦r tan (α + β) ✐♥ t❡r♠s ♦❢ tanα ❛♥❞ tanβ✳

❍✐♥t✿ ❯s❡ t❤❡ ❢♦r♠✉❧❛❡ ❢♦r sin (α + β) ❛♥❞ cos (α + β)

❊①❡r❝✐s❡ ✽✳✸✿ ❚r✐❣♦♥♦♠❡tr✐❝ ■❞❡♥t✐t✐❡s ✸

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✷✺✳✮

Pr♦✈❡ t❤❛t

sinθ + sin2θ

= tanθ

✭✽✳✸✵✮

1 + cosθ + cos2θ

■♥ ❢❛❝t✱ t❤✐s ✐❞❡♥t✐t② ✐s ♥♦t ✈❛❧✐❞ ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢ θ✳ ❲❤✐❝❤ ✈❛❧✉❡s ❛r❡ t❤♦s❡❄

❊①❡r❝✐s❡ ✽✳✹✿ ❚r✐❣♦♥♦♠❡tr✐❝ ❊q✉❛t✐♦♥s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✷✻✳✮

❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥ ❢♦r y ✇✐t❤♦✉t ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r✿

1 − siny − cos2y = −1

✭✽✳✸✶✮

sin2y − cosy

✽✳✷ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❚r✐❣ ❋✉♥❝t✐♦♥s ✭✷❉ ✫ ✸❉✮✱ ❖t❤❡r ●❡♦♠❡tr✐❡s✷

✽✳✷✳✶ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❚r✐❣♦♥♦♠❡tr✐❝ ❋✉♥❝t✐♦♥s

✽✳✷✳✶✳✶ Pr♦❜❧❡♠s ✐♥ ❚✇♦ ❉✐♠❡♥s✐♦♥s

❊①❡r❝✐s❡ ✽✳✺✿ Pr♦❜❧❡♠ ✐♥ ❚✇♦ ❉✐♠❡♥s✐♦♥s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✷✻✳✮

❋♦r t❤❡ ✜❣✉r❡ ❜❡❧♦✇✱ ✇❡ ❛r❡ ❣✐✈❡♥ t❤❛t BC = BD = x✳

❙❤♦✇ t❤❛t BC2 = 2x2 (1 + sinθ)✳

❋✐❣✉r❡ ✽✳✷

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✶✵✴✶✳✶✴❃✳

✶✷✵

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

✽✳✷✳✶✳✶✳✶

✶✳ ❋♦r t❤❡ ❞✐❛❣r❛♠ ♦♥ t❤❡ r✐❣❤t✱

❫

❛✳ ❋✐♥❞ A O C ✐♥ t❡r♠s ♦❢ θ✳

❜✳ ❋✐♥❞ ❛♥ ❡①♣r❡ss✐♦♥ ❢♦r✿

✶✳ cosθ

✷✳ sinθ

✸✳ sin2θ

❝✳ ❯s✐♥❣ t❤❡ ❛❜♦✈❡✱ s❤♦✇ t❤❛t sin2θ = 2sinθcosθ✳

❞✳ ◆♦✇ ❞♦ t❤❡ s❛♠❡ ❢♦r cos2θ ❛♥❞ tanθ✳

❋✐❣✉r❡ ✽✳✸

❫

✷✳ DC ✐s ❛ ❞✐❛♠❡t❡r ♦❢ ❝✐r❝❧❡ O ✇✐t❤ r❛❞✐✉s r✳ CA = r✱ AB = DE ❛♥❞ D O E = θ✳ ❙❤♦✇ t❤❛t cosθ = 1✳

4

❋✐❣✉r❡ ✽✳✹

✸✳ ❚❤❡ ✜❣✉r❡ ❜❡❧♦✇ s❤♦✇s ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ✇✐t❤ BC = AD ✳

CD

AB

❫

❛✳ ❙❤♦✇ t❤❛t t❤❡ ❛r❡❛ ♦❢ t❤❡ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧ ✐s DC · DA · sin D✳

❫

❫

❜✳ ❋✐♥❞ ❡①♣r❡ss✐♦♥s ❢♦r cos D ❛♥❞ cos B ✐♥ t❡r♠s ♦❢ t❤❡ q✉❛❞r✐❧❛t❡r❛❧ s✐❞❡s✳

❝✳ ❙❤♦✇ t❤❛t 2CA2 = CD2 + DA2 + AB2 + BC2✳

❞✳ ❙✉♣♣♦s❡ t❤❛t BC = 10✱ CD = 15✱ AD = 4 ❛♥❞ AB = 6✳ ❋✐♥❞ CA2✳

❫

❫

❡✳ ❋✐♥❞ t❤❡ ❛♥❣❧❡ D ✉s✐♥❣ ②♦✉r ❡①♣r❡ss✐♦♥ ❢♦r cos D✳ ❍❡♥❝❡ ✜♥❞ t❤❡ ❛r❡❛ ♦❢ ABCD✳

❋✐❣✉r❡ ✽✳✺

✽✳✷✳✶✳✷ Pr♦❜❧❡♠s ✐♥ ✸ ❞✐♠❡♥s✐♦♥s

❊①❡r❝✐s❡ ✽✳✻✿ ❍❡✐❣❤t ♦❢ t♦✇❡r

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✷✻✳✮

D ✐s t❤❡ t♦♣ ♦❢ ❛ t♦✇❡r ♦❢ ❤❡✐❣❤t h✳ ■ts ❜❛s❡ ✐s ❛t C✳ ❚❤❡ tr✐❛♥❣❧❡ ABC ❧✐❡s ♦♥ t❤❡ ❣r♦✉♥❞ ✭❛

✶✷✶

❫

❫

❫

❤♦r✐③♦♥t❛❧ ♣❧❛♥❡✮✳ ■❢ ✇❡ ❤❛✈❡ t❤❛t BC = b✱ D B A = α✱ D B C = x ❛♥❞ D C B = θ✱ s❤♦✇ t❤❛t

bsinαsinx

h =

✭✽✳✸✷✮

sin (x + θ)

❋✐❣✉r❡ ✽✳✻

✽✳✷✳✶✳✷✳✶

✶✳ ❚❤❡ ❧✐♥❡ BC r❡♣r❡s❡♥ts ❛ t❛❧❧ t♦✇❡r✱ ✇✐t❤ C ❛t ✐ts ❢♦♦t✳ ■ts ❛♥❣❧❡ ♦❢ ❡❧❡✈❛t✐♦♥ ❢r♦♠ D ✐s θ✳ ❲❡ ❛r❡ ❛❧s♦

❣✐✈❡♥ t❤❛t BA = AD = x✳

❋✐❣✉r❡ ✽✳✼

❛✳ ❋✐♥❞ t❤❡ ❤❡✐❣❤t ♦❢ t❤❡ t♦✇❡r BC ✐♥ t❡r♠s ♦❢ x✱ tanθ ❛♥❞ cos2α✳

❜✳ ❋✐♥❞ BC ✐❢ ✇❡ ❛r❡ ❣✐✈❡♥ t❤❛t x = 140m✱ α = 21◦ ❛♥❞ θ = 9◦✳

✽✳✷✳✷ ❖t❤❡r ●❡♦♠❡tr✐❡s

✽✳✷✳✷✳✶ ❚❛①✐❝❛❜ ●❡♦♠❡tr②

❚❛①✐❝❛❜ ❣❡♦♠❡tr②✱ ❝♦♥s✐❞❡r❡❞ ❜② ❍❡r♠❛♥♥ ▼✐♥❦♦✇s❦✐ ✐♥ t❤❡ ✶✾t❤ ❝❡♥t✉r②✱ ✐s ❛ ❢♦r♠ ♦❢ ❣❡♦♠❡tr② ✐♥ ✇❤✐❝❤

t❤❡ ✉s✉❛❧ ♠❡tr✐❝ ♦❢ ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr② ✐s r❡♣❧❛❝❡❞ ❜② ❛ ♥❡✇ ♠❡tr✐❝ ✐♥ ✇❤✐❝❤ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts

✐s t❤❡ s✉♠ ♦❢ t❤❡ ✭❛❜s♦❧✉t❡✮ ❞✐✛❡r❡♥❝❡s ♦❢ t❤❡✐r ❝♦♦r❞✐♥❛t❡s✳

✽✳✷✳✷✳✷ ▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡

❚❤❡ ♠❡tr✐❝ ✐♥ t❛①✐✲❝❛❜ ❣❡♦♠❡tr②✱ ✐s ❦♥♦✇♥ ❛s t❤❡ ▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡✱ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐♥ ❛♥ ❊✉❝❧✐❞❡❛♥

s♣❛❝❡ ✇✐t❤ ✜①❡❞ ❈❛rt❡s✐❛♥ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ❛s t❤❡ s✉♠ ♦❢ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♣r♦❥❡❝t✐♦♥s ♦❢ t❤❡ ❧✐♥❡ s❡❣♠❡♥t

❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ♦♥t♦ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

❋♦r ❡①❛♠♣❧❡✱ t❤❡ ▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥t P1 ✇✐t❤ ❝♦♦r❞✐♥❛t❡s (x1, y1) ❛♥❞ t❤❡ ♣♦✐♥t P2

❛t (x2, y2) ✐s

|x1 − x2| + |y1 − y2|

✭✽✳✸✸✮

✶✷✷

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

❋✐❣✉r❡ ✽✳✽✿ ▼❛♥❤❛tt❛♥ ❉✐st❛♥❝❡ ✭❞♦tt❡❞ ❛♥❞ s♦❧✐❞✮ ❝♦♠♣❛r❡❞ t♦ ❊✉❝❧✐❞❡❛♥ ❉✐st❛♥❝❡ ✭❞❛s❤❡❞✮✳ ■♥ ❡❛❝❤

√

❝❛s❡ t❤❡ ▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡ ✐s ✶✷ ✉♥✐ts✱ ✇❤✐❧❡ t❤❡ ❊✉❝❧✐❞❡❛♥ ❞✐st❛♥❝❡ ✐s 36

❚❤❡ ▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡ ❝❤❛♥❣❡s ✐❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ✐s r♦t❛t❡❞✱ ❜✉t ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ t❤❡ tr❛♥s✲

❧❛t✐♦♥ ♦❢ t❤❡ ❝♦♦r❞✐♥❛t❡ s②st❡♠ ♦r ✐ts r❡✢❡❝t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❝♦♦r❞✐♥❛t❡ ❛①✐s✳

▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡ ✐s ❛❧s♦ ❦♥♦✇♥ ❛s ❝✐t② ❜❧♦❝❦ ❞✐st❛♥❝❡ ♦r t❛①✐✲❝❛❜ ❞✐st❛♥❝❡✳ ■t ✐s ❣✐✈❡♥ t❤❡s❡ ♥❛♠❡s

❜❡❝❛✉s❡ ✐t ✐s t❤❡ s❤♦rt❡st ❞✐st❛♥❝❡ ❛ ❝❛r ✇♦✉❧❞ ❞r✐✈❡ ✐♥ ❛ ❝✐t② ❧❛✐❞ ♦✉t ✐♥ sq✉❛r❡ ❜❧♦❝❦s✳

❚❛①✐❝❛❜ ❣❡♦♠❡tr② s❛t✐s✜❡s ❛❧❧ ♦❢ ❊✉❝❧✐❞✬s ❛①✐♦♠s ❡①❝❡♣t ❢♦r t❤❡ s✐❞❡✲❛♥❣❧❡✲s✐❞❡ ❛①✐♦♠✱ ❛s ♦♥❡ ❝❛♥ ❣❡♥❡r❛t❡

t✇♦ tr✐❛♥❣❧❡s ✇✐t❤ t✇♦ s✐❞❡s ❛♥❞ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡♠ t❤❡ s❛♠❡ ❛♥❞ ❤❛✈❡ t❤❡♠ ♥♦t ❜❡ ❝♦♥❣r✉❡♥t✳ ■♥

♣❛rt✐❝✉❧❛r✱ t❤❡ ♣❛r❛❧❧❡❧ ♣♦st✉❧❛t❡ ❤♦❧❞s✳

❆ ❝✐r❝❧❡ ✐♥ t❛①✐❝❛❜ ❣❡♦♠❡tr② ❝♦♥s✐sts ♦❢ t❤♦s❡ ♣♦✐♥ts t❤❛t ❛r❡ ❛ ✜①❡❞ ▼❛♥❤❛tt❛♥ ❞✐st❛♥❝❡ ❢r♦♠ t❤❡ ❝❡♥t❡r✳

❚❤❡s❡ ❝✐r❝❧❡s ❛r❡ sq✉❛r❡s ✇❤♦s❡ s✐❞❡s ♠❛❦❡ ❛ 45◦ ❛♥❣❧❡ ✇✐t❤ t❤❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳

❚❤❡ ❣r❡❛t✲❝✐r❝❧❡ ❞✐st❛♥❝❡ ✐s t❤❡ s❤♦rt❡st ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ❛♥② t✇♦ ♣♦✐♥ts ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ ❛ s♣❤❡r❡

♠❡❛s✉r❡❞ ❛❧♦♥❣ ❛ ♣❛t❤ ♦♥ t❤❡ s✉r❢❛❝❡ ♦❢ t❤❡ s♣❤❡r❡ ✭❛s ♦♣♣♦s❡❞ t♦ ❣♦✐♥❣ t❤r♦✉❣❤ t❤❡ s♣❤❡r❡✬s ✐♥t❡r✐♦r✮✳

❇❡❝❛✉s❡ s♣❤❡r✐❝❛❧ ❣❡♦♠❡tr② ✐s r❛t❤❡r ❞✐✛❡r❡♥t ❢r♦♠ ♦r❞✐♥❛r② ❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✱ t❤❡ ❡q✉❛t✐♦♥s ❢♦r ❞✐st❛♥❝❡

t❛❦❡ ♦♥ ❛ ❞✐✛❡r❡♥t ❢♦r♠✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t✇♦ ♣♦✐♥ts ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ ❛ str❛✐❣❤t ❧✐♥❡

❢r♦♠ ♦♥❡ ♣♦✐♥t t♦ t❤❡ ♦t❤❡r✳ ❖♥ t❤❡ s♣❤❡r❡✱ ❤♦✇❡✈❡r✱ t❤❡r❡ ❛r❡ ♥♦ str❛✐❣❤t ❧✐♥❡s✳ ■♥ ♥♦♥✲❊✉❝❧✐❞❡❛♥ ❣❡♦♠❡tr②✱

str❛✐❣❤t ❧✐♥❡s ❛r❡ r❡♣❧❛❝❡❞ ✇✐t❤ ❣❡♦❞❡s✐❝s✳ ●❡♦❞❡s✐❝s ♦♥ t❤❡ s♣❤❡r❡ ❛r❡ t❤❡ ❣r❡❛t ❝✐r❝❧❡s ✭❝✐r❝❧❡s ♦♥ t❤❡ s♣❤❡r❡

✇❤♦s❡ ❝❡♥t❡rs ❛r❡ ❝♦✐♥❝✐❞❡♥t ✇✐t❤ t❤❡ ❝❡♥t❡r ♦❢ t❤❡ s♣❤❡r❡✮✳ ❚❤❡ s❤❛♣❡ ♦❢ t❤❡ ❊❛rt❤ ♠♦r❡ ❝❧♦s❡❧② r❡s❡♠❜❧❡s

❛ ✢❛tt❡♥❡❞ s♣❤❡r♦✐❞ ✇✐t❤ ❡①tr❡♠❡ ✈❛❧✉❡s ❢♦r t❤❡ r❛❞✐✉s ♦❢ ❝✉r✈❛t✉r❡✱ ♦r ❛r❝r❛❞✐✉s✱ ♦❢ ✻✸✸✺✳✹✸✼ ❦♠ ❛t t❤❡

❡q✉❛t♦r ✭✈❡rt✐❝❛❧❧②✮ ❛♥❞ ✻✸✾✾✳✺✾✷ ❦♠ ❛t t❤❡ ♣♦❧❡s✱ ❛♥❞ ❤❛✈✐♥❣ ❛♥ ❛✈❡r❛❣❡ ❣r❡❛t✲❝✐r❝❧❡ r❛❞✐✉s ♦❢ ✻✸✼✷✳✼✾✺ ❦♠✳

✶✷✸

✽✳✷✳✸ ❙✉♠♠❛r② ♦❢ t❤❡ ❚r✐❣♦♥♦♠❡rt✐❝ ❘✉❧❡s ❛♥❞ ■❞❡♥t✐t✐❡s

P②t❤❛❣♦r❡❛♥ ■❞❡♥t✐t②

❈♦❢✉♥❝t✐♦♥ ■❞❡♥t✐t✐❡s

❘❛t✐♦ ■❞❡♥t✐t✐❡s

cos2θ + sin2θ = 1

sin (90◦ − θ) = cosθ

tanθ = sinθ

cosθ

cos (90◦ − θ) = sinθ

❖❞❞✴❊✈❡♥ ■❞❡♥t✐t✐❡s

P❡r✐♦❞✐❝✐t② ■❞❡♥t✐t✐❡s

❉♦✉❜❧❡ ❛♥❣❧❡ ■❞❡♥t✐t✐❡s

sin (−θ) = −sinθ

sin (θ ± 360◦) = sinθ

sin (2θ) = 2sinθcosθ

cos (−θ) = cosθ

cos (θ ± 360◦) = cosθ

cos (2θ) = cos2θ − sin2θ

tan (−θ) = −tanθ

tan (θ ± 180◦) = tanθ

cos (2θ) = 2cos2θ − 1

tan (2θ) =

2tanθ

1−tan2θ

❆❞❞✐t✐♦♥✴❙✉❜tr❛❝t✐♦♥ ■❞❡♥t✐t✐❡s

❆r❡❛ ❘✉❧❡

❈♦s✐♥❡ r✉❧❡

sin (θ + φ) = sinθcosφ + cosθsinφ

Area = 1 bcsinA

a2 = b2 + c2 − 2bccosA

2

sin (θ − φ) = sinθcosφ − cosθsinφ

Area = 1 absinC

b2 = a2 + c2 − 2accosB

2

cos (θ + φ) = cosθcosφ − sinθsinφ

Area = 1 acsinB

c2 = a2 + b2 − 2abcosC

2

cos (θ − φ) = cosθcosφ + sinθsinφ

tan (θ + φ) = tanφ+tanθ

1−tanθtanφ

tan (θ − φ) = tanφ−tanθ

1+tanθtanφ

❙✐♥❡ ❘✉❧❡

sinA = sinB = sinC

a

b

c

❚❛❜❧❡ ✽✳✶

✽✳✷✳✹ ❊♥❞ ♦❢ ❈❤❛♣t❡r ❊①❡r❝✐s❡s

❉♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇✐t❤♦✉t ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r✳

✶✳ ❙✉♣♣♦s❡ cosθ = 0.7✳ ❋✐♥❞ cos2θ ❛♥❞ cos4θ✳

✷✳ ■❢ sinθ = 4✱ ❛❣❛✐♥ ✜♥❞ cos2θ ❛♥❞ cos4θ✳

7

✶✷✹

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

✸✳ ❲♦r❦ ♦✉t t❤❡ ❢♦❧❧♦✇✐♥❣✿

❛✳ cos15◦

❜✳ cos75◦

❝✳ tan105◦

❞✳ cos15◦

❡✳ cos3◦cos42◦ − sin3◦sin42◦

❢✳ 1 − 2sin2 (22.5◦)

✹✳ ❙♦❧✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s✿

❛✳ cos3θ · cosθ − sin3θ · sinθ = −12

❜✳ 3sinθ = 2cos2θ

✺✳ Pr♦✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐❞❡♥t✐t✐❡s

❛✳ sin3θ = 3sinθ−sin3θ

4

❜✳ cos2α 1 − tan2α = cos2α

❝✳ 4sinθ · cosθ · cos2θ = sin4θ

❞✳ 4cos3x − 3cosx = cos3x

❡✳ tany = sin2y

cos2y+1

✻✳ ✭❈❤❛❧❧❡♥❣❡ q✉❡st✐♦♥✦✮ ■❢ a + b + c = 180◦✱ ♣r♦✈❡ t❤❛t

sin3a + sin3b + sin3c = 3cos (a/2) cos (b/2) cos (c/2) + cos (3a/2) cos (3b/2) cos (3c/2)

✭✽✳✸✹✮

✶✷✺

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✽

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✽✳✶ ✭♣✳ ✶✶✾✮

❙t❡♣ ✶✳ ❲❡ ♦♥❧② ❦♥♦✇ t❤❡ ❡①❛❝t ✈❛❧✉❡s ♦❢ t❤❡ tr✐❣ ❢✉♥❝t✐♦♥s ❢♦r ❛ ❢❡✇ s♣❡❝✐❛❧ ❛♥❣❧❡s ✭30◦✱ 45◦✱ 60◦✱ ❡t❝✳✮✳ ❲❡

❝❛♥ s❡❡ t❤❛t 75◦ = 30◦ + 45◦✳ ❚❤✉s ✇❡ ❝❛♥ ✉s❡ ♦✉r ❞♦✉❜❧❡✲❛♥❣❧❡ ✐❞❡♥t✐t② ❢♦r sin (α + β) t♦ ❡①♣r❡ss

sin75◦ ✐♥ t❡r♠s ♦❢ ❦♥♦✇♥ tr✐❣ ❢✉♥❝t✐♦♥ ✈❛❧✉❡s✳

❙t❡♣ ✷✳

sin75◦

=

sin (45◦ + 30◦)

=

sin (45◦) cos (30◦) + cos (45◦) sin (30◦)

√

=

1

√

·

3 + 1 · 1

√

2

2

2

2

√

✭✽✳✸✺✮

=

3+1

√

2

2

√

√

=

3+1

√

×

2

√

2

2

2

√

√

2( 3+1)

=

4

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✽✳✷ ✭♣✳ ✶✶✾✮

❙t❡♣ ✶✳ ❲❡ ❝❛♥ ❡①♣r❡ss tan (α + β) ✐♥ t❡r♠s ♦❢ ❝♦s✐♥❡s ❛♥❞ s✐♥❡s✱ ❛♥❞ t❤❡♥ ✉s❡ t❤❡ ❞♦✉❜❧❡✲❛♥❣❧❡ ❢♦r♠✉❧❛s ❢♦r

t❤❡s❡✳ ❲❡ t❤❡♥ ♠❛♥✐♣✉❧❛t❡ t❤❡ r❡s✉❧t✐♥❣ ❡①♣r❡ss✐♦♥ ✐♥ ♦r❞❡r t♦ ❣❡t ✐t ✐♥ t❡r♠s ♦❢ tanα ❛♥❞ tanβ✳

❙t❡♣ ✷✳

tan (α + β)

=

sin(α+β)

cos(α+β)

=

sinα·cosβ+cosα·sinβ

cosα·cosβ−sinα·sinβ

sinα·cosβ

✭✽✳✸✻✮

+ cosα·sinβ

=

cosα·cosβ

cosα·cosβ

cosα·cosβ − sinα·sinβ

cosα·cosβ

cosα·cosβ

=

tanα+tanβ

1−tanα·tanβ

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✽✳✸ ✭♣✳ ✶✶✾✮

❙t❡♣ ✶✳ ❚❤❡ r✐❣❤t✲❤❛♥❞ s✐❞❡ ✭❘❍❙✮ ♦❢ t❤❡ ✐❞❡♥t✐t② ❝❛♥♥♦t ❜❡ s✐♠♣❧✐✜❡❞✳ ❚❤✉s ✇❡ s❤♦✉❧❞ tr② s✐♠♣❧✐❢② t❤❡ ❧❡❢t✲

❤❛♥❞ s✐❞❡ ✭▲❍❙✮✳ ❲❡ ❝❛♥ ❛❧s♦ ♥♦t✐❝❡ t❤❛t t❤❡ tr✐❣ ❢✉♥❝t✐♦♥ ♦♥ t❤❡ ❘❍❙ ❞♦❡s ♥♦t ❤❛✈❡ ❛ 2θ ❞❡♣❡♥❞❛♥❝❡✳

❚❤✉s ✇❡ ✇✐❧❧ ♥❡❡❞ t♦ ✉s❡ t❤❡ ❞♦✉❜❧❡✲❛♥❣❧❡ ❢♦r♠✉❧❛s t♦ s✐♠♣❧✐❢② t❤❡ sin2θ ❛♥❞ cos2θ ♦♥ t❤❡ ▲❍❙✳ ❲❡

❦♥♦✇ t❤❛t tanθ ✐s ✉♥❞❡✜♥❡❞ ❢♦r s♦♠❡ ❛♥❣❧❡s θ✳ ❚❤✉s t❤❡ ✐❞❡♥t✐t② ✐s ❛❧s♦ ✉♥❞❡✜♥❡❞ ❢♦r t❤❡s❡ θ✱ ❛♥❞

❤❡♥❝❡ ✐s ♥♦t ✈❛❧✐❞ ❢♦r t❤❡s❡ ❛♥❣❧❡s✳ ❆❧s♦✱ ❢♦r s♦♠❡ θ✱ ✇❡ ♠✐❣❤t ❤❛✈❡ ❞✐✈✐s✐♦♥ ❜② ③❡r♦ ✐♥ t❤❡ ▲❍❙✱ ✇❤✐❝❤

✐s ♥♦t ❛❧❧♦✇❡❞✳ ❚❤✉s t❤❡ ✐❞❡♥t✐t② ✇♦♥✬t ❤♦❧❞ ❢♦r t❤❡s❡ ❛♥❣❧❡s ❛❧s♦✳

❙t❡♣ ✷✳

LHS

=

sinθ+2 sinθ cosθ

1+cosθ+(2cos2θ−1)

=

sinθ(1+2cosθ)

cosθ(1+2cosθ)

=

sinθ

✭✽✳✸✼✮

cosθ

=

tanθ

=

RHS

❲❡ ❦♥♦✇ t❤❛t tanθ ✐s ✉♥❞❡✜♥❡❞ ✇❤❡♥ θ = 90◦ + 180◦n✱ ✇❤❡r❡ n ✐s ❛♥ ✐♥t❡❣❡r✳ ❚❤❡ ▲❍❙ ✐s ✉♥❞❡✜♥❡❞

✇❤❡♥ 1 + cosθ + cos2θ = 0✳ ❚❤✉s ✇❡ ♥❡❡❞ t♦ s♦❧✈❡ t❤✐s ❡q✉❛t✐♦♥✳

1 + cosθ + cos2θ

=

0

✭✽✳✸✽✮

⇒

cosθ (1 + 2cosθ)

=

0

❚❤❡ ❛❜♦✈❡ ❤❛s s♦❧✉t✐♦♥s ✇❤❡♥ cosθ = 0✱ ✇❤✐❝❤ ♦❝❝✉rs ✇❤❡♥ θ = 90◦ + 180◦n✱ ✇❤❡r❡ n ✐s ❛♥ ✐♥t❡❣❡r✳

❚❤❡s❡ ❛r❡ t❤❡ s❛♠❡ ✈❛❧✉❡s ✇❤❡♥ tanθ ✐s ✉♥❞❡✜♥❡❞✳ ■t ❛❧s♦ ❤❛s s♦❧✉t✐♦♥s ✇❤❡♥ 1 + 2cosθ = 0✳ ❚❤✐s ✐s

✶✷✻

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

tr✉❡ ✇❤❡♥ cosθ = −1✱ ❛♥❞ t❤✉s θ = ... − 240◦, −120◦, 120◦, 240◦, ...✳ ❚♦ s✉♠♠❛r✐s❡✱ t❤❡ ✐❞❡♥t✐t② ✐s ♥♦t

2

✈❛❧✐❞ ✇❤❡♥ θ = ... − 270◦, −240◦, −120◦, −90◦, 90◦, 120◦, 240◦, 270◦, ...

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✽✳✹ ✭♣✳ ✶✶✾✮

❙t❡♣ ✶✳ ❇❡❢♦r❡ ✇❡ ❛r❡ ❛❜❧❡ t♦ s♦❧✈❡ t❤❡ ❡q✉❛t✐♦♥✱ ✇❡ ✜rst ♥❡❡❞ t♦ s✐♠♣❧✐❢② t❤❡ ❧❡❢t✲❤❛♥❞ s✐❞❡✳ ❲❡ ❞♦ t❤✐s ❜②

✉s✐♥❣ t❤❡ ❞♦✉❜❧❡✲❛♥❣❧❡ ❢♦r♠✉❧❛s✳

❙t❡♣ ✷✳

1−siny−(1−2sin2y)

=

−1

2sinycosy−cosy

⇒

2sin2y−siny

=

−1

cosy(2siny−1)

⇒

siny(2siny−1)

=

−1

✭✽✳✸✾✮

cosy(2siny−1)

⇒

tany

=

−1

⇒

y = 135◦ + 180◦n; n ∈ Z

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✽✳✺ ✭♣✳ ✶✶✾✮

❫

❙t❡♣ ✶✳ ❲❡ ✇❛♥t CB✱ ❛♥❞ ✇❡ ❤❛✈❡ CD ❛♥❞ BD✳ ■❢ ✇❡ ❝♦✉❧❞ ❣❡t t❤❡ ❛♥❣❧❡ B D C✱ t❤❡♥ ✇❡ ❝♦✉❧❞ ✉s❡ t❤❡

❝♦s✐♥❡ r✉❧❡ t♦ ❞❡t❡r♠✐♥❡ BC✳ ❚❤✐s ✐s ♣♦ss✐❜❧❡✱ ❛s [U+25B5]ABD ✐s ❛ r✐❣❤t✲❛♥❣❧❡❞ tr✐❛♥❣❧❡✳ ❲❡ ❦♥♦✇

t❤✐s ❢r♦♠ ❝✐r❝❧❡ ❣❡♦♠❡tr②✱ t❤❛t ❛♥② tr✐❛♥❣❧❡ ❝✐r❝✉♠s❝r✐❜❡❞ ❜② ❛ ❝✐r❝❧❡ ✇✐t❤ ♦♥❡ s✐❞❡ ❣♦✐♥❣ t❤r♦✉❣❤ t❤❡

❫

❫

♦r✐❣✐♥✱ ✐s r✐❣❤t✲❛♥❣❧❡❞✳ ❆s ✇❡ ❤❛✈❡ t✇♦ ❛♥❣❧❡s ♦❢ [U+25B5]ABD✱ ✇❡ ❦♥♦✇ A D B ❛♥❞ ❤❡♥❝❡ B D C✳

❯s✐♥❣ t❤❡ ❝♦s✐♥❡ r✉❧❡✱ ✇❡ ❝❛♥ ❣❡t BC2✳

❙t❡♣ ✷✳

❫

A D B = 180◦ − θ − 90◦ = 90◦ − θ

✭✽✳✹✵✮

❚❤✉s

❫

❫

B D C

=

180◦ − A D B

=

180◦ − (90◦ − θ)

✭✽✳✹✶✮

=

90◦ + θ

◆♦✇ t❤❡ ❝♦s✐♥❡ r✉❧❡ ❣✐✈❡s

❫

BC2

=

CD2 + BD2 − 2 · CD · BD · cos

B D C

=

x2 + x2 − 2 · x2 · cos (90◦ + θ)

✭✽✳✹✷✮

=

2x2 + 2x2 [ sin (90◦) cos (θ) + sin (θ) cos (90◦)]

=

2x2 + 2x2 [ 1 · cos (θ) + sin (θ) · 0]

=

2x2 (1 − sinθ)

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✽✳✻ ✭♣✳ ✶✷✵✮

❙t❡♣ ✶✳ ❲❡ ❤❛✈❡ t❤❛t t❤❡ tr✐❛♥❣❧❡ ABD ✐s r✐❣❤t✲❛♥❣❧❡❞✳ ❚❤✉s ✇❡ ❝❛♥ r❡❧❛t❡ t❤❡ ❤❡✐❣❤t h ✇✐t❤ t❤❡ ❛♥❣❧❡ α

❛♥❞ ❡✐t❤❡r t❤❡ ❧❡♥❣t❤ BA ♦r BD ✭✉s✐♥❣ s✐♥❡s ♦r ❝♦s✐♥❡s✮✳ ❇✉t ✇❡ ❤❛✈❡ t✇♦ ❛♥❣❧❡s ❛♥❞ ❛ ❧❡♥❣t❤ ❢♦r

[U+25B5]BCD✱ ❛♥❞ t❤✉s ❝❛♥ ✇♦r❦ ♦✉t ❛❧❧ t❤❡ r❡♠❛✐♥✐♥❣ ❧❡♥❣t❤s ❛♥❞ ❛♥❣❧❡s ♦❢ t❤✐s tr✐❛♥❣❧❡✳ ❲❡ ❝❛♥

t❤✉s ✇♦r❦ ♦✉t BD✳

❙t❡♣ ✷✳ ❲❡ ❤❛✈❡ t❤❛t

h

=

sinα

BD

✭✽✳✹✸✮

⇒

h

=

BDsinα

✶✷✼

◆♦✇ ✇❡ ♥❡❡❞ BD ✐♥ t❡r♠s ♦❢ t❤❡ ❣✐✈❡♥ ❛♥❣❧❡s ❛♥❞ ❧❡♥❣t❤ b✳ ❈♦♥s✐❞❡r✐♥❣ t❤❡ tr✐❛♥❣❧❡ BCD✱ ✇❡ s❡❡

t❤❛t ✇❡ ❝❛♥ ✉s❡ t❤❡ s✐♥❡ r✉❧❡✳

0

❫ 1

sin@BDCA

sinθ

=

BD

b

✭✽✳✹✹✮

⇒

BD

=

bsinθ

0

❫ 1

sin@bDCA

❫

❇✉t D B C = 180◦ − α − θ✱ ❛♥❞

sin (180◦ − α − θ)

=

−sin (−α − θ)

✭✽✳✹✺✮

=

sin (α + θ)

❙♦

BD

=

bsinθ

0

❫ 1

sin@DBCA

✭✽✳✹✻✮

=

bsinθ

sin(α+θ)

✶✷✽

❈❍❆P❚❊❘ ✽✳ ❚❘■●❖◆❖▼❊❚❘❨

❈❤❛♣t❡r ✾

❙t❛t✐st✐❝s

✾✳✶ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥✱ ❙❛♠♣❧✐♥❣✱ ❋✉♥❝t✐♦♥ ❋✐tt✐♥❣ ✫ ❘❡❣r❡ss✐♦♥

❆♥❛❧②s✐s✶

✾✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

■♥ t❤✐s ❝❤❛♣t❡r✱ ②♦✉ ✇✐❧❧ ✉s❡ t❤❡ ♠❡❛♥✱ ♠❡❞✐❛♥✱ ♠♦❞❡ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ ❛ s❡t ♦❢ ❞❛t❛ t♦ ✐❞❡♥t✐❢②

✇❤❡t❤❡r t❤❡ ❞❛t❛ ✐s ♥♦r♠❛❧❧② ❞✐str✐❜✉t❡❞ ♦r ✇❤❡t❤❡r ✐t ✐s s❦❡✇❡❞✳ ❨♦✉ ✇✐❧❧ ❧❡❛r♥ ♠♦r❡ ❛❜♦✉t ♣♦♣✉❧❛t✐♦♥s

❛♥❞ s❡❧❡❝t✐♥❣ ❞✐✛❡r❡♥t ❦✐♥❞s ♦❢ s❛♠♣❧❡s ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❜✐❛s✳ ❨♦✉ ✇✐❧❧ ✇♦r❦ ✇✐t❤ ❧✐♥❡s ♦❢ ❜❡st ✜t✱ ❛♥❞ ❧❡❛r♥

❤♦✇ t♦ ✜♥❞ ❛ r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥ ❛♥❞ ❛ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t✳ ❨♦✉ ✇✐❧❧ ❛♥❛❧②s❡ t❤❡s❡ ♠❡❛s✉r❡s ✐♥ ♦r❞❡r t♦

❞r❛✇ ❝♦♥❝❧✉s✐♦♥s ❛♥❞ ♠❛❦❡ ♣r❡❞✐❝t✐♦♥s✳

✾✳✶✳✷ ❆ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥

✾✳✶✳✷✳✶ ■♥✈❡st✐❣❛t✐♦♥ ✿

❨♦✉ ❛r❡ ❣✐✈❡♥ ❛ t❛❜❧❡ ♦❢ ❞❛t❛ ❜❡❧♦✇✳

✼✺ ✻✼ ✼✵ ✼✶ ✼✶ ✼✸ ✼✹ ✼✺

✽✵ ✼✺ ✼✼ ✼✽ ✼✽ ✼✽ ✼✽ ✼✾

✾✶ ✽✶ ✽✷ ✽✷ ✽✸ ✽✻ ✽✻ ✽✼

❚❛❜❧❡ ✾✳✶

✶✳ ❈❛❧❝✉❧❛t❡ t❤❡ ♠❡❛♥✱ ♠❡❞✐❛♥✱ ♠♦❞❡ ❛♥❞ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ❞❛t❛✳

✷✳ ❲❤❛t ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡ ❞❛t❛ ✐s ✇✐t❤✐♥ ♦♥❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥❄

✸✳ ❉r❛✇ ❛ ❤✐st♦❣r❛♠ ♦❢ t❤❡ ❞❛t❛ ✉s✐♥❣ ✐♥t❡r✈❛❧s 60 ≤ x < 64✱ 64 ≤ x < 68✱ ❡t❝✳

✹✳ ❏♦✐♥ t❤❡ ♠✐❞♣♦✐♥ts ♦❢ t❤❡ ❜❛rs t♦ ❢♦r♠ ❛ ❢r❡q✉❡♥❝② ♣♦❧②❣♦♥✳

■❢ ❧❛r❣❡ ♥✉♠❜❡rs ♦❢ ❞❛t❛ ❛r❡ ❝♦❧❧❡❝t❡❞ ❢r♦♠ ❛ ♣♦♣✉❧❛t✐♦♥✱ t❤❡ ❣r❛♣❤ ✇✐❧❧ ♦❢t❡♥ ❤❛✈❡ ❛ ❜❡❧❧ s❤❛♣❡✳ ■❢ t❤❡ ❞❛t❛

✇❛s✱ s❛②✱ ❡①❛♠✐♥❛t✐♦♥ r❡s✉❧ts✱ ❛ ❢❡✇ ❧❡❛r♥❡rs ✉s✉❛❧❧② ❣❡t ✈❡r② ❤✐❣❤ ♠❛r❦s✱ ❛ ❢❡✇ ✈❡r② ❧♦✇ ♠❛r❦s ❛♥❞ ♠♦st

❣❡t ❛ ♠❛r❦ ✐♥ t❤❡ ♠✐❞❞❧❡ r❛♥❣❡✳ ❲❡ s❛② ❛ ❞✐str✐❜✉t✐♦♥ ✐s ♥♦r♠❛❧ ✐❢

• t❤❡ ♠❡❛♥✱ ♠❡❞✐❛♥ ❛♥❞ ♠♦❞❡ ❛r❡ ❡q✉❛❧✳

• ✐t ✐s s②♠♠❡tr✐❝ ❛r♦✉♥❞ t❤❡ ♠❡❛♥✳

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✵✺✴✶✳✶✴❃✳

✶✷✾

✶✸✵

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

• ±68% ♦❢ t❤❡ s❛♠♣❧❡ ❧✐❡s ✇✐t❤✐♥ ♦♥❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ ♠❡❛♥✱ 95% ✇✐t❤✐♥ t✇♦ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s

❛♥❞ 99% ✇✐t❤✐♥ t❤r❡❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s ♦❢ t❤❡ ♠❡❛♥✳

❋✐❣✉r❡ ✾✳✶

❲❤❛t ❤❛♣♣❡♥s ✐❢ t❤❡ t❡st ✇❛s ✈❡r② ❡❛s② ♦r ✈❡r② ❞✐✣❝✉❧t❄ ❚❤❡♥ t❤❡ ❞✐str✐❜✉t✐♦♥ ♠❛② ♥♦t ❜❡ s②♠♠❡tr✐❝❛❧✳

■❢ ❡①tr❡♠❡❧② ❤✐❣❤ ♦r ❡①tr❡♠❡❧② ❧♦✇ s❝♦r❡s ❛r❡ ❛❞❞❡❞ t♦ ❛ ❞✐str✐❜✉t✐♦♥✱ t❤❡♥ t❤❡ ♠❡❛♥ t❡♥❞s t♦ s❤✐❢t t♦✇❛r❞s

t❤❡s❡ s❝♦r❡s ❛♥❞ t❤❡ ❝✉r✈❡ ❜❡❝♦♠❡s s❦❡✇❡❞✳

■❢ t❤❡ t❡st ✇❛s ✈❡r② ❞✐✣❝✉❧t✱ t❤❡ ♠❡❛♥ s❝♦r❡ ✐s s❤✐❢t❡❞ t♦ t❤❡ ❧❡❢t✳ ■♥ t❤✐s ❝❛s❡✱ ✇❡ s❛② t❤❡ ❞✐str✐❜✉t✐♦♥

✐s ♣♦s✐t✐✈❡❧② s❦❡✇❡❞✱ ♦r s❦❡✇❡❞ r✐❣❤t✳ ■❢ ✐t ✇❛s ✈❡r② ❡❛s②✱ t❤❡♥ ♠❛♥② ❧❡❛r♥❡rs ✇♦✉❧❞ ❣❡t ❤✐❣❤ s❝♦r❡s✱ ❛♥❞

t❤❡ ♠❡❛♥ ♦❢ t❤❡ ❞✐str✐❜✉t✐♦♥ ✇♦✉❧❞ ❜❡ s❤✐❢t❡❞ t♦ t❤❡ r✐❣❤t✳ ❲❡ s❛② t❤❡ ❞✐str✐❜✉t✐♦♥ ✐s ♥❡❣❛t✐✈❡❧② s❦❡✇❡❞✱ ♦r

s❦❡✇❡❞ ❧❡❢t✳

❋✐❣✉r❡ ✾✳✷

❋✐❣✉r❡ ✾✳✸

✶✸✶

✾✳✶✳✷✳✷ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥

✶✳ ●✐✈❡♥ t❤❡ ♣❛✐rs ♦❢ ♥♦r♠❛❧ ❝✉r✈❡s ❜❡❧♦✇✱ s❦❡t❝❤ t❤❡ ❣r❛♣❤s ♦♥ t❤❡ s❛♠❡ s❡t ♦❢ ❛①❡s ❛♥❞ s❤♦✇ ❛♥② r❡❧❛t✐♦♥

❜❡t✇❡❡♥ t❤❡♠✳ ❆♥ ✐♠♣♦rt❛♥t ♣♦✐♥t t♦ r❡♠❡♠❜❡r ✐s t❤❛t t❤❡ ❛r❡❛ ❜❡♥❡❛t❤ t❤❡ ❝✉r✈❡ ❝♦rr❡s♣♦♥❞s t♦

✶✵✵✪✳

❛✳ ▼❡❛♥ ❂ ✽✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ❂ ✹ ❛♥❞ ▼❡❛♥ ❂ ✹✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ❂ ✽

❜✳ ▼❡❛♥ ❂ ✽✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ❂ ✹ ❛♥❞ ▼❡❛♥ ❂ ✶✻✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ❂ ✹

❝✳ ▼❡❛♥ ❂ ✽✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ❂ ✹ ❛♥❞ ▼❡❛♥ ❂ ✽✱ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ❂ ✽

✷✳ ❆❢t❡r ❛ ❝❧❛ss t❡st✱ t❤❡ ❢♦❧❧♦✇✐♥❣ s❝♦r❡s ✇❡r❡ r❡❝♦r❞❡❞✿

❚❡st ❙❝♦r❡

❋r❡q✉❡♥❝②

✸

✶

✹

✼

✺

✶✹

✻

✷✶

✼

✶✹

✽

✻

✾

✶

❚♦t❛❧

✻✹

▼❡❛♥

✻

❙t❛♥❞❛r❞ ❉❡✈✐❛t✐♦♥ ✶✱✷

❚❛❜❧❡ ✾✳✷

❛✳ ❉r❛✇ t❤❡ ❤✐st♦❣r❛♠ ♦❢ t❤❡ r❡s✉❧ts✳

❜✳ ❏♦✐♥ t❤❡ ♠✐❞♣♦✐♥ts ♦❢ ❡❛❝❤ ❜❛r ❛♥❞ ❞r❛✇ ❛ ❢r❡q✉❡♥❝② ♣♦❧②❣♦♥✳

❝✳ ❲❤❛t ♠❛r❦ ♠✉st ♦♥❡ ♦❜t❛✐♥ ✐♥ ♦r❞❡r t♦ ❜❡ ✐♥ t❤❡ t♦♣ ✷✪ ♦❢ t❤❡ ❝❧❛ss❄

❞✳ ❆♣♣r♦①✐♠❛t❡❧② ✽✹✪ ♦❢ t❤❡ ♣✉♣✐❧s ♣❛ss❡❞ t❤❡ t❡st✳ ❲❤❛t ✇❛s t❤❡ ♣❛ss ♠❛r❦❄

❡✳ ■s t❤❡ ❞✐str✐❜✉t✐♦♥ ♥♦r♠❛❧ ♦r s❦❡✇❡❞❄

✸✳ ■♥ ❛ r♦❛❞ s❛❢❡t② st✉❞②✱ t❤❡ s♣❡❡❞ ♦❢ ✶✼✺ ❝❛rs ✇❛s ♠♦♥✐t♦r❡❞ ❛❧♦♥❣ ❛ s♣❡❝✐✜❝ str❡t❝❤ ♦❢ ❤✐❣❤✇❛② ✐♥ ♦r❞❡r

t♦ ✜♥❞ ♦✉t ✇❤❡t❤❡r t❤❡r❡ ❡①✐st❡❞ ❛♥② ❧✐♥❦ ❜❡t✇❡❡♥ ❤✐❣❤ s♣❡❡❞ ❛♥❞ t❤❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❛❝❝✐❞❡♥ts ❛❧♦♥❣

t❤❡ r♦✉t❡✳ ❆ ❢r❡q✉❡♥❝② t❛❜❧❡ ♦❢ t❤❡ r❡s✉❧ts ✐s ❞r❛✇♥ ✉♣ ❜❡❧♦✇✳

❙♣❡❡❞ ✭❦♠✳❤−1✮ ◆✉♠❜❡r ♦❢ ❝❛rs ✭❋r❡q✉❡♥❝②✮

✺✵

✶✾

✻✵

✷✽

✼✵

✷✸

✽✵

✺✻

✾✵

✷✵

✶✵✵

✶✻

✶✶✵

✽

✶✷✵

✺

✶✸✷

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

❚❛❜❧❡ ✾✳✸

❚❤❡ ♠❡❛♥ s♣❡❡❞ ✇❛s ❞❡t❡r♠✐♥❡❞ t♦ ❜❡ ❛r♦✉♥❞ ✽✷ ❦♠✳❤−1 ✇❤✐❧❡ t❤❡ ♠❡❞✐❛♥ s♣❡❡❞ ✇❛s ✇♦r❦❡❞ ♦✉t t♦

❜❡ ❛r♦✉♥❞ ✽✹✱✺ ❦♠✳❤−1✳

❛✳ ❉r❛✇ ❛ ❢r❡q✉❡♥❝② ♣♦❧②❣♦♥ t♦ ✈✐s✉❛❧✐s❡ t❤❡ ❞❛t❛ ✐♥ t❤❡ t❛❜❧❡ ❛❜♦✈❡✳

❜✳ ■s t❤✐s ❞✐str✐❜✉t✐♦♥ s②♠♠❡tr✐❝❛❧ ♦r s❦❡✇❡❞ ❧❡❢t ♦r r✐❣❤t❄ ●✐✈❡ ❛ r❡❛s♦♥ ❢r♦ ②♦✉r ❛♥s✇❡r✳

✾✳✶✳✸ ❊①tr❛❝t✐♥❣ ❛ ❙❛♠♣❧❡ P♦♣✉❧❛t✐♦♥

❙✉♣♣♦s❡ ②♦✉ ❛r❡ tr②✐♥❣ t♦ ✜♥❞ ♦✉t ✇❤❛t ♣❡r❝❡♥t❛❣❡ ♦❢ ❙♦✉t❤ ❆❢r✐❝❛✬s ♣♦♣✉❧❛t✐♦♥ ♦✇♥s ❛ ❝❛r✳ ❖♥❡ ✇❛② ♦❢

❞♦✐♥❣ t❤✐s ♠✐❣❤t ❜❡ t♦ s❡♥❞ q✉❡st✐♦♥♥❛✐r❡s t♦ ♣❡♦♣❧❡s ❤♦♠❡s✱ ❛s❦✐♥❣ t❤❡♠ ✇❤❡t❤❡r t❤❡② ♦✇♥ ❛ ❝❛r✳ ❍♦✇❡✈❡r✱

②♦✉ q✉✐❝❦❧② r✉♥ ✐♥t♦ ❛ ♣r♦❜❧❡♠✿ ②♦✉ ❝❛♥♥♦t ❤♦♣❡ t♦ s❡♥❞ ❡✈❡r② ♣❡rs♦♥ ✐♥ t❤❡ ❝♦✉♥tr② ❛ q✉❡st✐♦♥♥❛✐r❡✱ ✐t

✇♦✉❧❞ ❜❡ ❢❛r t♦ ❡①♣❡♥s✐✈❡✳ ❆❧s♦✱ ♥♦t ❡✈❡r②♦♥❡ ✇♦✉❧❞ r❡♣❧②✳ ❚❤❡ ❜❡st ②♦✉ ❝❛♥ ❞♦ ✐s s❡♥❞ ✐t t♦ ❛ ❢❡✇ ♣❡♦♣❧❡✱

s❡❡ ✇❤❛t ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡s❡ ♦✇♥ ❛ ❝❛r✱ ❛♥❞ t❤❡♥ ✉s❡ t❤✐s t♦ ❡st✐♠❛t❡ ✇❤❛t ♣❡r❝❡♥t❛❣❡ ♦❢ t❤❡ ❡♥t✐r❡ ❝♦✉♥tr②

♦✇♥ ❝❛rs✳ ❚❤✐s s♠❛❧❧❡r ❣r♦✉♣ ♦❢ ♣❡♦♣❧❡ ✐s ❝❛❧❧❡❞ t❤❡ s❛♠♣❧❡ ♣♦♣✉❧❛t✐♦♥✳

❚❤❡ s❛♠♣❧❡ ♣♦♣✉❧❛t✐♦♥ ♠✉st ❜❡ ❝❛r❡❢✉❧❧② ❝❤♦s❡♥✱ ✐♥ ♦r❞❡r t♦ ❛✈♦✐❞ ❜✐❛s❡❞ r❡s✉❧ts✳ ❍♦✇ ❞♦ ✇❡ ❞♦ t❤✐s❄

❋✐rst✱ ✐t ♠✉st ❜❡ r❡♣r❡s❡♥t❛t✐✈❡✳ ■❢ ❛❧❧ ♦❢ ♦✉r s❛♠♣❧❡ ♣♦♣✉❧❛t✐♦♥ ❝♦♠❡s ❢r♦♠ ❛ ✈❡r② r✐❝❤ ❛r❡❛✱ t❤❡♥ ❛❧♠♦st

❛❧❧ ✇✐❧❧ ❤❛✈❡ ❝❛rs✳ ❇✉t ✇❡ ♦❜✈✐♦✉s❧② ❝❛♥♥♦t ❝♦♥❝❧✉❞❡ ❢r♦♠ t❤✐s t❤❛t ❛❧♠♦st ❡✈❡r②♦♥❡ ✐♥ t❤❡ ❝♦✉♥tr② ❤❛s ❛

❝❛r✦ ❲❡ ♥❡❡❞ t♦ s❡♥❞ t❤❡ q✉❡st✐♦♥♥❛✐r❡ t♦ r✐❝❤ ❛s ✇❡❧❧ ❛s ♣♦♦r ♣❡♦♣❧❡✳

❙❡❝♦♥❞❧②✱ t❤❡ s✐③❡ ♦❢ t❤❡ s❛♠♣❧❡ ♣♦♣✉❧❛t✐♦♥ ♠✉st ❜❡ ❧❛r❣❡ ❡♥♦✉❣❤✳ ■t ✐s ♥♦ ❣♦♦❞ ❤❛✈✐♥❣ ❛ s❛♠♣❧❡

♣♦♣✉❧❛t✐♦♥ ❝♦♥s✐st✐♥❣ ♦❢ ♦♥❧② t✇♦ ♣❡♦♣❧❡✱ ❢♦r ❡①❛♠♣❧❡✳ ❇♦t❤ ♠❛② ✈❡r② ✇❡❧❧ ♥♦t ❤❛✈❡ ❝❛rs✳ ❇✉t ✇❡ ♦❜✈✐♦✉s❧②

❝❛♥♥♦t ❝♦♥❝❧✉❞❡ t❤❛t ♥♦ ♦♥❡ ✐♥ t❤❡ ❝♦✉♥tr② ❤❛s ❛ ❝❛r✦ ❚❤❡ ❧❛r❣❡r t❤❡ s❛♠♣❧❡ ♣♦♣✉❧❛t✐♦♥ s✐③❡✱ t❤❡ ♠♦r❡ ❧✐❦❡❧②

✐t ✐s t❤❛t t❤❡ st❛t✐st✐❝s ♦❢ ♦✉r s❛♠♣❧❡ ♣♦♣✉❧❛t✐♦♥ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ st❛t✐st✐❝s ♦❢ t❤❡ ❡♥t✐r❡ ♣♦♣✉❧❛t✐♦♥✳

❙♦ ❤♦✇ ❞♦❡s ♦♥❡ ❡♥s✉r❡ t❤❛t ♦♥❡s s❛♠♣❧❡ ✐s r❡♣r❡s❡♥t❛t✐✈❡❄ ❚❤❡r❡ ❛r❡ ❛ ✈❛r✐❡t② ♦❢ ♠❡t❤♦❞s ❛✈❛✐❧❛❜❧❡✱

✇❤✐❝❤ ✇❡ ✇✐❧❧ ❧♦♦❦ ❛t ♥♦✇✳

✶✳ ❘❛♥❞♦♠ ❙❛♠♣❧✐♥❣✳ ❊✈❡r② ♣❡rs♦♥ ✐♥ t❤❡ ❝♦✉♥tr② ❤❛s ❛♥ ❡q✉❛❧ ❝❤❛♥❝❡ ♦❢ ❜❡✐♥❣ s❡❧❡❝t❡❞✳ ■t ✐s ✉♥❜✐❛s❡❞

❛♥❞ ❛❧s♦ ✐♥❞❡♣❡♥❞❛♥t✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ s❡❧❡❝t✐♦♥ ♦❢ ♦♥❡ ♣❡rs♦♥ ❤❛s ♥♦ ❡✛❡❝t ♦♥ t❤❡ s❡❧❡❝t✐♦♥ ♦♥

❛♥♦t❤❡r✳ ❖♥❡ ✇❛② ♦❢ ❞♦✐♥❣ t❤✐s ✇♦✉❧❞ ❜❡ t♦ ❣✐✈❡ ❡❛❝❤ ♣❡rs♦♥ ✐♥ t❤❡ ❝♦✉♥tr② ❛ ♥✉♠❜❡r✱ ❛♥❞ t❤❡♥ ❛s❦

❛ ❝♦♠♣✉t❡r t♦ ❣✐✈❡ ✉s ❛ ❧✐st ♦❢ r❛♥❞♦♠ ♥✉♠❜❡rs✳ ❲❡ ❝♦✉❧❞ t❤❡♥ s❡♥❞ t❤❡ q✉❡st✐♦♥♥❛✐r❡ t♦ t❤❡ ♣❡♦♣❧❡

❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ r❛♥❞♦♠ ♥✉♠❜❡rs✳

✷✳ ❙②st❡♠❛t✐❝ ❙❛♠♣❧✐♥❣✳ ❆❣❛✐♥ ❣✐✈❡ ❡✈❡r② ♣❡rs♦♥ ✐♥ t❤❡ ❝♦✉♥tr② ❛ ♥✉♠❜❡r✱ ❛♥❞ t❤❡♥✱ ❢♦r ❡①❛♠♣❧❡✱

s❡❧❡❝t ❡✈❡r② ❤✉♥❞r❡❞t❤ ♣❡rs♦♥ ♦♥ t❤❡ ❧✐st✳ ❙♦ ♣❡rs♦♥ ✇✐t❤ ♥✉♠❜❡r ✶ ✇♦✉❧❞ ❜❡ s❡❧❡❝t❡❞✱ ♣❡rs♦♥ ✇✐t❤

♥✉♠❜❡r ✶✵✵ ✇♦✉❧❞ ❜❡ s❡❧❡❝t❡❞✱ ♣❡rs♦♥ ✇✐t❤ ♥✉♠❜❡r ✷✵✵ ✇♦✉❧❞ ❜❡ s❡❧❡❝t❡❞✱ ❡t❝✳

✸✳ ❙tr❛t✐✜❡❞ ❙❛♠♣❧✐♥❣✳ ❲❡ ❝♦♥s✐❞❡r ❞✐✛❡r❡♥t s✉❜❣r♦✉♣s ♦❢ t❤❡ ♣♦♣✉❧❛t✐♦♥✱ ❛♥❞ t❛❦❡ r❛♥❞♦♠ s❛♠♣❧❡s

❢r♦♠ t❤❡s❡✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝❛♥ ❞✐✈✐❞❡ t❤❡ ♣♦♣✉❧❛t✐♦♥ ✐♥t♦ ♠❛❧❡ ❛♥❞ ❢❡♠❛❧❡✱ ❞✐✛❡r❡♥t ❛❣❡s✱ ♦r ✐♥t♦

❞✐✛❡r❡♥t ✐♥❝♦♠❡ r❛♥❣❡s✳

✹✳ ❈❧✉st❡r ❙❛♠♣❧✐♥❣✳ ❍❡r❡ t❤❡ s❛♠♣❧❡ ✐s ❝♦♥❝❡♥tr❛t❡❞ ✐♥ ♦♥❡ ❛r❡❛✳ ❋♦r ❡①❛♠♣❧❡✱ ✇❡ ❝♦♥s✐❞❡r ❛❧❧ t❤❡

♣❡♦♣❧❡ ❧✐✈✐♥❣ ✐♥ ♦♥❡ ✉r❜❛♥ ❛r❡❛✳

✾✳✶✳✸✳✶ ❙❛♠♣❧✐♥❣

✶✳ ❉✐s❝✉ss t❤❡ ❛❞✈❛♥t❛❣❡s✱ ❞✐s❛❞✈❛♥t❛❣❡s ❛♥❞ ♣♦ss✐❜❧❡ ❜✐❛s ✇❤❡♥ ✉s✐♥❣

❛✳ s②st❡♠❛t✐❝ s❛♠♣❧✐♥❣

❜✳ r❛♥❞♦♠ s❛♠♣❧✐♥❣

❝✳ ❝❧✉st❡r s❛♠♣❧✐♥❣

✷✳ ❙✉❣❣❡st ❛ s✉✐t❛❜❧❡ s❛♠♣❧✐♥❣ ♠❡t❤♦❞ t❤❛t ❝♦✉❧❞ ❜❡ ✉s❡❞ t♦ ♦❜t❛✐♥ ✐♥❢♦r♠❛t✐♦♥ ♦♥✿

❛✳ ♣❛ss❡♥❣❡rs ✈✐❡✇s ♦♥ ❛✈❛✐❧❛❜✐❧✐t② ♦❢ ❛ ❧♦❝❛❧ t❛①✐ s❡r✈✐❝❡✳

❜✳ ✈✐❡✇s ♦❢ ❧❡❛r♥❡rs ♦♥ s❝❤♦♦❧ ♠❡❛❧s✳

❝✳ ❞❡❢❡❝ts ✐♥ ❛♥ ✐t❡♠ ♠❛❞❡ ✐♥ ❛ ❢❛❝t♦r②✳

✶✸✸

❞✳ ♠❡❞✐❝❛❧ ❝♦sts ♦❢ ❡♠♣❧♦②❡❡s ✐♥ ❛ ❧❛r❣❡ ❝♦♠♣❛♥②✳

✸✳ 2% ♦❢ ❛ ❝❡rt❛✐♥ ♠❛❣❛③✐♥❡s✬ s✉❜s❝r✐❜❡rs ✐s r❛♥❞♦♠❧② s❡❧❡❝t❡❞✳ ❚❤❡ r❛♥❞♦♠ ♥✉♠❜❡r ✶✻ ♦✉t ♦❢ ✺✵✱ ✐s

s❡❧❡❝t❡❞✳ ❚❤❡♥ s✉❜s❝r✐❜❡rs ✇✐t❤ ♥✉♠❜❡rs ✶✻✱ ✻✻✱ ✶✶✻✱ ✶✻✻✱ ... ❛r❡ ❝❤♦s❡♥ ❛s ❛ s❛♠♣❧❡✳ ❲❤❛t ❦✐♥❞ ♦❢

s❛♠♣❧✐♥❣ ✐s t❤✐s❄

✾✳✶✳✹ ❋✉♥❝t✐♦♥ ❋✐tt✐♥❣ ❛♥❞ ❘❡❣r❡ss✐♦♥ ❆♥❛❧②s✐s

■♥ ●r❛❞❡ ✶✶ ✇❡ r❡❝♦r❞❡❞ t✇♦ s❡ts ♦❢ ❞❛t❛ ✭❜✐✈❛r✐❛t❡ ❞❛t❛✮ ♦♥ ❛ s❝❛tt❡r ♣❧♦t ❛♥❞ t❤❡♥ ✇❡ ❞r❡✇ ❛ ❧✐♥❡ ♦❢ ❜❡st ✜t

❛s ❝❧♦s❡ t♦ ❛s ♠❛♥② ♦❢ t❤❡ ❞❛t❛ ✐t❡♠s ❛s ♣♦ss✐❜❧❡✳ ❘❡❣r❡ss✐♦♥ ❛♥❛❧②s✐s ✐s ❛ ♠❡t❤♦❞ ♦❢ ✜♥❞✐♥❣ ♦✉t ❡①❛❝t❧② ✇❤✐❝❤

❢✉♥❝t✐♦♥ ❜❡st ✜ts ❛ ❣✐✈❡♥ s❡t ♦❢ ❞❛t❛✳ ❲❡ ❝❛♥ ✜♥❞ ♦✉t t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❧✐♥❡ ❜② ❞r❛✇✐♥❣ ❛♥❞

❡st✐♠❛t✐♥❣✱ ♦r ❜② ✉s✐♥❣ ❛♥ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞ ❝❛❧❧❡❞ ✏t❤❡ ❧❡❛st sq✉❛r❡s ♠❡t❤♦❞✑✱ ❛✈❛✐❧❛❜❧❡ ♦♥ ♠♦st s❝✐❡♥t✐✜❝

❫

❝❛❧❝✉❧❛t♦rs✳ ❚❤❡ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥ ✐s ✇r✐tt❡♥ y= a + bx ✭✇❡ s❛② ②✲❤❛t✮ ♦r y = A + Bx✳ ❖❢ ❝♦✉rs❡

t❤❡s❡ ❛r❡ ❜♦t❤ ✈❛r✐❛t✐♦♥s ♦❢ ❛ ♠♦r❡ ❢❛♠✐❧✐❛r ❡q✉❛t✐♦♥ y = mx + c✳

❙✉♣♣♦s❡ ②♦✉ ❛r❡ ❞♦✐♥❣ ❛♥ ❡①♣❡r✐♠❡♥t ✇✐t❤ ✇❛s❤✐♥❣ ❞✐s❤❡s✳ ❨♦✉ ❝♦✉♥t ❤♦✇ ♠❛♥② ❞✐s❤❡s ②♦✉ ❜❡❣✐♥ ✇✐t❤✱

❛♥❞ t❤❡♥ ✜♥❞ ♦✉t ❤♦✇ ❧♦♥❣ ✐t t❛❦❡s t♦ ✜♥✐s❤ ✇❛s❤✐♥❣ t❤❡♠✳ ❙♦ ②♦✉ ♣❧♦t t❤❡ ❞❛t❛ ♦♥ ❛ ❣r❛♣❤ ♦❢ t✐♠❡ t❛❦❡♥

✈❡rs✉s ♥✉♠❜❡r ♦❢ ❞✐s❤❡s✳ ❚❤✐s ✐s ♣❧♦tt❡❞ ❜❡❧♦✇✳

❋✐❣✉r❡ ✾✳✹

■❢ t ✐s t❤❡ t✐♠❡ t❛❦❡♥✱ ❛♥❞ d t❤❡ ♥✉♠❜❡r ♦❢ ❞✐s❤❡s✱ t❤❡♥ ✐t ❧♦♦❦s ❛s t❤♦✉❣❤ t ✐s ♣r♦♣♦rt✐♦♥❛❧ t♦ d✱ ✐❡✳

t = m · d✱ ✇❤❡r❡ m ✐s t❤❡ ❝♦♥st❛♥t ♦❢ ♣r♦♣♦rt✐♦♥❛❧✐t②✳ ❚❤❡r❡ ❛r❡ t✇♦ q✉❡st✐♦♥s t❤❛t ✐♥t❡r❡st ✉s ♥♦✇✳

✶✳ ❍♦✇ ❞♦ ✇❡ ✜♥❞ m❄ ❖♥❡ ✇❛② ②♦✉ ❤❛✈❡ ❛❧r❡❛❞② ❧❡❛r♥t✱ ✐s t♦ ❞r❛✇ ❛ ❧✐♥❡ ♦❢ ❜❡st✲✜t t❤r♦✉❣❤ t❤❡ ❞❛t❛

♣♦✐♥ts✱ ❛♥❞ t❤❡♥ ♠❡❛s✉r❡ t❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐♥❡✳ ❇✉t t❤✐s ✐s ♥♦t t❡rr✐❜❧② ♣r❡❝✐s❡✳ ■s t❤❡r❡ ❛ ❜❡tt❡r

✇❛② ♦❢ ❞♦✐♥❣ ✐t❄

✷✳ ❍♦✇ ✇❡❧❧ ❞♦❡s ♦✉r ❧✐♥❡ ♦❢ ❜❡st ✜t r❡❛❧❧② ✜t ♦✉r ❞❛t❛❄ ■❢ t❤❡ ♣♦✐♥ts ♦♥ ♦✉r ♣❧♦t ❞♦♥✬t ❛❧❧ ❧✐❡ ❝❧♦s❡ t♦

t❤❡ ❧✐♥❡ ♦❢ ❜❡st ✜t✱ ❜✉t ❛r❡ s❝❛tt❡r❡❞ ❡✈❡r②✇❤❡r❡✱ t❤❡♥ t❤❡ ✜t ✐s ♥♦t ✬❣♦♦❞✬✱ ❛♥❞ ♦✉r ❛ss✉♠♣t✐♦♥ t❤❛t

t = m · d ♠✐❣❤t ❜❡ ✐♥❝♦rr❡❝t✳ ❈❛♥ ✇❡ ✜♥❞ ❛ q✉❛♥t✐t❛t✐✈❡ ♠❡❛s✉r❡ ♦❢ ❤♦✇ ✇❡❧❧ ♦✉r ❧✐♥❡ r❡❛❧❧② ✜ts t❤❡

❞❛t❛❄

■♥ t❤✐s ❝❤❛♣t❡r✱ ✇❡ ❛♥s✇❡r ❜♦t❤ ♦❢ t❤❡s❡ q✉❡st✐♦♥s✱ ✉s✐♥❣ t❤❡ t❡❝❤♥✐q✉❡s ♦❢ r❡❣r❡ss✐♦♥ ❛♥❛❧②s✐s✳

P❤❡t s✐♠✉❧❛t✐♦♥ ❢♦r ❝✉r✈❡ ✜tt✐♥❣

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❝✉r✈❡✲✜tt✐♥❣✳s✇❢❃

❋✐❣✉r❡ ✾✳✺

❊①❡r❝✐s❡ ✾✳✶✿ ❋✐tt✐♥❣ ❜② ❤❛♥❞

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✹✶✳✮

❯s❡ t❤❡ ❞❛t❛ ❣✐✈❡♥ t♦ ❞r❛✇ ❛ s❝❛tt❡r ♣❧♦t ❛♥❞ ❧✐♥❡ ♦❢ ❜❡st ✜t✳ ◆♦✇ ✇r✐t❡ ❞♦✇♥ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡

❧✐♥❡ t❤❛t ❜❡st s❡❡♠s t♦ ✜t t❤❡ ❞❛t❛✳

✶✸✹

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

① ✶✱✵ ✷✱✹ ✸✱✶ ✹✱✾ ✺✱✻ ✻✱✷

② ✷✱✺ ✷✱✽ ✸✱✵ ✹✱✽ ✺✱✶ ✺✱✸

❚❛❜❧❡ ✾✳✹

✾✳✷ ▲❡❛st ❙q✉❛r❡s✱ ❈❛❧❝✉❧❛t♦r ❲♦r❦✱ ❈♦rr❡❧❛t✐♦♥ ❈♦❡✣❝✐❡♥ts✷

✾✳✷✳✶ ❚❤❡ ▼❡t❤♦❞ ♦❢ ▲❡❛st ❙q✉❛r❡s

❲❡ ♥♦✇ ❝♦♠❡ t♦ ❛ ♠♦r❡ ❛❝❝✉r❛t❡ ♠❡t❤♦❞ ♦❢ ✜♥❞✐♥❣ t❤❡ ❧✐♥❡ ♦❢ ❜❡st✲✜t✳ ❚❤❡ ♠❡t❤♦❞ ✐s ✈❡r② s✐♠♣❧❡✳ ❙✉♣♣♦s❡

✇❡ ❣✉❡ss ❛ ❧✐♥❡ ♦❢ ❜❡st✲✜t✳ ❚❤❡♥ ❛t ❛t ❡✈❡r② ❞❛t❛ ♣♦✐♥t✱ ✇❡ ✜♥❞ t❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❞❛t❛ ♣♦✐♥t ❛♥❞ t❤❡

❧✐♥❡✳ ■❢ t❤❡ ❧✐♥❡ ✜tt❡❞ t❤❡ ❞❛t❛ ♣❡r❢❡❝t❧②✱ t❤✐s ❞✐st❛♥❝❡ s❤♦✉❧❞ ❜❡ ③❡r♦ ❢♦r ❛❧❧ t❤❡ ❞❛t❛ ♣♦✐♥ts✳ ❚❤❡ ✇♦rs❡ t❤❡

✜t✱ t❤❡ ❧❛r❣❡r t❤❡ ❞✐✛❡r❡♥❝❡s✳ ❲❡ t❤❡♥ sq✉❛r❡ ❡❛❝❤ ♦❢ t❤❡s❡ ❞✐st❛♥❝❡s✱ ❛♥❞ ❛❞❞ t❤❡♠ ❛❧❧ t♦❣❡t❤❡r✳

❋✐❣✉r❡ ✾✳✻

❚❤❡ ❜❡st✲✜t ❧✐♥❡ ✐s t❤❡♥ t❤❡ ❧✐♥❡ t❤❛t ♠✐♥✐♠✐s❡s t❤❡ s✉♠ ♦❢ t❤❡ sq✉❛r❡❞ ❞✐st❛♥❝❡s✳

❙✉♣♣♦s❡ ✇❡ ❤❛✈❡ ❛ ❞❛t❛ s❡t ♦❢ n ♣♦✐♥ts {(x1; y1) , (x2; y2) , ..., (xn, yn)}✳ ❲❡ ❛❧s♦ ❤❛✈❡ ❛ ❧✐♥❡ f (x) = mx+c

t❤❛t ✇❡ ❛r❡ tr②✐♥❣ t♦ ✜t t♦ t❤❡ ❞❛t❛✳ ❚❤❡ ❞✐st❛♥❝❡ ❜❡t✇❡❡♥ t❤❡ ✜rst ❞❛t❛ ♣♦✐♥t ❛♥❞ t❤❡ ❧✐♥❡✱ ❢♦r ❡①❛♠♣❧❡✱ ✐s

❞✐st❛♥❝❡ = y1 − f (x) = y1 − (mx + c)

✭✾✳✶✮

❲❡ ♥♦✇ sq✉❛r❡ ❡❛❝❤ ♦❢ t❤❡s❡ ❞✐st❛♥❝❡s ❛♥❞ ❛❞❞ t❤❡♠ t♦❣❡t❤❡r✳ ▲❡ts ❝❛❧❧ t❤✐s s✉♠ S (m, c)✳ ❚❤❡♥ ✇❡ ❤❛✈❡

t❤❛t

S (m, c)

=

(y1 − f (x1))2 + (y2 − f (x2))2 + ... + (yn − f (xn))2

✭✾✳✷✮

=

n

(y

i=1

i − f (xi))2

❚❤✉s ♦✉r ♣r♦❜❧❡♠ ✐s t♦ ✜♥❞ t❤❡ ✈❛❧✉❡ ♦❢ m ❛♥❞ c s✉❝❤ t❤❛t S (m, c) ✐s ♠✐♥✐♠✐s❡❞✳ ▲❡t ✉s ❝❛❧❧ t❤❡s❡ ♠✐♥✐♠✐s✐♥❣

✈❛❧✉❡s m0 ❛♥❞ c0✳ ❚❤❡♥ t❤❡ ❧✐♥❡ ♦❢ ❜❡st✲✜t ✐s f (x) = m0x + c0✳ ❲❡ ❝❛♥ ✜♥❞ m0 ❛♥❞ c0 ✉s✐♥❣ ❝❛❧❝✉❧✉s✱ ❜✉t

✐t ✐s tr✐❝❦②✱ ❛♥❞ ✇❡ ✇✐❧❧ ❥✉st ❣✐✈❡ ②♦✉ t❤❡ r❡s✉❧t✱ ✇❤✐❝❤ ✐s t❤❛t

n Pn

x

x Pn

y

m

i=1

i yi −Pn

i=1

i

i=1

i

0

=

n Pn

(x

x

i=1

i )2 −(Pn

i=1

i )2

✭✾✳✸✮

c

n

n

0

=

1

y

x

n

i=1

i − m0

n

i=0

i = y − m0x

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✷✹✴✶✳✶✴❃✳

✶✸✺

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ r❡❣r❡ss✐♦♥ ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴●❆♠③✇■❦●❋❣❊✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✾✳✼

❊①❡r❝✐s❡ ✾✳✷✿ ▼❡t❤♦❞ ♦❢ ▲❡❛st ❙q✉❛r❡s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✹✶✳✮

■♥ t❤❡ t❛❜❧❡ ❜❡❧♦✇✱ ✇❡ ❤❛✈❡ t❤❡ r❡❝♦r❞s ♦❢ t❤❡ ♠❛✐♥t❡♥❛♥❝❡ ❝♦sts ✐♥ ❘❛♥❞s✱ ❝♦♠♣❛r❡❞ ✇✐t❤ t❤❡ ❛❣❡

♦❢ t❤❡ ❛♣♣❧✐❛♥❝❡ ✐♥ ♠♦♥t❤s✳ ❲❡ ❤❛✈❡ ❞❛t❛ ❢♦r ✺ ❛♣♣❧✐❛♥❝❡s✳

❛♣♣❧✐❛♥❝❡ ✶

✷

✸

✹

✺

❛❣❡ ✭x✮

✺

✶✵

✶✺

✷✵

✸✵

❝♦st ✭y✮

✾✵ ✶✹✵ ✷✺✵ ✸✵✵ ✸✽✵

❚❛❜❧❡ ✾✳✺

❋✐♥❞ t❤❡ ❧✐♥❡ ♦❢ ❜❡st ✜t ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❧❡❛st sq✉❛r❡s✳

✾✳✷✳✷ ❯s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r

❊①❡r❝✐s❡ ✾✳✸✿ ❯s✐♥❣ t❤❡ ❙❤❛r♣ ❊▲✲✺✸✶❱❍ ❝❛❧❝✉❧❛t♦r

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✹✶✳✮

❋✐♥❞ ❛ r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛t❛✿

❉❛②s ✭x✮

✶

✷

✸

✹

✺

●r♦✇t❤ ✐♥ ♠ ✭y✮ ✶✱✵✵ ✷✱✺✵ ✷✱✼✺ ✸✱✵✵ ✸✱✺✵

❚❛❜❧❡ ✾✳✻

❊①❡r❝✐s❡ ✾✳✹✿ ❯s✐♥❣ t❤❡ ❈❆❙■❖ ❢①✲✽✷❊❙ ◆❛t✉r❛❧ ❉✐s♣❧❛② ❝❛❧❝✉❧❛t♦r

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳

✶✹✷✳✮

❯s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r ❞❡t❡r♠✐♥❡ t❤❡ ❧❡❛st sq✉❛r❡s ❧✐♥❡ ♦❢ ❜❡st ✜t ❢♦r t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❛t❛ s❡t ♦❢ ♠❛r❦s✳

▲❡❛r♥❡r

✶

✷

✸

✹

✺

❈❤❡♠✐str② ✭✪✮

✺✷ ✺✺ ✽✻ ✼✶ ✹✺

❆❝❝♦✉♥t✐♥❣ ✭✪✮ ✹✽ ✻✹ ✾✺ ✼✾ ✺✵

❚❛❜❧❡ ✾✳✼

❋♦r ❛ ❈❤❡♠✐str② ♠❛r❦ ♦❢ ✻✺✪✱ ✇❤❛t ♠❛r❦ ❞♦❡s t❤❡ ❧❡❛st sq✉❛r❡s ❧✐♥❡ ♣r❡❞✐❝t ❢♦r ❆❝❝♦✉♥t✐♥❣❄

✶✸✻

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

✾✳✷✳✷✳✶

✶✳ ❚❤❡ t❛❜❧❡ ❜❡❧♦✇ ❧✐sts t❤❡ ❡①❛♠ r❡s✉❧ts ❢♦r ✺ st✉❞❡♥ts ✐♥ t❤❡ s✉❜❥❡❝ts ♦❢ ❙❝✐❡♥❝❡ ❛♥❞ ❇✐♦❧✲

♦❣②✳

▲❡❛r♥❡r

✶

✷

✸

✹

✺

❙❝✐❡♥❝❡ ✪

✺✺ ✻✻ ✼✹ ✾✷ ✹✼

❇✐♦❧♦❣② ✪ ✹✽ ✺✾ ✻✽ ✽✹ ✺✸

❚❛❜❧❡ ✾✳✽

❛✳ ❯s❡ t❤❡ ❢♦r♠✉❧❛❡ t♦ ✜♥❞ t❤❡ r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥ ❝♦❡✣❝✐❡♥ts a ❛♥❞ b✳

❜✳ ❉r❛✇ ❛ s❝❛tt❡r ♣❧♦t ♦❢ t❤❡ ❞❛t❛ ♦♥ ❣r❛♣❤ ♣❛♣❡r✳

❝✳ ◆♦✇ ✉s❡ ❛❧❣❡❜r❛ t♦ ✜♥❞ ❛ ♠♦r❡ ❛❝❝✉r❛t❡ ❡q✉❛t✐♦♥✳

✷✳ ❋♦♦t❧❡♥❣t❤s ❛♥❞ ❤❡✐❣❤ts ♦❢ ✼ st✉❞❡♥ts ❛r❡ ❣✐✈❡♥ ✐♥ t❤❡ t❛❜❧❡ ❜❡❧♦✇✳

❍❡✐❣❤t ✭❝♠✮

✶✼✵ ✶✻✸ ✶✸✶ ✶✽✶ ✶✹✻ ✶✸✹ ✶✻✻

❋♦♦t❧❡♥❣t❤ ✭❝♠✮ ✷✼

✷✸

✷✵

✷✽

✷✷

✷✵

✷✹

❚❛❜❧❡ ✾✳✾

❛✳ ❉r❛✇ ❛ s❝❛tt❡r ♣❧♦t ♦❢ t❤❡ ❞❛t❛ ♦♥ ❣r❛♣❤ ♣❛♣❡r✳

❜✳ ■♥❞❡♥t✐❢② ❛♥❞ ❞❡s❝r✐❜❡ ❛♥② tr❡♥❞s s❤♦✇♥ ✐♥ t❤❡ s❝❛tt❡r ♣❧♦t✳

❝✳ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧❡❛st sq✉❛r❡s ❧✐♥❡ ❜② ✉s✐♥❣ ❛❧❣❡❜r❛✐❝ ♠❡t❤♦❞s ❛♥❞ ❞r❛✇ t❤❡ ❧✐♥❡ ♦♥ ②♦✉r

❣r❛♣❤✳

❞✳ ❯s❡ ②♦✉r ❡q✉❛t✐♦♥ t♦ ♣r❡❞✐❝t t❤❡ ❤❡✐❣❤t ♦❢ ❛ st✉❞❡♥t ✇✐t❤ ❢♦♦t❧❡♥❣t❤ ✷✶✱✻ ❝♠✳

❡✳ ❯s❡ ②♦✉r ❡q✉❛t✐♦♥ t♦ ♣r❡❞✐❝t t❤❡ ❢♦♦t❧❡♥❣t❤ ♦❢ ❛ st✉❞❡♥t ✶✼✻ ❝♠ t❛❧❧✳

✸✳ ❘❡♣❡❛t t❤❡ ❞❛t❛ ✐♥ q✉❡st✐♦♥ ✷ ❛♥❞ ✜♥❞ t❤❡ r❡❣r❡ss✐♦♥ ❧✐♥❡ ✉s✐♥❣ ❛ ❝❛❧❝✉❧❛t♦r

✾✳✷✳✸ ❈♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥ts

❖♥❝❡ ✇❡ ❤❛✈❡ ❛♣♣❧✐❡❞ r❡❣r❡ss✐♦♥ ❛♥❛❧②s✐s t♦ ❛ s❡t ♦❢ ❞❛t❛✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❤❛✈❡ ❛ ♥✉♠❜❡r t❤❛t t❡❧❧s ✉s ❡①❛❝t❧②

❤♦✇ ✇❡❧❧ t❤❡ ❞❛t❛ ✜ts t❤❡ ❢✉♥❝t✐♦♥✳ ❆ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t✱ r✱ ✐s ❛ t♦♦❧ t❤❛t t❡❧❧s ✉s t♦ ✇❤❛t ❞❡❣r❡❡ t❤❡r❡

✐s ❛ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t✇♦ s❡ts ♦❢ ❞❛t❛✳ ❚❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t r ∈ [−1; 1] ✇❤❡♥ r = −1✱ t❤❡r❡ ✐s ❛

♣❡r❢❡❝t ♥❡❣❛t✐✈❡ r❡❧❛t✐♦♥s❤✐♣✱ ✇❤❡♥ r = 0✱ t❤❡r❡ ✐s ♥♦ r❡❧❛t✐♦♥s❤✐♣ ❛♥❞ r = 1 ✐s ❛ ♣❡r❢❡❝t ♣♦s✐t✐✈❡ ❝♦rr❡❧❛t✐♦♥✳

❋✐❣✉r❡ ✾✳✽

❋✐❣✉r❡ ✾✳✾

❋✐❣✉r❡ ✾✳✶✵

❋✐❣✉r❡ ✾✳✶✶

❋✐❣✉r❡ ✾✳✶✷

P♦s✐t✐✈❡✱ str♦♥❣

P♦s✐t✐✈❡✱

❢❛✐r❧②

P♦s✐t✐✈❡✱ ✇❡❛❦

◆♦ ❛ss♦❝✐❛t✐♦♥

◆❡❣❛t✐✈❡✱

❢❛✐r❧②

str♦♥❣

str♦♥❣

r ≈ 0, 9

r ≈ 0, 7

r ≈ 0, 4

r = 0

r ≈ −0, 7

✶✸✼

❚❛❜❧❡ ✾✳✶✵

❲❡ ♦❢t❡♥ ✉s❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t r2 ✐♥ ♦r❞❡r t♦ ❡①❛♠✐♥❡ t❤❡ str❡♥❣t❤ ♦❢ t❤❡ ❝♦rr❡❧❛t✐♦♥ ♦♥❧②✳

■♥ t❤✐s ❝❛s❡✿

r2 = 0

♥♦ ❝♦rr❡❧❛t✐♦♥

✵ < r2 < ✵✱✷✺

✈❡r② ✇❡❛❦

✵✱✷✺ < r2 < ✵✱✺ ✇❡❛❦

✵✱✺ < r2 < ✵✱✼✺ ♠♦❞❡r❛t❡

✵✱✼✺ < r2 < ✵✱✾ str♦♥❣

✵✱✾ < r2 < ✶

✈❡r② str♦♥❣

r2 = 1

♣❡r❢❡❝t ❝♦rr❡❧❛t✐♦♥

❚❛❜❧❡ ✾✳✶✶

❚❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ✉s✐♥❣ t❤❡ ❢♦r♠✉❧❛

1

x − x

y − y

r =

✭✾✳✹✮

n − 1

sx

sy

• ✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❞❛t❛ ♣♦✐♥ts✱

• sx ✐s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ x✲✈❛❧✉❡s ❛♥❞

• sy ✐s t❤❡ st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥ ♦❢ t❤❡ y✲✈❛❧✉❡s✳

❚❤✐s ✐s ❦♥♦✇♥ ❛s t❤❡ P❡❛rs♦♥✬s ♣r♦❞✉❝t ♠♦♠❡♥t ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t✳ ■t ✐s ❛ ❧♦♥❣ ❝❛❧❝✉❧❛t✐♦♥ ❛♥❞ ♠✉❝❤

❡❛s✐❡r t♦ ❞♦ ♦♥ t❤❡ ❝❛❧❝✉❧❛t♦r ✇❤❡r❡ ②♦✉ s✐♠♣❧② ❢♦❧❧♦✇ t❤❡ ♣r♦❝❡❞✉r❡ ❢♦r t❤❡ r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥✱ ❛♥❞ ❣♦ ♦♥

t♦ ✜♥❞ r✳

✾✳✷✳✹ ❊①❡r❝✐s❡s

✶✳ ❇❡❧♦✇ ✐s ❛ ❧✐st ♦❢ ❞❛t❛ ❝♦♥❝❡r♥✐♥❣ ✶✷ ❝♦✉♥tr✐❡s ❛♥❞ t❤❡✐r r❡s♣❡❝t✐✈❡ ❝❛r❜♦♥ ❞✐♦①✐❞❡ ✭❈❖2✮ ❡♠♠✐ss✐♦♥

❧❡✈❡❧s ♣❡r ♣❡rs♦♥ ❛♥❞ t❤❡ ❣r♦ss ❞♦♠❡st✐❝ ♣r♦❞✉❝t ✭●❉P ✲ ❛ ♠❡❛s✉r❡ ♦❢ ♣r♦❞✉❝ts ♣r♦❞✉❝❡❞ ❛♥❞ s❡r✈✐❝❡s

❞❡❧✐✈❡r❡❞ ✇✐t❤✐♥ ❛ ❝♦✉♥tr② ✐♥ ❛ ②❡❛r✮ ♣❡r ♣❡rs♦♥✳

✶✸✽

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

❈❖2 ❡♠♠✐ss✐♦♥s ♣❡r ❝❛♣✐t❛ ✭x✮ ●❉P ♣❡r ❝❛♣✐t❛ ✭y✮

❙♦✉t❤ ❆❢r✐❝❛

✽✱✶

✸ ✾✸✽

❚❤❛✐❧❛♥❞

✷✱✺

✷ ✼✶✷

■t❛❧②

✼✱✸

✷✵ ✾✹✸

❆✉str❛❧✐❛

✶✼✱✵

✷✸ ✽✾✸

❈❤✐♥❛

✷✱✺

✽✶✻

■♥❞✐❛

✵✱✾

✹✻✸

❈❛♥❛❞❛

✶✻✱✵

✷✷ ✺✸✼

❯♥✐t❡❞ ❑✐♥❣❞♦♠ ✾✱✵

✷✶ ✼✽✺

❯♥✐t❡❞ ❙t❛t❡s

✶✾✱✾

✸✶ ✽✵✻

❙❛✉❞✐ ❆r❛❜✐❛

✶✶✱✵

✻ ✽✺✸

■r❛♥

✸✱✽

✶ ✹✾✸

■♥❞♦♥❡s✐❛

✶✱✷

✾✽✻

❚❛❜❧❡ ✾✳✶✷

❛✳ ❉r❛✇ ❛ s❝❛tt❡r ♣❧♦t ♦❢ t❤❡ ❞❛t❛ s❡t ❛♥❞ ②♦✉r ❡st✐♠❛t❡ ♦❢ ❛ ❧✐♥❡ ♦❢ ❜❡st ✜t✳

❜✳ ❈❛❧❝✉❧❛t❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♦❢ r❡❣r❡ss✐♦♥ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❧❡❛st sq✉❛r❡s✳

❝✳ ❉r❛✇ t❤❡ r❡❣r❡ss✐♦♥ ❧✐♥❡ ❡q✉❛t✐♦♥ ♦♥t♦ t❤❡ ❣r❛♣❤✳

❞✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t r✳

❡✳ ❲❤❛t ❝♦♥❝❧✉s✐♦♥ ❝❛♥ ②♦✉ r❡❛❝❤✱ r❡❣❛r❞✐♥❣ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ❈❖2 ❡♠✐ss✐♦♥ ❛♥❞ ●❉P ♣❡r

❝❛♣✐t❛ ❢♦r t❤❡ ❝♦✉♥tr✐❡s ✐♥ t❤❡ ❞❛t❛ s❡t❄

✷✳ ❆ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❞❛t❛ ♦♥ t❤❡ ♣❡❝✉❧✐❛r ✐♥✈❡st✐❣❛t✐♦♥ ✐♥t♦ ❛ ❢♦♦t s✐③❡ ❛♥❞ ❤❡✐❣❤t ♦❢ st✉❞❡♥ts ✇❛s r❡❝♦r❞❡❞

✐♥ t❤❡ t❛❜❧❡ ❜❡❧♦✇✳ ❆♥s✇❡r t❤❡ q✉❡st✐♦♥s t♦ ❢♦❧❧♦✇✳

▲❡♥❣t❤ ♦❢ r✐❣❤t ❢♦♦t ✭❝♠✮ ❍❡✐❣❤t ✭❝♠✮

✷✺✱✺

✶✻✸✱✸

✷✻✱✶

✶✻✹✱✾

✷✸✱✼

✶✻✺✱✺

✷✻✱✹

✶✼✸✱✼

✷✼✱✺

✶✼✹✱✹

✷✹

✶✺✻

✷✷✱✻

✶✺✺✱✸

✷✼✱✶

✶✻✾✱✸

❚❛❜❧❡ ✾✳✶✸

❛✳ ❉r❛✇ ❛ s❝❛tt❡r ♣❧♦t ♦❢ t❤❡ ❞❛t❛ s❡t ❛♥❞ ②♦✉r ❡st✐♠❛t❡ ♦❢ ❛ ❧✐♥❡ ♦❢ ❜❡st ✜t✳

❜✳ ❈❛❧❝✉❧❛t❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♦❢ r❡❣r❡ss✐♦♥ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ ❧❡❛st sq✉❛r❡s ♦r ②♦✉r ❝❛❧❝✉❧❛t♦r✳

❝✳ ❉r❛✇ t❤❡ r❡❣r❡ss✐♦♥ ❧✐♥❡ ❡q✉❛t✐♦♥ ♦♥t♦ t❤❡ ❣r❛♣❤✳

❞✳ ❈❛❧❝✉❧❛t❡ t❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t r✳

✶✸✾

❡✳ ❲❤❛t ❝♦♥❝❧✉s✐♦♥ ❝❛♥ ②♦✉ r❡❛❝❤✱ r❡❣❛r❞✐♥❣ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ r✐❣❤t ❢♦♦t

❛♥❞ ❤❡✐❣❤t ♦❢ t❤❡ st✉❞❡♥ts ✐♥ t❤❡ ❞❛t❛ s❡t❄

✸✳ ❆ ❝❧❛ss ✇r♦t❡ t✇♦ t❡sts✱ ❛♥❞ t❤❡ ♠❛r❦s ❢♦r ❡❛❝❤ ✇❡r❡ r❡❝♦r❞❡❞ ✐♥ t❤❡ t❛❜❧❡ ❜❡❧♦✇✳ ❋✉❧❧ ♠❛r❦s ✐♥ t❤❡

✜rst t❡st ✇❛s ✺✵✱ ❛♥❞ t❤❡ s❡❝♦♥❞ t❡st ✇❛s ♦✉t ♦❢ ✸✵✳

❛✳ ■s t❤❡r❡ ❛ str♦♥❣ ❛ss♦❝✐❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♠❛r❦s ❢♦r t❤❡ ✜rst ❛♥❞ s❡❝♦♥❞ t❡st❄ ❙❤♦✇ ✇❤② ♦r ✇❤②

♥♦t✳

❜✳ ❖♥❡ ♦❢ t❤❡ ❧❡❛r♥❡rs ✭✐♥ r♦✇ ✶✽✮ ❞✐❞ ♥♦t ✇r✐t❡ t❤❡ s❡❝♦♥❞ t❡st✳ ●✐✈❡♥ t❤❡✐r ♠❛r❦ ❢♦r t❤❡ ✜rst t❡st✱

❝❛❧❝✉❧❛t❡ ❛♥ ❡①♣❡❝t❡❞ ♠❛r❦ ❢♦r t❤❡ s❡❝♦♥❞ t❡st✳

▲❡❛r♥❡r ❚❡st ✶

❚❡st ✷

✭❋✉❧❧ ♠❛r❦s✿ ✺✵✮ ✭❋✉❧❧ ♠❛r❦s✿ ✸✵✮

✶

✹✷

✷✺

✷

✸✷

✶✾

✸

✸✶

✷✵

✹

✹✷

✷✻

✺

✸✺

✷✸

✻

✷✸

✶✹

✼

✹✸

✷✹

✽

✷✸

✶✷

✾

✷✹

✶✹

✶✵

✶✺

✶✵

✶✶

✶✾

✶✶

✶✷

✶✸

✶✵

✶✸

✸✻

✷✷

✶✹

✷✾

✶✼

✶✺

✷✾

✶✼

✶✻

✷✺

✶✻

✶✼

✷✾

✶✽

✶✽

✶✼

✶✾

✸✵

✶✾

✷✵

✷✽

✶✼

❚❛❜❧❡ ✾✳✶✹

✹✳ ❆ ❢❛st ❢♦♦❞ ❝♦♠♣❛♥② ♣r♦❞✉❝❡s ❤❛♠❜✉r❣❡rs✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❤❛♠❜✉r❣❡rs ♠❛❞❡✱ ❛♥❞ t❤❡ ❝♦sts ❛r❡

r❡❝♦r❞❡❞ ♦✈❡r ❛ ✇❡❡❦✳

✶✹✵

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

❍❛♠❜✉r❣❡rs ♠❛❞❡ ❈♦sts

✹✾✺

❘✷✸✽✷

✺✺✵

❘✷✹✹✷

✺✶✺

❘✷✹✽✹

✺✵✵

❘✷✹✵✵

✹✽✵

❘✷✸✼✵

✺✸✵

❘✷✹✹✽

✺✽✺

❘✷✽✵✺

❚❛❜❧❡ ✾✳✶✺

❛✳ ❋✐♥❞ t❤❡ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ t❤❛t ❜❡st ✜ts t❤❡ ❞❛t❛✳

❜✳ ■❢ t❤❡ t♦t❛❧ ❝♦st ✐♥ ❛ ❞❛② ✐s ❘✷✺✵✵✱ ❡st✐♠❛t❡ t❤❡ ♥✉♠❜❡r ♦❢ ❤❛♠❜✉r❣❡rs ♣r♦❞✉❝❡❞✳

❝✳ ❲❤❛t ✐s t❤❡ ❝♦st ♦❢ ✹✾✵ ❤❛♠❜✉r❣❡rs❄

✺✳ ❚❤❡ ♣r♦✜ts ♦❢ ❛ ♥❡✇ s❤♦♣ ❛r❡ r❡❝♦r❞❡❞ ♦✈❡r t❤❡ ✜rst ✻ ♠♦♥t❤s✳ ❚❤❡ ♦✇♥❡r ✇❛♥ts t♦ ♣r❡❞✐❝t ❤✐s ❢✉t✉r❡

s❛❧❡s✳ ❚❤❡ ♣r♦✜ts s♦ ❢❛r ❤❛✈❡ ❜❡❡♥ ❘✾✵ ✵✵✵ ✱ ❘✾✸ ✵✵✵✱ ❘✾✾ ✺✵✵✱ ❘✶✵✷ ✵✵✵✱ ❘✶✵✶ ✸✵✵✱ ❘✶✵✾ ✵✵✵✳

❛✳ ❋♦r t❤❡ ♣r♦✜t ❞❛t❛✱ ❝❛❧❝✉❧❛t❡ t❤❡ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥✳

❜✳ ●✐✈❡ ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ♣r♦✜ts ❢♦r t❤❡ ♥❡①t t✇♦ ♠♦♥t❤s✳

❝✳ ❚❤❡ ♦✇♥❡r ✇❛♥ts ❛ ♣r♦✜t ♦❢ ❘✶✸✵ ✵✵✵✳ ❊st✐♠❛t❡ ❤♦✇ ♠❛♥② ♠♦♥t❤s t❤✐s ✇✐❧❧ t❛❦❡✳

✻✳ ❆ ❝♦♠♣❛♥② ♣r♦❞✉❝❡s s✇❡❡ts ✉s✐♥❣ ❛ ♠❛❝❤✐♥❡ ✇❤✐❝❤ r✉♥s ❢♦r ❛ ❢❡✇ ❤♦✉rs ♣❡r ❞❛②✳ ❚❤❡ ♥✉♠❜❡r ♦❢ ❤♦✉rs

r✉♥♥✐♥❣ t❤❡ ♠❛❝❤✐♥❡ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ s✇❡❡ts ♣r♦❞✉❝❡❞ ❛r❡ r❡❝♦r❞❡❞✳

▼❛❝❤✐♥❡ ❤♦✉rs ❙✇❡❡ts ♣r♦❞✉❝❡❞

✸✱✽✵

✷✼✺

✹✱✷✸

✷✽✼

✹✱✸✼

✷✾✶

✹✱✶✵

✷✽✶

✹✱✶✼

✷✽✻

❚❛❜❧❡ ✾✳✶✻

❋✐♥❞ t❤❡ ❧✐♥❡❛r r❡❣r❡ss✐♦♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ❞❛t❛✱ ❛♥❞ ❡st✐♠❛t❡ t❤❡ ♠❛❝❤✐♥❡ ❤♦✉rs ♥❡❡❞❡❞ t♦ ♠❛❦❡ ✸✵✵

s✇❡❡ts✳

✶✹✶

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✾

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✾✳✶ ✭♣✳ ✶✸✸✮

❙t❡♣ ✶✳ ❚❤❡ ✜rst st❡♣ ✐s t♦ ❞r❛✇ t❤❡ ❣r❛♣❤✳ ❚❤✐s ✐s s❤♦✇♥ ❜❡❧♦✇✳

❋✐❣✉r❡ ✾✳✶✸

❙t❡♣ ✷✳ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ✐s

y = mx + c

✭✾✳✺✮

❋r♦♠ t❤❡ ❣r❛♣❤ ✇❡ ❤❛✈❡ ❞r❛✇♥✱ ✇❡ ❡st✐♠❛t❡ t❤❡ ②✲✐♥t❡r❝❡♣t t♦ ❜❡ ✶✱✺✳ ❲❡ ❡st✐♠❛t❡ t❤❛t y = 3, 5 ✇❤❡♥

x = 3✳ ❙♦ ✇❡ ❤❛✈❡ t❤❛t ♣♦✐♥ts (3; 3, 5) ❛♥❞ (0; 1, 5) ❧✐❡ ♦♥ t❤❡ ❧✐♥❡✳ ❚❤❡ ❣r❛❞✐❡♥t ♦❢ t❤❡ ❧✐♥❡✱ m✱ ✐s ❣✐✈❡♥

❜②

m

=

y2−y1

x2−x1

=

3,5−1,5

✭✾✳✻✮

3−0

=

2

3

❙♦ ✇❡ ✜♥❛❧❧② ❤❛✈❡ t❤❛t t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♦❢ ❜❡st ✜t ✐s

2

y =

x + 1, 5

✭✾✳✼✮

3

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✾✳✷ ✭♣✳ ✶✸✺✮

❙t❡♣ ✶✳ ❛♣♣❧✐❛♥❝❡ x y

xy

x2

✶

✺

✾✵

✹✺✵

✷✺

✷

✶✵ ✶✹✵

✶✹✵✵

✶✵✵

✸

✶✺ ✷✺✵

✸✼✺✵

✷✷✺

✹

✷✵ ✸✵✵

✻✵✵✵

✹✵✵

✺

✸✵ ✸✽✵

✶✶✹✵✵ ✾✵✵

❚♦t❛❧

✽✵ ✶✶✻✵ ✷✸✵✵✵ ✶✻✺✵

❚❛❜❧❡ ✾✳✶✼

❙t❡♣ ✷✳

m0

=

n P xy−P x P y = 5×23000−80×1160 = 12

n P x2−(P x)2

5×1650−802

c0

=

y − bx = 1160 − 12×80 = 40

✭✾✳✽✮

5

5

❫

∴

y= 40 + 12x

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✾✳✸ ✭♣✳ ✶✸✺✮

✶✹✷

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

❙t❡♣ ✶✳ ❯s✐♥❣ ②♦✉r ❝❛❧❝✉❧❛t♦r✱ ❝❤❛♥❣❡ t❤❡ ♠♦❞❡ ❢r♦♠ ♥♦r♠❛❧ t♦ ✏❙t❛t xy✑✳ ❚❤✐s ♠♦❞❡ ❡♥❛❜❧❡s ②♦✉ t♦ t②♣❡ ✐♥

❜✐✈❛r✐❛t❡ ❞❛t❛✳

❙t❡♣ ✷✳ ❑❡② ✐♥ t❤❡ ❞❛t❛ ❛s ❢♦❧❧♦✇s✿

✶ (x, y) ✶

❉❆❚❆ n ❂ ✶

✷ (x, y) ✷✱✺

❉❆❚❆ n ❂ ✷

✸ (x, y) ✷✱✼✺ ❉❆❚❆ n ❂ ✸

✹ (x, y) ✸✱✵

❉❆❚❆ n ❂ ✹

✺ (x, y) ✸✱✺

❉❆❚❆ n ❂ ✺

❚❛❜❧❡ ✾✳✶✽

❙t❡♣ ✸✳ ❆s❦ ❢♦r t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❝♦❡✣❝✐❡♥ts a ❛♥❞ b✳

❘❈▲ a ❣✐✈❡s a = ✵✱✾

❘❈▲ b

❣✐✈❡s b = ✵✱✺✺

❚❛❜❧❡ ✾✳✶✾

❫

∴y= 0, 9 + 0, 55x

✭✾✳✾✮

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✾✳✹ ✭♣✳ ✶✸✺✮

❙t❡♣ ✶✳ ❙✇✐t❝❤ ♦♥ t❤❡ ❝❛❧❝✉❧❛t♦r✳ Pr❡ss ❬▼❖❉❊❪ ❛♥❞ t❤❡♥ s❡❧❡❝t ❙❚❆❚ ❜② ♣r❡ss✐♥❣ ❬✷❪✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ s❝r❡❡♥

✇✐❧❧ ❛♣♣❡❛r✿

✶ ✶✲❱❆❘

✷ ❆ ✰ ❇❳

✸ ❴ ✰ ❈❳2

✹ ❧♥ ❳

✺ ❡ ❫ ❳

✻ ❆ ✳ ❇ ❫ ❳

✼ ❆ ✳ ❳ ❫ ❇ ✽ ✶✴❳

❚❛❜❧❡ ✾✳✷✵

◆♦✇ ♣r❡ss ❬✷❪ ❢♦r ❧✐♥❡❛r r❡❣r❡ss✐♦♥✳ ❨♦✉r s❝r❡❡♥ s❤♦✉❧❞ ❧♦♦❦ s♦♠❡t❤✐♥❣ ❧✐❦❡ t❤✐s✿

①

②

✶

✷

✸

❚❛❜❧❡ ✾✳✷✶

❙t❡♣ ✷✳ Pr❡ss ❬✺✷❪ ❛♥❞ t❤❡♥ ❬❂❪ t♦ ❡♥t❡r t❤❡ ✜rst ♠❛r❦ ✉♥❞❡r x✳ ❚❤❡♥ ❡♥t❡r t❤❡ ♦t❤❡r ✈❛❧✉❡s✱ ✐♥ t❤❡ s❛♠❡ ✇❛②✱

❢♦r t❤❡ x✲✈❛r✐❛❜❧❡ ✭t❤❡ ❈❤❡♠✐str② ♠❛r❦s✮ ✐♥ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡② ❛r❡ ❣✐✈❡♥ ✐♥ t❤❡ ❞❛t❛ s❡t✳ ❚❤❡♥

♠♦✈❡ t❤❡ ❝✉rs♦r ❛❝r♦ss ❛♥❞ ✉♣ ❛♥❞ ❡♥t❡r ✹✽ ✉♥❞❡r y ♦♣♣♦s✐t❡ ✺✷ ✐♥ t❤❡ x✲❝♦❧✉♠♥✳ ❈♦♥t✐♥✉❡ t♦ ❡♥t❡r t❤❡

♦t❤❡r y✲✈❛❧✉❡s ✭t❤❡ ❆❝❝♦✉♥t✐♥❣ ♠❛r❦s✮ ✐♥ ♦r❞❡r s♦ t❤❛t t❤❡② ♣❛✐r ♦✛ ❝♦rr❡❝t❧② ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣

x✲✈❛❧✉❡s✳

✶✹✸

①

②

✶

✺✷

✷

✺✺

✸

❚❛❜❧❡ ✾✳✷✷

❚❤❡♥ ♣r❡ss ❬❆❈❪✳ ❚❤❡ s❝r❡❡♥ ❝❧❡❛rs ❜✉t t❤❡ ❞❛t❛ r❡♠❛✐♥s st♦r❡❞✳

✶✿ ❚②♣❡ ✷✿ ❉❛t❛

✸✿ ❊❞✐t

✹✿ ❙✉♠

✺✿ ❱❛r

✻✿ ▼✐♥▼❛①

✼✿ ❘❡❣

❚❛❜❧❡ ✾✳✷✸

◆♦✇ ♣r❡ss ❬❙❍■❋❚❪❬✶❪ t♦ ❣❡t t❤❡ st❛ts ❝♦♠♣✉t❛t✐♦♥s s❝r❡❡♥ s❤♦✇♥ ❜❡❧♦✇✳ ❈❤♦♦s❡ ❘❡❣r❡ss✐♦♥ ❜② ♣r❡ss✐♥❣

❬✼❪✳

✶✿ ❆ ✷✿ ❇

❫

✸✿ r

✹✿ x

❫

✺✿ y

❚❛❜❧❡ ✾✳✷✹

❙t❡♣ ✸✳

❛✳ Pr❡ss ❬✶❪ ❛♥❞ ❬❂❪ t♦ ❣❡t t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ y✲✐♥t❡r❝❡♣t✱ a = −5, 065.. = −5, 07✭t♦ ✷ ❞✳♣✳✮ ❋✐♥❛❧❧②✱

t♦ ❣❡t t❤❡ s❧♦♣❡✱ ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❦❡② s❡q✉❡♥❝❡✿ ❬❙❍■❋❚❪❬✶❪❬✼❪❬✷❪❬❂❪✳ ❚❤❡ ❝❛❧❝✉❧❛t♦r ❣✐✈❡s b =

❫

1, 169.. = 1, 17✭t♦ ✷ ❞✳♣✳✮ ❚❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ ♦❢ r❡❣r❡ss✐♦♥ ✐s t❤✉s✿ y= −5, 07 + 1, 17x

❜✳ Pr❡ss ❬❆❈❪❬✻✺❪❬❙❍■❋❚❪❬✶❪❬✼❪❬✺❪❬❂❪ ❚❤✐s ❣✐✈❡s ❛ ✭♣r❡❞✐❝t❡❞✮ ❆❝❝♦✉♥t✐♥❣ ♠❛r❦ ♦❢ ❫= 70, 94 = 71✪

✶✹✹

❈❍❆P❚❊❘ ✾✳ ❙❚❆❚■❙❚■❈❙

❈❤❛♣t❡r ✶✵

❈♦♠❜✐♥❛t✐♦♥s ❛♥❞ ♣❡r♠✉t❛t✐♦♥s

✶✵✳✶ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ◆♦t❛t✐♦♥✶

✶✵✳✶✳✶ ■♥tr♦❞✉❝t✐♦♥

▼❛t❤❡♠❛t✐❝s ❡❞✉❝❛t✐♦♥ ❜❡❣❛♥ ✇✐t❤ ❝♦✉♥t✐♥❣✳ ■♥ t❤❡ ❜❡❣✐♥♥✐♥❣✱ ✜♥❣❡rs✱ ❜❡❛♥s ❛♥❞ ❜✉tt♦♥s ✇❡r❡ ✉s❡❞ t♦ ❤❡❧♣

✇✐t❤ ❝♦✉♥t✐♥❣✱ ❜✉t t❤❡s❡ ❛r❡ ♦♥❧② ♣r❛❝t✐❝❛❧ ❢♦r s♠❛❧❧ ♥✉♠❜❡rs✳ ❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ❛ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ✐t❡♠s

♠✉st ❜❡ ❝♦✉♥t❡❞❄

❚❤✐s ❝❤❛♣t❡r ❢♦❝✉s❡s ♦♥ ❤♦✇ t♦ ✉s❡ ♠❛t❤❡♠❛t✐❝❛❧ t❡❝❤♥✐q✉❡s t♦ ❝♦✉♥t ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ✐t❡♠s✳

✶✵✳✶✳✷ ❈♦✉♥t✐♥❣

❆♥ ✐♠♣♦rt❛♥t ❛s♣❡❝t ♦❢ ♣r♦❜❛❜✐❧✐t② t❤❡♦r② ✐s t❤❡ ❛❜✐❧✐t② t♦ ❞❡t❡r♠✐♥❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s

✇❤❡♥ ♠✉❧t✐♣❧❡ ❡✈❡♥ts ❛r❡ ❝♦♥s✐❞❡r❡❞✳

❋♦r ❡①❛♠♣❧❡✱ ✇❤❛t ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ✇❤❡♥ ❛ ❞✐❡ ✐s r♦❧❧❡❞ ❛♥❞ t❤❡♥ ❛ ❝♦✐♥ ✐s t♦ss❡❞❄

❚❤❡ r♦❧❧ ♦❢ ❛ ❞✐❡ ❤❛s s✐① ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ✭✶✱ ✷✱ ✸✱ ✹✱ ✺ ♦r ✻✮ ❛♥❞ t❤❡ t♦ss ♦❢ ❛ ❝♦✐♥✱ ✷ ♦✉t❝♦♠❡s ✭❤❡❛❞ ♦r

t❛✐❧s✮✳ ❈♦✉♥t✐♥❣ t❤❡ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s ❝❛♥ ❜❡ t❡❞✐♦✉s✳

✶✵✳✶✳✷✳✶ ▼❛❦✐♥❣ ❛ ▲✐st

❚❤❡ s✐♠♣❧❡st ♠❡t❤♦❞ ♦❢ ❝♦✉♥t✐♥❣ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ✐s ❜② ♠❛❦✐♥❣ ❛ ❧✐st✿

✶❍✱ ✶❚✱ ✷❍✱ ✷❚✱ ✸❍✱ ✸❚✱ ✹❍✱ ✹❚✱ ✺❍✱ ✺❚✱ ✻❍✱ ✻❚

♦r ❞r❛✇✐♥❣ ✉♣ ❛ t❛❜❧❡✿

✶❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✵✵✴✶✳✶✴❃✳

✶✹✺

✶✹✻

❈❍❆P❚❊❘ ✶✵✳ ❈❖▼❇■◆❆❚■❖◆❙ ❆◆❉ P❊❘▼❯❚❆❚■❖◆❙

❞✐❡ ❝♦✐♥

✶

❍

✶

❚

✷

❍

✷

❚

✸

❍

✸

❚

✹

❍

✹

❚

✺

❍

✺

❚

✻

❍

✻

❚

❚❛❜❧❡ ✶✵✳✶

❇♦t❤ t❤❡s❡ ♠❡t❤♦❞s r❡s✉❧t ✐♥ ✶✷ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s✱ ❜✉t ❜♦t❤ t❤❡s❡ ♠❡t❤♦❞s ❤❛✈❡ ❛ ❧♦t ♦❢ r❡♣❡t✐t✐♦♥✳

▼❛②❜❡ t❤❡r❡ ✐s ❛ s♠❛rt❡r ✇❛② t♦ ✇r✐t❡ ❞♦✇♥ t❤❡ r❡s✉❧t❄

✶✵✳✶✳✷✳✷ ❚r❡❡ ❉✐❛❣r❛♠s

❖♥❡ ♠❡t❤♦❞ ♦❢ ❡❧✐♠✐♥❛t✐♥❣ s♦♠❡ ♦❢ t❤❡ r❡♣❡t✐t✐♦♥ ✐s t♦ ✉s❡ tr❡❡ ❞✐❛❣r❛♠s✳ ❚r❡❡ ❞✐❛❣r❛♠s ❛r❡ ❛ ❣r❛♣❤✐❝❛❧

♠❡t❤♦❞ ♦❢ ❧✐st✐♥❣ ❛❧❧ ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡✈❡♥ts ❢r♦♠ ❛ r❛♥❞♦♠ ❡①♣❡r✐♠❡♥t✳

❋✐❣✉r❡ ✶✵✳✶✿ ❊①❛♠♣❧❡ ♦❢ ❛ tr❡❡ ❞✐❛❣r❛♠✳ ❊❛❝❤ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡ ✐s ❛ ❜r❛♥❝❤ ♦❢ t❤❡ tr❡❡✳

✶✵✳✶✳✸ ◆♦t❛t✐♦♥

✶✵✳✶✳✸✳✶ ❚❤❡ ❋❛❝t♦r✐❛❧ ◆♦t❛t✐♦♥

❋♦r ❛♥ ✐♥t❡❣❡r n✱ t❤❡ ♥♦t❛t✐♦♥ n! ✭r❡❛❞ n ❢❛❝t♦r✐❛❧✮ r❡♣r❡s❡♥ts✿

n × (n − 1) × (n − 2) × ... × 3 × 2 × 1

✭✶✵✳✶✮

✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥✿ 0! = 1✳

❚❤❡ ❢❛❝t♦r✐❛❧ ♥♦t❛t✐♦♥ ✇✐❧❧ ❜❡ ✉s❡❞ ♦❢t❡♥ ✐♥ t❤✐s ❝❤❛♣t❡r✳

✶✹✼

✶✵✳✷ ❋✉♥❞❛♠❡♥t❛❧ ❈♦✉♥t✐♥❣ Pr✐♥❝✐♣❧❡ ❛♥❞ ❈♦♠❜✐♥❛t✐♦♥s✷

✶✵✳✷✳✶ ❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❈♦✉♥t✐♥❣ Pr✐♥❝✐♣❧❡

❚❤❡ ✉s❡ ♦❢ ❧✐sts✱ t❛❜❧❡s ❛♥❞ tr❡❡ ❞✐❛❣r❛♠s ✐s ♦♥❧② ❢❡❛s✐❜❧❡ ❢♦r ❡✈❡♥ts ✇✐t❤ ❛ ❢❡✇ ♦✉t❝♦♠❡s✳ ❲❤❡♥ t❤❡ ♥✉♠❜❡r

♦❢ ♦✉t❝♦♠❡s ❣r♦✇s✱ ✐t ✐s ♥♦t ♣r❛❝t✐❝❛❧ t♦ ❧✐st t❤❡ ❞✐✛❡r❡♥t ♣♦ss✐❜✐❧✐t✐❡s ❛♥❞ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦✉♥t✐♥❣ ♣r✐♥❝✐♣❧❡

✐s ✉s❡❞✳

❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦✉♥t✐♥❣ ♣r✐♥❝✐♣❧❡ ❞❡s❝r✐❜❡s ❤♦✇ t♦ ❞❡t❡r♠✐♥❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ♦❢ ❛

s❡r✐❡s ♦❢ ❡✈❡♥ts✳

❙✉♣♣♦s❡ t❤❛t t✇♦ ❡①♣❡r✐♠❡♥ts t❛❦❡ ♣❧❛❝❡✳ ❚❤❡ ✜rst ❡①♣❡r✐♠❡♥t ❤❛s n1 ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s✱ ❛♥❞ t❤❡ s❡❝♦♥❞

❤❛s n2 ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s✳ ❚❤❡r❡❢♦r❡✱ t❤❡ ✜rst ❡①♣❡r✐♠❡♥t✱ ❢♦❧❧♦✇❡❞ ❜② t❤❡ s❡❝♦♥❞ ❡①♣❡r✐♠❡♥t✱ ✇✐❧❧ ❤❛✈❡ ❛

t♦t❛❧ ♦❢ n1 × n2 ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s✳ ❚❤✐s ✐❞❡❛ ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐s❡❞ t♦ m ❡①♣❡r✐♠❡♥ts ❛s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢

♦✉t❝♦♠❡s ❢♦r m ❡①♣❡r✐♠❡♥ts ✐s✿

m

n1 × n2 × n3 × ... × nm =

ni

✭✶✵✳✷✮

i=1

✐s t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❡q✉✐✈❛❧❡♥t ♦❢

✳

◆♦t❡✿ t❤❡ ♦r❞❡r ✐♥ ✇❤✐❝❤ t❤❡ ❡①♣❡r✐♠❡♥ts ❛r❡ ❞♦♥❡ ❞♦❡s ♥♦t ❛✛❡❝t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣♦ss✐❜❧❡ ♦✉t❝♦♠❡s✳

❊①❡r❝✐s❡ ✶✵✳✶✿ ▲✉♥❝❤ ❙♣❡❝✐❛❧

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✺✸✳✮

❆ t❛❦❡✲❛✇❛② ❤❛s ❛ ✹✲♣✐❡❝❡ ❧✉♥❝❤ s♣❡❝✐❛❧ ✇❤✐❝❤ ❝♦♥s✐sts ♦❢ ❛ s❛♥❞✇✐❝❤✱ s♦✉♣✱ ❞❡ss❡rt ❛♥❞ ❞r✐♥❦ ❢♦r

❘✷✺✳✵✵✳ ❚❤❡② ♦✛❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤♦✐❝❡s ❢♦r ✿

❙❛♥❞✇✐❝❤✿ ❝❤✐❝❦❡♥ ♠❛②♦♥♥❛✐s❡✱ ❝❤❡❡s❡ ❛♥❞ t♦♠❛t♦✱ t✉♥❛✱ ❛♥❞ ❤❛♠ ❛♥❞ ❧❡tt✉❝❡

❙♦✉♣✿ t♦♠❛t♦✱ ❝❤✐❝❦❡♥ ♥♦♦❞❧❡✱ ✈❡❣❡t❛❜❧❡

❉❡ss❡rt✿ ✐❝❡✲❝r❡❛♠✱ ♣✐❡❝❡ ♦❢ ❝❛❦❡

❉r✐♥❦✿ t❡❛✱ ❝♦✛❡❡✱ ❝♦❦❡✱ ❋❛♥t❛ ❛♥❞ ❙♣r✐t❡✳

❍♦✇ ♠❛♥② ♣♦ss✐❜❧❡ ♠❡❛❧s ❛r❡ t❤❡r❡❄

✶✵✳✷✳✷ ❈♦♠❜✐♥❛t✐♦♥s

❚❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦✉♥t✐♥❣ ♣r✐♥❝✐♣❧❡ ❞❡s❝r✐❜❡s ❤♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♦✉t❝♦♠❡s ✇❤❡♥ ♠✉❧t✐♣❧❡

✐♥❞❡♣❡♥❞❡♥t ❡✈❡♥ts ❛r❡ ♣❡r❢♦r♠❡❞ t♦❣❡t❤❡r✳

❆ ♠♦r❡ ❝♦♠♣❧❡① ♣r♦❜❧❡♠ ✐s ❞❡t❡r♠✐♥✐♥❣ ❤♦✇ ♠❛♥② ❝♦♠❜✐♥❛t✐♦♥s t❤❡r❡ ❛r❡ ♦❢ s❡❧❡❝t✐♥❣ ❛ ❣r♦✉♣ ♦❢ ♦❜❥❡❝ts

❢r♦♠ ❛ s❡t✳ ▼❛t❤❡♠❛t✐❝❛❧❧②✱ ❛ ❝♦♠❜✐♥❛t✐♦♥ ✐s ❞❡✜♥❡❞ ❛s ❛♥ ✉♥✲♦r❞❡r❡❞ ❝♦❧❧❡❝t✐♦♥ ♦❢ ✉♥✐q✉❡ ❡❧❡♠❡♥ts✱ ♦r ♠♦r❡

❢♦r♠❛❧❧②✱ ❛ s✉❜s❡t ♦❢ ❛ s❡t✳ ❋♦r ❡①❛♠♣❧❡✱ s✉♣♣♦s❡ ②♦✉ ❤❛✈❡ ✜❢t②✲t✇♦ ♣❧❛②✐♥❣ ❝❛r❞s✱ ❛♥❞ s❡❧❡❝t ✜✈❡ ❝❛r❞s✳ ❚❤❡

✜✈❡ ❝❛r❞s ✇♦✉❧❞ ❢♦r♠ ❛ ❝♦♠❜✐♥❛t✐♦♥ ❛♥❞ ✇♦✉❧❞ ❜❡ ❛ s✉❜s❡t ♦❢ t❤❡ s❡t ♦❢ ✺✷ ❝❛r❞s✳

■♥ ❛ s❡t✱ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ s❡t ❞♦❡s ♥♦t ♠❛tt❡r✳ ❚❤❡s❡ ❛r❡ r❡♣r❡s❡♥t❡❞ ✉s✉❛❧❧② ✇✐t❤ ❝✉r❧②

❜r❛❝❡s✳ ❋♦r ❡①❛♠♣❧❡ {2, 4, 6} ✐s ❛ s✉❜s❡t ♦❢ t❤❡ s❡t {1, 2, 3, 4, 5, 6}✳ ❙✐♥❝❡ t❤❡ ♦r❞❡r ♦❢ t❤❡ ❡❧❡♠❡♥ts ❞♦❡s ♥♦t

♠❛tt❡r✱ ♦♥❧② t❤❡ s♣❡❝✐✜❝ ❡❧❡♠❡♥ts ❛r❡ ♦❢ ✐♥t❡r❡st✳ ❚❤❡r❡❢♦r❡✱

{2, 4, 6} = {6, 4, 2}

✭✶✵✳✸✮

❛♥❞ {1, 1, 1} ✐s t❤❡ s❛♠❡ ❛s {1} ❜❡❝❛✉s❡ ✐♥ ❛ s❡t t❤❡ ❡❧❡♠❡♥ts ❞♦♥✬t ✉s✉❛❧❧② ❛♣♣❡❛r ♠♦r❡ t❤❛♥ ♦♥❝❡✳

❙♦ ✐♥ s✉♠♠❛r② ✇❡ ❝❛♥ s❛② t❤❡ ❢♦❧❧♦✇✐♥❣✿ ●✐✈❡♥ S✱ t❤❡ s❡t ♦❢ ❛❧❧ ♣♦ss✐❜❧❡ ✉♥✐q✉❡ ❡❧❡♠❡♥ts✱ ❛ ❝♦♠❜✐♥❛t✐♦♥

✐s ❛ s✉❜s❡t ♦❢ t❤❡ ❡❧❡♠❡♥ts ♦❢ S✳ ❚❤❡ ♦r❞❡r ♦❢ t❤❡ ❡❧❡♠❡♥ts ✐♥ ❛ ❝♦♠❜✐♥❛t✐♦♥ ✐s ♥♦t ✐♠♣♦rt❛♥t ✭t✇♦ ❧✐sts ✇✐t❤

t❤❡ s❛♠❡ ❡❧❡♠❡♥ts ✐♥ ❞✐✛❡r❡♥t ♦r❞❡rs ❛r❡ ❝♦♥s✐❞❡r❡❞ t♦ ❜❡ t❤❡ s❛♠❡ ❝♦♠❜✐♥❛t✐♦♥✮✳ ❆❧s♦✱ t❤❡ ❡❧❡♠❡♥ts ❝❛♥♥♦t

❜❡ r❡♣❡❛t❡❞ ✐♥ ❛ ❝♦♠❜✐♥❛t✐♦♥ ✭❡✈❡r② ❡❧❡♠❡♥t ❛♣♣❡❛rs ♦♥❝❡✮✳

✷❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✵✻✴✶✳✶✴❃✳

✶✹✽

❈❍❆P❚❊❘ ✶✵✳ ❈❖▼❇■◆❆❚■❖◆❙ ❆◆❉ P❊❘▼❯❚❆❚■❖◆❙

✶✵✳✷✳✷✳✶ ❈♦✉♥t✐♥❣ ❈♦♠❜✐♥❛t✐♦♥s

❈❛❧❝✉❧❛t✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s t❤❛t ❝❡rt❛✐♥ ♣❛tt❡r♥s ❝❛♥ ❜❡ ❢♦r♠❡❞ ✐s t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ❝♦♠❜✐♥❛t♦r✐❝s✱ t❤❡

st✉❞② ♦❢ ❝♦♠❜✐♥❛t✐♦♥s✳ ▲❡t S ❜❡ ❛ s❡t ✇✐t❤ n ♦❜❥❡❝ts✳ ❈♦♠❜✐♥❛t✐♦♥s ♦❢ r ♦❜❥❡❝ts ❢r♦♠ t❤✐s s❡t S ❛r❡ s✉❜s❡ts

♦❢ S ❤❛✈✐♥❣ r ❡❧❡♠❡♥ts ❡❛❝❤ ✭✇❤❡r❡ t❤❡ ♦r❞❡r ♦❢ ❧✐st✐♥❣ t❤❡ ❡❧❡♠❡♥ts ❞♦❡s ♥♦t ❞✐st✐♥❣✉✐s❤ t✇♦ s✉❜s❡ts✮✳

✶✵✳✷✳✷✳✶✳✶ ❈♦♠❜✐♥❛t✐♦♥ ✇✐t❤♦✉t ❘❡♣❡t✐t✐♦♥

❲❤❡♥ t❤❡ ♦r❞❡r ❞♦❡s ♥♦t ♠❛tt❡r✱ ❜✉t ❡❛❝❤ ♦❜❥❡❝t ❝❛♥ ❜❡ ❝❤♦s❡♥ ♦♥❧② ♦♥❝❡✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠❜✐♥❛t✐♦♥s ✐s✿

n!

n

=

✭✶✵✳✹✮

r! (n − r)!

r

✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts ❢r♦♠ ✇❤✐❝❤ ②♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥❞ r ✐s t❤❡ ♥✉♠❜❡r t♦ ❜❡ ❝❤♦s❡♥✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ②♦✉ ❤❛✈❡ ✶✵ ♥✉♠❜❡rs ❛♥❞ ✇✐s❤ t♦ ❝❤♦♦s❡ ✺ ②♦✉ ✇♦✉❧❞ ❤❛✈❡ ✶✵✦✴✭✺✦✭✶✵ ✲ ✺✮✦✮ ❂ ✷✺✷ ✇❛②s

t♦ ❝❤♦♦s❡✳

❋♦r ❡①❛♠♣❧❡ ❤♦✇ ♠❛♥② ♣♦ss✐❜❧❡ ✺ ❝❛r❞ ❤❛♥❞s ❛r❡ t❤❡r❡ ✐♥ ❛ ❞❡❝❦ ♦❢ ❝❛r❞s ✇✐t❤ ✺✷ ❝❛r❞s❄

✺✷✦ ✴ ✭✺✦✭✺✷✲✺✮✦✮ ❂ 2 598 960 ❝♦♠❜✐♥❛t✐♦♥s

✶✵✳✷✳✷✳✶✳✷ ❈♦♠❜✐♥❛t✐♦♥ ✇✐t❤ ❘❡♣❡t✐t✐♦♥

❲❤❡♥ t❤❡ ♦r❞❡r ❞♦❡s ♥♦t ♠❛tt❡r ❛♥❞ ❛♥ ♦❜❥❡❝t ❝❛♥ ❜❡ ❝❤♦s❡♥ ♠♦r❡ t❤❛♥ ♦♥❝❡✱ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❝♦♠❜✐♥❛✲

t✐♦♥s ✐s✿

(n + r − 1)!

n + r − 1

n + r − 1

=

=

✭✶✵✳✺✮

r! (n − 1)!

r

n − 1

✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts ❢r♦♠ ✇❤✐❝❤ ②♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥❞ r ✐s t❤❡ ♥✉♠❜❡r t♦ ❜❡ ❝❤♦s❡♥✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ②♦✉ ❤❛✈❡ t❡♥ t②♣❡s ♦❢ ❞♦♥✉ts t♦ ❝❤♦♦s❡ ❢r♦♠ ❛♥❞ ②♦✉ ✇❛♥t t❤r❡❡ ❞♦♥✉ts t❤❡r❡ ❛r❡ ✭✶✵ ✰

✸ ✲ ✶✮✦ ✴ ✸✦✭✶✵ ✲ ✶✮✦ ❂ ✷✷✵ ✇❛②s t♦ ❝❤♦♦s❡✳

◆♦t❡ t❤❛t ✐♥ t❤✐s ✈✐❞❡♦ ♣❡r♠✉t❛t✐♦♥s ❛r❡ ♠❡♥t✐♦♥❡❞✱ ②♦✉ ✇✐❧❧ ❝♦✈❡r ♣❡r♠✉t❛t✐♦♥s ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ♣r♦❜❛❜✐❧✐t② ✲ ✶

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❜❈①▼❤♥❝❘✼P❯✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✶✵✳✷

✶✵✳✷✳✷✳✷ ❈♦♠❜✐♥❛t♦r✐❝s ❛♥❞ Pr♦❜❛❜✐❧✐t②

❈♦♠❜✐♥❛t♦r✐❝s ✐s q✉✐t❡ ✉s❡❢✉❧ ✐♥ t❤❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ♣r♦❜❛❜✐❧✐t✐❡s ♦❢ ❡✈❡♥ts✱ ❛s ✐t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡t❡r♠✐♥❡

❡①❛❝t❧② ❤♦✇ ♠❛♥② ♦✉t❝♦♠❡s ❛r❡ ♣♦ss✐❜❧❡ ✐♥ ❛ ❣✐✈❡♥ ❡①♣❡r✐♠❡♥t✳

❊①❡r❝✐s❡ ✶✵✳✷✿ Pr♦❜❛❜✐❧✐t②

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✺✸✳✮

❆t ❛ s❝❤♦♦❧✱ ❧❡❛r♥❡rs ❡❛❝❤ ♣❧❛② ✷ s♣♦rts✳ ❚❤❡② ❝❛♥ ❝❤♦♦s❡ ❢r♦♠ ♥❡t❜❛❧❧✱ ❜❛s❦❡t❜❛❧❧✱ s♦❝❝❡r✱ ❛t❤✲

❧❡t✐❝s✱ s✇✐♠♠✐♥❣✱ ♦r t❡♥♥✐s✳ ❲❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t ❛ ❧❡❛r♥❡r ♣❧❛②s s♦❝❝❡r ❛♥❞ ❡✐t❤❡r ♥❡t❜❛❧❧✱

❜❛s❦❡t❜❛❧❧ ♦r t❡♥♥✐s❄

✶✹✾

✶✵✳✸ P❡r♠✉t❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✸

✶✵✳✸✳✶ P❡r♠✉t❛t✐♦♥s

❚❤❡ ❝♦♥❝❡♣t ♦❢ ❛ ❝♦♠❜✐♥❛t✐♦♥ ❞✐❞ ♥♦t ❝♦♥s✐❞❡r t❤❡ ♦r❞❡r ♦❢ t❤❡ ❡❧❡♠❡♥ts ♦❢ t❤❡ s✉❜s❡t t♦ ❜❡ ✐♠♣♦rt❛♥t✳ ❆

♣❡r♠✉t❛t✐♦♥ ✐s ❛ ❝♦♠❜✐♥❛t✐♦♥ ✇✐t❤ t❤❡ ♦r❞❡r ♦❢ ❛ s❡❧❡❝t✐♦♥ ❢r♦♠ ❛ ❣r♦✉♣ ❜❡✐♥❣ ✐♠♣♦rt❛♥t✳ ❋♦r ❡①❛♠♣❧❡✱ ❢♦r

t❤❡ s❡t {1, 2, 3, 4, 5, 6}✱ t❤❡ ❝♦♠❜✐♥❛t✐♦♥ {1, 2, 3} ✇♦✉❧❞ ❜❡ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ❝♦♠❜✐♥❛t✐♦♥ {3, 2, 1}✱ ❜✉t t❤❡s❡

t✇♦ ❝♦♠❜✐♥❛t✐♦♥s ❛r❡ ❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥s✱ ❜❡❝❛✉s❡ t❤❡ ❡❧❡♠❡♥ts ✐♥ t❤❡ s❡t ❛r❡ ♦r❞❡r❡❞ ❞✐✛❡r❡♥t❧②✳

▼♦r❡ ❢♦r♠❛❧❧②✱ ❛ ♣❡r♠✉t❛t✐♦♥ ✐s ❛♥ ♦r❞❡r❡❞ ❧✐st ✇✐t❤♦✉t r❡♣❡t✐t✐♦♥s✱ ♣❡r❤❛♣s ♠✐ss✐♥❣ s♦♠❡ ❡❧❡♠❡♥ts✳

❚❤✐s ♠❡❛♥s t❤❛t {1, 2, 2, 3, 4, 5, 6} ❛♥❞ {1, 2, 4, 5, 5, 6} ❛r❡ ♥♦t ♣❡r♠✉t❛t✐♦♥s ♦❢ t❤❡ s❡t {1, 2, 3, 4, 5, 6}✳

◆♦✇ s✉♣♣♦s❡ ②♦✉ ❤❛✈❡ t❤❡s❡ ♦❜❥❡❝ts✿

✶✱ ✷✱ ✸

❍❡r❡ ✐s ❛ ❧✐st ♦❢ ❛❧❧ ♣❡r♠✉t❛t✐♦♥s ♦❢ ❛❧❧ t❤r❡❡ ♦❜❥❡❝ts✿

✶ ✷ ✸❀ ✶ ✸ ✷❀ ✷ ✶ ✸❀ ✷ ✸ ✶❀ ✸ ✶ ✷❀ ✸ ✷ ✶✳

✶✵✳✸✳✶✳✶ ❈♦✉♥t✐♥❣ P❡r♠✉t❛t✐♦♥s

▲❡t S ❜❡ ❛ s❡t ✇✐t❤ n ♦❜❥❡❝ts✳ P❡r♠✉t❛t✐♦♥s ♦❢ r ♦❜❥❡❝ts ❢r♦♠ t❤✐s s❡t S r❡❢❡r t♦ s❡q✉❡♥❝❡s ♦❢ r ❞✐✛❡r❡♥t

❡❧❡♠❡♥ts ♦❢ S ✭✇❤❡r❡ t✇♦ s❡q✉❡♥❝❡s ❛r❡ ❝♦♥s✐❞❡r❡❞ ❞✐✛❡r❡♥t ✐❢ t❤❡② ❝♦♥t❛✐♥ t❤❡ s❛♠❡ ❡❧❡♠❡♥ts ❜✉t ✐♥ ❛

❞✐✛❡r❡♥t ♦r❞❡r✮✳ ❋♦r♠✉❧❛s ❢♦r t❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ❛♥❞ ❝♦♠❜✐♥❛t✐♦♥s ❛r❡ r❡❛❞✐❧② ❛✈❛✐❧❛❜❧❡ ❛♥❞

✐♠♣♦rt❛♥t t❤r♦✉❣❤♦✉t ❝♦♠❜✐♥❛t♦r✐❝s✳

■t ✐s ❡❛s② t♦ ❝♦✉♥t t❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ s✐③❡ r ✇❤❡♥ ❝❤♦s❡♥ ❢r♦♠ ❛ s❡t ♦❢ s✐③❡ n ✭✇✐t❤ r ≤ n✮✳

✶✳ ❙❡❧❡❝t t❤❡ ✜rst ♠❡♠❜❡r ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ♦✉t ♦❢ n ❝❤♦✐❝❡s✱ ❜❡❝❛✉s❡ t❤❡r❡ ❛r❡ n ❞✐st✐♥❝t ❡❧❡♠❡♥ts ✐♥

t❤❡ s❡t✳

✷✳ ◆❡①t✱ s✐♥❝❡ ♦♥❡ ♦❢ t❤❡ n ❡❧❡♠❡♥ts ❤❛s ❛❧r❡❛❞② ❜❡❡♥ ✉s❡❞✱ t❤❡ s❡❝♦♥❞ ♠❡♠❜❡r ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ❤❛s

(n − 1) ❡❧❡♠❡♥ts t♦ ❝❤♦♦s❡ ❢r♦♠ t❤❡ r❡♠❛✐♥✐♥❣ s❡t✳

✸✳ ❚❤❡ t❤✐r❞ ♠❡♠❜❡r ♦❢ t❤❡ ♣❡r♠✉t❛t✐♦♥ ❝❛♥ ❜❡ ✜❧❧❡❞ ✐♥ (n − 2) ✇❛②s s✐♥❝❡ ✷ ❤❛✈❡ ❜❡❡♥ ✉s❡❞ ❛❧r❡❛❞②✳

✹✳ ❚❤✐s ♣❛tt❡r♥ ❝♦♥t✐♥✉❡s ✉♥t✐❧ t❤❡r❡ ❛r❡ r ♠❡♠❜❡rs ♦♥ t❤❡ ♣❡r♠✉t❛t✐♦♥✳ ❚❤✐s ♠❡❛♥s t❤❛t t❤❡ ❧❛st ♠❡♠❜❡r

❝❛♥ ❜❡ ✜❧❧❡❞ ✐♥ (n − (r − 1)) = (n − r + 1) ✇❛②s✳

✺✳ ❙✉♠♠❛r✐③✐♥❣✱ ✇❡ ✜♥❞ t❤❛t t❤❡r❡ ✐s ❛ t♦t❛❧ ♦❢

n (n − 1) (n − 2) ... (n − r + 1)

✭✶✵✳✻✮

❞✐✛❡r❡♥t ♣❡r♠✉t❛t✐♦♥s ♦❢ r ♦❜❥❡❝ts✱ t❛❦❡♥ ❢r♦♠ ❛ ♣♦♦❧ ♦❢ n ♦❜❥❡❝ts✳ ❚❤✐s ♥✉♠❜❡r ✐s ❞❡♥♦t❡❞ ❜② P (n, r)

❛♥❞ ❝❛♥ ❜❡ ✇r✐tt❡♥ ✐♥ ❢❛❝t♦r✐❛❧ ♥♦t❛t✐♦♥ ❛s✿

n!

P (n, r) =

.

✭✶✵✳✼✮

(n − r)!

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ✇❡ ❤❛✈❡ ❛ t♦t❛❧ ♦❢ ✺ ❡❧❡♠❡♥ts✱ t❤❡ ✐♥t❡❣❡rs {1, 2, 3, 4, 5}✱ ❤♦✇ ♠❛♥② ✇❛②s ❛r❡ t❤❡r❡ ❢♦r

❛ ♣❡r♠✉t❛t✐♦♥ ♦❢ t❤r❡❡ ❡❧❡♠❡♥ts t♦ ❜❡ s❡❧❡❝t❡❞ ❢r♦♠ t❤✐s s❡t❄ ■♥ t❤✐s ❝❛s❡✱ n = 5 ❛♥❞ r = 3✳ ❚❤❡♥✱

P (5, 3) = 5!/7! = 60!✳

❑❤❛♥ ❛❝❛❞❡♠② ✈✐❞❡♦ ♦♥ ♣r♦❜❛❜✐❧✐t② ✲ ✷

❚❤✐s ♠❡❞✐❛ ♦❜❥❡❝t ✐s ❛ ❋❧❛s❤ ♦❜❥❡❝t✳ P❧❡❛s❡ ✈✐❡✇ ♦r ❞♦✇♥❧♦❛❞ ✐t ❛t

❁❤tt♣✿✴✴✇✇✇✳②♦✉t✉❜❡✳❝♦♠✴✈✴❳q◗❚❳❲✼❳❢❨❆✫r❡❧❂✵✫❤❧❂❡♥❴❯❙✫❢❡❛t✉r❡❂♣❧❛②❡r❴❡♠❜❡❞❞❡❞✫✈❡rs✐♦♥❂✸❃

❋✐❣✉r❡ ✶✵✳✸

✸❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✵✸✴✶✳✶✴❃✳

✶✺✵

❈❍❆P❚❊❘ ✶✵✳ ❈❖▼❇■◆❆❚■❖◆❙ ❆◆❉ P❊❘▼❯❚❆❚■❖◆❙

❊①❡r❝✐s❡ ✶✵✳✸✿ P❡r♠✉t❛t✐♦♥s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✺✸✳✮

❙❤♦✇ t❤❛t ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ n ♦❜❥❡❝ts ❤❛s n! ♣❡r♠✉t❛t✐♦♥s✳

✶✵✳✸✳✶✳✶✳✶ P❡r♠✉t❛t✐♦♥ ✇✐t❤ ❘❡♣❡t✐t✐♦♥

❲❤❡♥ ♦r❞❡r ♠❛tt❡rs ❛♥❞ ❛♥ ♦❜❥❡❝t ❝❛♥ ❜❡ ❝❤♦s❡♥ ♠♦r❡ t❤❛♥ ♦♥❝❡ t❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢

♣❡r♠✉t❛t✐♦♥s ✐s✿

nr

✭✶✵✳✽✮

✇❤❡r❡ n ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♦❜❥❡❝ts ❢r♦♠ ✇❤✐❝❤ ②♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥❞ r ✐s t❤❡ ♥✉♠❜❡r t♦ ❜❡ ❝❤♦s❡♥✳

❋♦r ❡①❛♠♣❧❡✱ ✐❢ ②♦✉ ❤❛✈❡ t❤❡ ❧❡tt❡rs ❆✱ ❇✱ ❈✱ ❛♥❞ ❉ ❛♥❞ ②♦✉ ✇✐s❤ t♦ ❞✐s❝♦✈❡r t❤❡ ♥✉♠❜❡r ♦❢ ✇❛②s ♦❢

❛rr❛♥❣✐♥❣ t❤❡♠ ✐♥ t❤r❡❡ ❧❡tt❡r ♣❛tt❡r♥s ✭tr✐❣r❛♠s✮ ②♦✉ ✜♥❞ t❤❛t t❤❡r❡ ❛r❡ 43 ♦r ✻✹ ✇❛②s✳ ❚❤✐s ✐s ❜❡❝❛✉s❡ ❢♦r

t❤❡ ✜rst s❧♦t ②♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥② ♦❢ t❤❡ ❢♦✉r ✈❛❧✉❡s✱ ❢♦r t❤❡ s❡❝♦♥❞ s❧♦t ②♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥② ♦❢ t❤❡ ❢♦✉r✱ ❛♥❞

❢♦r t❤❡ ✜♥❛❧ s❧♦t ②♦✉ ❝❛♥ ❝❤♦♦s❡ ❛♥② ♦❢ t❤❡ ❢♦✉r ❧❡tt❡rs✳ ▼✉❧t✐♣❧②✐♥❣ t❤❡♠ t♦❣❡t❤❡r ❣✐✈❡s t❤❡ t♦t❛❧✳

✶✵✳✸✳✷ ❆♣♣❧✐❝❛t✐♦♥s

✶✵✳✸✳✷✳✶ ❚❤❡ ❇✐♥♦♠✐❛❧ ❚❤❡♦r❡♠

■♥ ♠❛t❤❡♠❛t✐❝s✱ t❤❡ ❜✐♥♦♠✐❛❧ t❤❡♦r❡♠ ✐s ❛♥ ✐♠♣♦rt❛♥t ❢♦r♠✉❧❛ ❣✐✈✐♥❣ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ ♣♦✇❡rs ♦❢ s✉♠s✳ ■ts

s✐♠♣❧❡st ✈❡rs✐♦♥ r❡❛❞s

n

n

(x + y)n =

xkyn−k

✭✶✵✳✾✮

k

k=0

❲❤❡♥❡✈❡r n ✐s ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✱ t❤❡ ♥✉♠❜❡rs

n

n!

=

✭✶✵✳✶✵✮

k

k! (n − k)!

❛r❡ t❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ✭t❤❡ ❝♦❡✣❝✐❡♥ts ✐♥ ❢r♦♥t ♦❢ t❤❡ ♣♦✇❡rs✮✳

❋♦r ❡①❛♠♣❧❡✱ ❤❡r❡ ❛r❡ t❤❡ ❝❛s❡s ♥ ❂ ✷✱ ♥ ❂ ✸ ❛♥❞ ♥ ❂ ✹✿

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2y + 3xy2 + y3

✭✶✵✳✶✶✮

(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

❚❤❡ ❝♦❡✣❝✐❡♥ts ❢♦r♠ ❛ tr✐❛♥❣❧❡✱ ✇❤❡r❡ ❡❛❝❤ ♥✉♠❜❡r ✐s t❤❡ s✉♠ ♦❢ t❤❡ t✇♦ ♥✉♠❜❡rs ❛❜♦✈❡ ✐t✿

❋✐❣✉r❡ ✶✵✳✹

❚❤✐s ❢♦r♠✉❧❛✱ ❛♥❞ t❤❡ tr✐❛♥❣✉❧❛r ❛rr❛♥❣❡♠❡♥t ♦❢ t❤❡ ❜✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts✱ ❛r❡ ♦❢t❡♥ ❛ttr✐❜✉t❡❞ t♦ ❇❧❛✐s❡

P❛s❝❛❧ ✇❤♦ ❞❡s❝r✐❜❡❞ t❤❡♠ ✐♥ t❤❡ ✶✼t❤ ❝❡♥t✉r②✳ ■t ✇❛s✱ ❤♦✇❡✈❡r✱ ❦♥♦✇♥ t♦ t❤❡ ❈❤✐♥❡s❡ ♠❛t❤❡♠❛t✐❝✐❛♥ ❨❛♥❣

❍✉✐ ✐♥ t❤❡ ✶✸t❤ ❝❡♥t✉r②✱ t❤❡ ❡❛r❧✐❡r P❡rs✐❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥ ❖♠❛r ❑❤❛②②➹➼♠ ✐♥ t❤❡ ✶✶t❤ ❝❡♥t✉r②✱ ❛♥❞ t❤❡

❡✈❡♥ ❡❛r❧✐❡r ■♥❞✐❛♥ ♠❛t❤❡♠❛t✐❝✐❛♥ P✐♥❣❛❧❛ ✐♥ t❤❡ ✸r❞ ❝❡♥t✉r② ❇❈✳

✶✺✶

❊①❡r❝✐s❡ ✶✵✳✹✿ ◆✉♠❜❡r P❧❛t❡s

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✺✸✳✮

❚❤❡ ♥✉♠❜❡r ♣❧❛t❡ ♦♥ ❛ ❝❛r ❝♦♥s✐sts ♦❢ ❛♥② ✸ ❧❡tt❡rs ♦❢ t❤❡ ❛❧♣❤❛❜❡t ✭❡①❝❧✉❞✐♥❣ t❤❡ ✈♦✇❡❧s ❛♥❞ ✬◗✬✮✱

❢♦❧❧♦✇❡❞ ❜② ❛♥② ✸ ❞✐❣✐ts ✭✵ t♦ ✾✮✳ ❋♦r ❛ ❝❛r ❝❤♦s❡♥ ❛t r❛♥❞♦♠✱ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡

♥✉♠❜❡r ♣❧❛t❡ st❛rts ✇✐t❤ ❛ ✬❨✬ ❛♥❞ ❡♥❞s ✇✐t❤ ❛♥ ♦❞❞ ❞✐❣✐t❄

❊①❡r❝✐s❡ ✶✵✳✺✿ ❋❛❝t♦r✐❛❧

✭❙♦❧✉t✐♦♥ ♦♥ ♣✳ ✶✺✸✳✮

❙❤♦✇ t❤❛t

n!

= n

✭✶✵✳✶✷✮

(n − 1)!

✶✵✳✸✳✸ ❊①❡r❝✐s❡s

✶✳ ❚s❤❡♣♦ ❛♥❞ ❙❛❧❧② ❣♦ t♦ ❛ r❡st❛✉r❛♥t✱ ✇❤❡r❡ t❤❡ ♠❡♥✉ ✐s✿

❙t❛rt❡r

▼❛✐♥ ❈♦✉rs❡

❉❡ss❡rt

❈❤✐❝❦❡♥ ✇✐♥❣s

❇❡❡❢ ❜✉r❣❡r

❈❤♦❝♦❧❛t❡ ✐❝❡ ❝r❡❛♠

▼✉s❤r♦♦♠ s♦✉♣ ❈❤✐❝❦❡♥ ❜✉r❣❡r

❙tr❛✇❜❡rr② ✐❝❡ ❝r❡❛♠

●r❡❡❦ s❛❧❛❞

❈❤✐❝❦❡♥ ❝✉rr②

❆♣♣❧❡ ❝r✉♠❜❧❡

▲❛♠❜ ❝✉rr②

❈❤♦❝♦❧❛t❡ ♠♦✉ss❡

❱❡❣❡t❛❜❧❡ ❧❛s❛❣♥❡

❚❛❜❧❡ ✶✵✳✷

❛✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❝♦♠❜✐♥❛t✐♦♥s ✭♦❢ st❛rt❡r✱ ♠❛✐♥ ❝♦✉rs❡✱ ❛♥❞ ❞❡ss❡rt✮ ❝❛♥ ❚s❤❡♣♦ ❤❛✈❡❄

❜✳ ❙❛❧❧② ❞♦❡s♥✬t ❧✐❦❡ ❝❤✐❝❦❡♥✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❝♦♠❜✐♥❛t✐♦♥s ❝❛♥ s❤❡ ❤❛✈❡❄

✷✳ ❋♦✉r ❝♦✐♥s ❛r❡ t❤r♦✇♥✱ ❛♥❞ t❤❡ ♦✉t❝♦♠❡s r❡❝♦r❞❡❞✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ✇❛②s ❛r❡ t❤❡r❡ ♦❢ ❣❡tt✐♥❣ t❤r❡❡

❤❡❛❞s❄ ❋✐rst ✇r✐t❡ ♦✉t t❤❡ ♣♦ss✐❜✐❧✐t✐❡s✱ ❛♥❞ t❤❡♥ ✉s❡ t❤❡ ❢♦r♠✉❧❛ ❢♦r ❝♦♠❜✐♥❛t✐♦♥s✳

✸✳ ❚❤❡ ❛♥s✇❡rs ✐♥ ❛ ♠✉❧t✐♣❧❡ ❝❤♦✐❝❡ t❡st ❝❛♥ ❜❡ ❆✱ ❇✱ ❈✱ ❉✱ ♦r ❊✳ ■♥ ❛ t❡st ♦❢ ✶✷ q✉❡st✐♦♥s✱ ❤♦✇ ♠❛♥②

❞✐✛❡r❡♥t ✇❛②s ❛r❡ t❤❡r❡ ♦❢ ❛♥s✇❡r✐♥❣ t❤❡ t❡st❄

✹✳ ❆ ❣✐r❧ ❤❛s ✹ ❞r❡ss❡s✱ ✷ ♥❡❝❦❧❛❝❡s✱ ❛♥❞ ✸ ❤❛♥❞❜❛❣s✳

❛✳ ❍♦✇ ♠❛♥② ❞✐✛❡r❡♥t ❝❤♦✐❝❡s ♦❢ ♦✉t✜t ✭❞r❡ss✱ ♥❡❝❦❧❛❝❡ ❛♥❞ ❤❛♥❞❜❛❣✮ ❞♦❡s s❤❡ ❤❛✈❡❄

❜✳ ❙❤❡ ♥♦✇ ❜✉②s t✇♦ ♣❛✐rs ♦❢ s❤♦❡s✳ ❍♦✇ ♠❛♥② ❝❤♦✐❝❡s ♦❢ ♦✉t✜t ✭❞r❡ss✱ ♥❡❝❦❧❛❝❡✱ ❤❛♥❞❜❛❣ ❛♥❞ s❤♦❡s✮

❞♦❡s s❤❡ ♥♦✇ ❤❛✈❡❄

✺✳ ■♥ ❛ s♦❝❝❡r t♦✉r♥❛♠❡♥t ♦❢ ✾ t❡❛♠s✱ ❡✈❡r② t❡❛♠ ♣❧❛②s ❡✈❡r② ♦t❤❡r t❡❛♠✳

❛✳ ❍♦✇ ♠❛♥② ♠❛t❝❤❡s ❛r❡ t❤❡r❡ ✐♥ t❤❡ t♦✉r♥❛♠❡♥t❄

❜✳ ■❢ t❤❡r❡ ❛r❡ ✺ ❜♦②s✬ t❡❛♠s ❛♥❞ ✹ ❣✐r❧s✬ t❡❛♠s✱ ✇❤❛t ✐s t❤❡ ♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ✜rst ♠❛t❝❤ ✇✐❧❧ ❜❡

♣❧❛②❡❞ ❜❡t✇❡❡♥ ✷ ❣✐r❧s✬ t❡❛♠s❄

✻✳ ❚❤❡ ❧❡tt❡rs ♦❢ t❤❡ ✇♦r❞ ✬❇▲❯❊✬ ❛r❡ r❡❛rr❛♥❣❡❞ r❛♥❞♦♠❧②✳ ❍♦✇ ♠❛♥② ♥❡✇ ✇♦r❞s ✭❛ ✇♦r❞ ✐s ❛♥②

❝♦♠❜✐♥❛t✐♦♥ ♦❢ ❧❡tt❡rs✮ ❝❛♥ ❜❡ ♠❛❞❡❄

✼✳ ❚❤❡ ❧❡tt❡rs ♦❢ t❤❡ ✇♦r❞ ✬❈❍❊▼■❙❚❘❨✬ ❛r❡ ❛rr❛♥❣❡❞ r❛♥❞♦♠❧② t♦ ❢♦r♠ ❛ ♥❡✇ ✇♦r❞✳ ❲❤❛t ✐s t❤❡

♣r♦❜❛❜✐❧✐t② t❤❛t t❤❡ ✇♦r❞ ✇✐❧❧ st❛rt ❛♥❞ ❡♥❞ ✇✐t❤ ❛ ✈♦✇❡❧❄

✽✳ ❚❤❡r❡ ❛r❡ ✷ ❍✐st♦r② ❝❧❛ss❡s✱ ✺ ❆❝❝♦✉♥t✐♥❣ ❝❧❛ss❡s✱ ❛♥❞ ✹ ▼❛t❤❡♠❛t✐❝s ❝❧❛ss❡s ❛t s❝❤♦♦❧✳ ▲✉❦❡ ✇❛♥ts t♦

❞♦ ❛❧❧ t❤r❡❡ s✉❜❥❡❝ts✳ ❍♦✇ ♠❛♥② ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❝❧❛ss❡s ❛r❡ t❤❡r❡❄

✾✳ ❆ s❝❤♦♦❧ ♥❡t❜❛❧❧ t❡❛♠ ❤❛s ✽ ♠❡♠❜❡rs✳ ❍♦✇ ♠❛♥② ✇❛②s ❛r❡ t❤❡r❡ t♦ ❝❤♦♦s❡ ❛ ❝❛♣t❛✐♥✱ ✈✐❝❡✲❝❛♣t❛✐♥✱

❛♥❞ r❡s❡r✈❡❄

✶✺✷

❈❍❆P❚❊❘ ✶✵✳ ❈❖▼❇■◆❆❚■❖◆❙ ❆◆❉ P❊❘▼❯❚❆❚■❖◆❙

✶✵✳ ❆ ❝❧❛ss ❤❛s ✶✺ ❜♦②s ❛♥❞ ✶✵ ❣✐r❧s✳ ❆ ❞❡❜❛t✐♥❣ t❡❛♠ ♦❢ ✹ ❜♦②s ❛♥❞ ✻ ❣✐r❧s ♠✉st ❜❡ ❝❤♦s❡♥✳ ❍♦✇ ♠❛♥②

✇❛②s ❝❛♥ t❤✐s ❜❡ ❞♦♥❡❄

✶✶✳ ❆ s❡❝r❡t ♣✐♥ ♥✉♠❜❡r ✐s ✸ ❝❤❛r❛❝t❡rs ❧♦♥❣✱ ❛♥❞ ❝❛♥ ✉s❡ ❛♥② ❞✐❣✐t ✭✵ t♦ ✾✮ ♦r ❛♥② ❧❡tt❡r ♦❢ t❤❡ ❛❧♣❤❛❜❡t✳

❘❡♣❡❛t❡❞ ❝❤❛r❛❝t❡rs ❛r❡ ❛❧❧♦✇❡❞✳ ❍♦✇ ♠❛♥② ♣♦ss✐❜❧❡ ❝♦♠❜✐♥❛t✐♦♥s ❛r❡ t❤❡r❡❄

✶✺✸

❙♦❧✉t✐♦♥s t♦ ❊①❡r❝✐s❡s ✐♥ ❈❤❛♣t❡r ✶✵

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✶✵✳✶ ✭♣✳ ✶✹✼✮

❙t❡♣ ✶✳ ❚❤❡r❡ ❛r❡ ✹ ♣❛rts✿ s❛♥❞✇✐❝❤✱ s♦✉♣✱ ❞❡ss❡rt ❛♥❞ ❞r✐♥❦✳

❙t❡♣ ✷✳ ▼❡❛❧ ❝♦♠♣♦♥❡♥t ❙❛♥❞✇✐❝❤ ❙♦✉♣ ❉❡ss❡rt ❉r✐♥❦

◆✉♠❜❡r ♦❢ ❝❤♦✐❝❡s ✹

✸

✷

✺

❚❛❜❧❡ ✶✵✳✸

❙t❡♣ ✸✳

4 × 3 × 2 × 5 = 120

✭✶✵✳✶✸✮

❙♦ t❤❡r❡ ❛r❡ ✶✷✵ ♣♦ss✐❜❧❡ ♠❡❛❧s✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✶✵✳✷ ✭♣✳ ✶✹✽✮

❙t❡♣ ✶✳ ❲❡ ❝♦✉♥t t❤❡ ❡✈❡♥ts✿ s♦❝❝❡r ❛♥❞ ♥❡t❜❛❧❧✱ s♦❝❝❡r ❛♥❞ ❜❛s❦❡t❜❛❧❧✱ s♦❝❝❡r ❛♥❞ t❡♥♥✐s✳ ❚❤✐s ❣✐✈❡s t❤r❡❡

❝❤♦✐❝❡s✳

❙t❡♣ ✷✳ ❚❤❡r❡ ❛r❡ ✻ s♣♦rts t♦ ❝❤♦♦s❡ ❢r♦♠ ❛♥❞ ✇❡ ❝❤♦♦s❡ ✷ s♣♦rts✳ ❚❤❡r❡ ❛r❡

6

= 6!/ (2! (6 − 2)!) ❂ ✶✺ ❝❤♦✐❝❡s✳

2

❙t❡♣ ✸✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ✇❡ ❛r❡ ❝♦✉♥t✐♥❣✱ ❞✐✈✐❞❡❞ ❜② t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❤♦✐❝❡s✳

Pr♦❜❛❜✐❧✐t② ❂ 3 ❂ 1 ❂ ✵✱✷

15

5

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✶✵✳✸ ✭♣✳ ✶✺✵✮

❙t❡♣ ✶✳ Pr♦♦❢✿ ❈♦♥str✉❝t✐♥❣ ❛♥ ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ n ♦❜❥❡❝ts ✐s ❡q✉✐✈❛❧❡♥t t♦ ❝❤♦♦s✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❝❝✉♣✐❡❞

❜② t❤❡ ✜rst ♦❜❥❡❝t✱ t❤❡♥ ❝❤♦♦s✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ♦❢ t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✱ ❛♥❞ s♦ ♦♥✱ ✉♥t✐❧ ✇❡ ❤❛✈❡ ❝❤♦s❡♥

t❤❡ ♣♦s✐t✐♦♥ ♦❢ ❡❛❝❤ ♦❢ ♦✉r n ♦❜❥❡❝ts✳

❙t❡♣ ✷✳ ❚❤❡r❡ ❛r❡ ♥ ✇❛②s t♦ ❝❤♦♦s❡ ❛ ♣♦s✐t✐♦♥ ❢♦r t❤❡ ✜rst ♦❜❥❡❝t✳ ❖♥❝❡ ✐ts ♣♦s✐t✐♦♥ ✐s ✜①❡❞✱ ✇❡ ❝❛♥ ❝❤♦♦s❡ ❢r♦♠

(n − 1) ♣♦ss✐❜❧❡ ♣♦s✐t✐♦♥s ❢♦r t❤❡ s❡❝♦♥❞ ♦❜❥❡❝t✳ ❲✐t❤ t❤❡ ✜rst t✇♦ ♣❧❛❝❡❞✱ t❤❡r❡ ❛r❡ (n − 2) r❡♠❛✐♥✐♥❣

♣♦ss✐❜❧❡ ♣♦s✐t✐♦♥s ❢♦r t❤❡ t❤✐r❞ ♦❜❥❡❝t❀ ❛♥❞ s♦ ♦♥✳ ❚❤❡r❡ ❛r❡ ♦♥❧② t✇♦ ♣♦s✐t✐♦♥s t♦ ❝❤♦♦s❡ ❢r♦♠ ❢♦r t❤❡

♣❡♥✉❧t✐♠❛t❡ ♦❜❥❡❝t✱ ❛♥❞ t❤❡ nth ♦❜❥❡❝t ✇✐❧❧ ♦❝❝✉♣② t❤❡ ❧❛st r❡♠❛✐♥✐♥❣ ♣♦s✐t✐♦♥✳

❙t❡♣ ✸✳ ❚❤❡r❡❢♦r❡✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❝♦✉♥t✐♥❣ ♣r✐♥❝✐♣❧❡✱ t❤❡r❡ ❛r❡

n (n − 1) (n − 2) ...2 × 1 = n!

✭✶✵✳✶✹✮

✇❛②s ♦❢ ❝♦♥str✉❝t✐♥❣ ❛♥ ♦r❞❡r❡❞ s❡q✉❡♥❝❡ ♦❢ n ♦❜❥❡❝ts✳

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✶✵✳✹ ✭♣✳ ✶✺✵✮

❙t❡♣ ✶✳ ❚❤❡ ♥✉♠❜❡r ♣❧❛t❡ st❛rts ✇✐t❤ ❛ ✬❨✬✱ s♦ t❤❡r❡ ✐s ♦♥❧② ✶ ❝❤♦✐❝❡ ❢♦r t❤❡ ✜rst ❧❡tt❡r✱ ❛♥❞ ❡♥❞s ✇✐t❤ ❛♥ ❡✈❡♥

❞✐❣✐t✱ s♦ t❤❡r❡ ❛r❡ ✺ ❝❤♦✐❝❡s ❢♦r t❤❡ ❧❛st ❞✐❣✐t ✭✶✱ ✸✱ ✺✱ ✼✱ ✾✮✳

❙t❡♣ ✷✳ ❯s❡ t❤❡ ❝♦✉♥t✐♥❣ ♣r✐♥❝✐♣❧❡✳ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ♦t❤❡r ❧❡tt❡rs✱ t❤❡r❡ ❛r❡ ✷✵ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ✭✷✻ ✐♥ t❤❡

❛❧♣❤❛❜❡t✱ ♠✐♥✉s ✺ ✈♦✇❡❧s ❛♥❞ ✬◗✬✮ ❛♥❞ ✶✵ ♣♦ss✐❜❧❡ ❝❤♦✐❝❡s ❢♦r ❡❛❝❤ ♦❢ t❤❡ ♦t❤❡r ❞✐❣✐ts✳

◆✉♠❜❡r ♦❢ ❡✈❡♥ts ❂ 1 × 20 × 20 × 10 × 10 × 5 = 200 000

❙t❡♣ ✸✳ ❯s❡ t❤❡ ❝♦✉♥t✐♥❣ ♣r✐♥❝✐♣❧❡✳ ❚❤✐s t✐♠❡✱ t❤❡ ✜rst ❧❡tt❡r ❛♥❞ ❧❛st ❞✐❣✐t ❝❛♥ ❜❡ ❛♥②t❤✐♥❣✳

❚♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❤♦✐❝❡s ❂ 20 × 20 × 20 × 10 × 10 × 10 = 8 000 000

❙t❡♣ ✹✳ ❚❤❡ ♣r♦❜❛❜✐❧✐t② ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❡✈❡♥ts ✇❡ ❛r❡ ❝♦✉♥t✐♥❣✱ ❞✐✈✐❞❡❞ ❜② t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❤♦✐❝❡s✳

Pr♦❜❛❜✐❧✐t② ❂ 200 000 = 1 = 0, 025

8 000 000

40

❙♦❧✉t✐♦♥ t♦ ❊①❡r❝✐s❡ ✶✵✳✺ ✭♣✳ ✶✺✶✮

✶✺✹

❈❍❆P❚❊❘ ✶✵✳ ❈❖▼❇■◆❆❚■❖◆❙ ❆◆❉ P❊❘▼❯❚❆❚■❖◆❙

❙t❡♣ ✶✳ ▼❡t❤♦❞ ✶✿ ❊①♣❛♥❞ t❤❡ ❢❛❝t♦r✐❛❧ ♥♦t❛t✐♦♥✳

n!

n × (n − 1) × (n − 2) × ... × 2 × 1

=

✭✶✵✳✶✺✮

(n − 1)!

(n − 1) × (n − 2) × ... × 2 × 1

❈❛♥❝❡❧❧✐♥❣ t❤❡ ❝♦♠♠♦♥ ❢❛❝t♦r ♦❢ (n − 1) × (n − 2) × ... × 2 × 1 ♦♥ t❤❡ t♦♣ ❛♥❞ ❜♦tt♦♠ ❧❡❛✈❡s n✳

❙♦ n! = n

(n−1)!

❙t❡♣ ✷✳ ▼❡t❤♦❞ ✷✿ ❲❡ ❦♥♦✇ t❤❛t P (n, r) = n! ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♣❡r♠✉t❛t✐♦♥s ♦❢ r ♦❜❥❡❝ts✱ t❛❦❡♥ ❢r♦♠ ❛

(n−r)!

♣♦♦❧ ♦❢ n ♦❜❥❡❝ts✳ ■♥ t❤✐s ❝❛s❡✱ r = 1✳ ❚♦ ❝❤♦♦s❡ ✶ ♦❜❥❡❝t ❢r♦♠ n ♦❜❥❡❝ts✱ t❤❡r❡ ❛r❡ n ❝❤♦✐❝❡s✳

❙♦ n! = n

(n−1)!

●▲❖❙❙❆❘❨

✶✺✺

●❧♦ss❛r②

❆ ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡

❆♥ ❛r✐t❤♠❡t✐❝ ✭♦r ❧✐♥❡❛r✮ s❡q✉❡♥❝❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ ✇❤✐❝❤ ❡❛❝❤ ♥❡✇ t❡r♠ ✐s

❝❛❧❝✉❧❛t❡❞ ❜② ❛❞❞✐♥❣ ❛ ❝♦♥st❛♥t ✈❛❧✉❡ t♦ t❤❡ ♣r❡✈✐♦✉s t❡r♠

❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡

❆♥ ❛r✐t❤♠❡t✐❝ ✭♦r ❧✐♥❡❛r✮ s❡q✉❡♥❝❡ ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ♥✉♠❜❡rs ✐♥ ✇❤✐❝❤ ❡❛❝❤ ♥❡✇ t❡r♠ ✐s

❝❛❧❝✉❧❛t❡❞ ❜② ❛❞❞✐♥❣ ❛ ❝♦♥st❛♥t ✈❛❧✉❡ t♦ t❤❡ ♣r❡✈✐♦✉s t❡r♠✿

an = an−1 + d

✭✶✳✸✮

✇❤❡r❡

• an r❡♣r❡s❡♥ts t❤❡ ♥❡✇ t❡r♠✱ t❤❡ nth✲t❡r♠✱ t❤❛t ✐s ❝❛❧❝✉❧❛t❡❞❀

• an−1 r❡♣r❡s❡♥ts t❤❡ ♣r❡✈✐♦✉s t❡r♠✱ t❤❡ (n − 1)th✲t❡r♠❀

• d r❡♣r❡s❡♥ts s♦♠❡ ❝♦♥st❛♥t✳

❉ ❉❡r✐✈❛t✐✈❡

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ f (x) ✐s ✇r✐tt❡♥ ❛s f✬ (x) ❛♥❞ ✐s ❞❡✜♥❡❞ ❜②✿

f (x + h) − f (x)

f ✬ (x) = lim

✭✺✳✻✮

h→0

h

❋ ❋❛❝t♦r ❚❤❡♦r❡♠

❋♦r ❛♥② ♣♦❧②♥♦♠✐❛❧✱ f (x)✱ ❢♦r ❛❧❧ ✈❛❧✉❡s ♦❢ a ✇❤✐❝❤ s❛t✐s❢② f (a) = 0✱ (x − a) ✐s ❛ ❢❛❝t♦r ♦❢ f (x)✳

❖r✱ ♠♦r❡ ❝♦♥❝✐s❡❧②✿

f (x) = (x − a) q (x)

✭✸✳✶✮

✐s ❛ ♣♦❧②♥♦♠✐❛❧✳

■♥ ♦t❤❡r ✇♦r❞s✿ ■❢ t❤❡ r❡♠❛✐♥❞❡r ✇❤❡♥ ❞✐✈✐❞✐♥❣ f (x) ❜② (x − a) ✐s ③❡r♦✱ t❤❡♥ (x − a) ✐s ❛ ❢❛❝t♦r

♦❢ f (x)✳

❙♦ ✐❢ f − b = 0✱ t❤❡♥ (ax + b) ✐s ❛ ❢❛❝t♦r ♦❢ f (x)✳

a

● ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s

❆ ❣❡♦♠❡tr✐❝ s❡q✉❡♥❝❡ ✐s ❛ s❡q✉❡♥❝❡ ✐♥ ✇❤✐❝❤ ❡✈❡r② ♥✉♠❜❡r ✐♥ t❤❡ s❡q✉❡♥❝❡ ✐s ❡q✉❛❧ t♦ t❤❡

♣r❡✈✐♦✉s ♥✉♠❜❡r ✐♥ t❤❡ s❡q✉❡♥❝❡✱ ♠✉❧t✐♣❧✐❡❞ ❜② ❛ ❝♦♥st❛♥t ♥✉♠❜❡r✳

▲ ▲♦❣❛r✐t❤♠s

■❢ an = x✱ t❤❡♥✿ log (x) = n✱ ✇❤❡r❡ a > 0❀ a = 1 ❛♥❞ x > 0✳

a

✶✺✻

■◆❉❊❳

■♥❞❡① ♦❢ ❑❡②✇♦r❞s ❛♥❞ ❚❡r♠s

❑❡②✇♦r❞s ❛r❡ ❧✐st❡❞ ❜② t❤❡ s❡❝t✐♦♥ ✇✐t❤ t❤❛t ❦❡②✇♦r❞ ✭♣❛❣❡ ♥✉♠❜❡rs ❛r❡ ✐♥ ♣❛r❡♥t❤❡s❡s✮✳ ❑❡②✇♦r❞s

❞♦ ♥♦t ♥❡❝❡ss❛r✐❧② ❛♣♣❡❛r ✐♥ t❤❡ t❡①t ♦❢ t❤❡ ♣❛❣❡✳ ❚❤❡② ❛r❡ ♠❡r❡❧② ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❛t s❡❝t✐♦♥✳ ❊①✳

❛♣♣❧❡s✱ ➓ ✶✳✶ ✭✶✮ ❚❡r♠s ❛r❡ r❡❢❡r❡♥❝❡❞ ❜② t❤❡ ♣❛❣❡ t❤❡② ❛♣♣❡❛r ♦♥✳ ❊①✳ ❛♣♣❧❡s✱ ✶

❆ ❆r✐t❤♠❡t✐❝ ❙❡q✉❡♥❝❡✱ ✶✶✱ ✶✷

▲ ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✱ ➓ ✻✳✶✭✽✺✮✱ ➓ ✻✳✷✭✽✽✮

▲♦❣❛r✐t❤♠s✱ ➓ ✭✶✮✱ ✷

❈ ❈♦♠❜✐♥❛t✐♦♥s✱ ➓ ✶✵✳✶✭✶✹✺✮✱ ➓ ✶✵✳✷✭✶✹✼✮✱

➓ ✶✵✳✸✭✶✹✾✮

▼ ▼❛t❤❡♠❛t✐❝s✱ ➓ ✭✶✮✱ ➓ ✶✵✳✶✭✶✹✺✮

❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s✱ ➓ ✸✳✶✭✹✺✮✱ ➓ ✸✳✷✭✹✼✮

▼❛t❤s✱ ➓ ✶✳✶✭✶✶✮✱ ➓ ✶✳✷✭✶✼✮✱ ➓ ✷✳✶✭✷✾✮✱

➓ ✷✳✷✭✸✺✮✱ ➓ ✸✳✶✭✹✺✮✱ ➓ ✸✳✷✭✹✼✮✱ ➓ ✹✳✶✭✺✸✮✱

❉ ❉❡r✐✈❛t✐✈❡✱ ✻✼

➓ ✹✳✷✭✺✹✮✱ ➓ ✺✳✶✭✻✶✮✱ ➓ ✺✳✷✭✻✼✮✱ ➓ ✺✳✸✭✼✷✮✱

❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s✱ ➓ ✺✳✶✭✻✶✮✱ ➓ ✺✳✷✭✻✼✮✱

➓ ✻✳✶✭✽✺✮✱ ➓ ✻✳✷✭✽✽✮✱ ➓ ✼✳✶✭✾✸✮✱ ➓ ✼✳✷✭✶✵✸✮✱

➓ ✺✳✸✭✼✷✮

➓ ✽✳✶✭✶✶✺✮✱ ➓ ✽✳✷✭✶✶✾✮✱ ➓ ✾✳✶✭✶✷✾✮✱ ➓ ✾✳✷✭✶✸✹✮✱

❋ ❋❛❝t♦r ❚❤❡♦r❡♠✱ ✹✺

➓ ✶✵✳✷✭✶✹✼✮✱ ➓ ✶✵✳✸✭✶✹✾✮

❋❛❝t♦r✐s✐♥❣✱ ➓ ✸✳✶✭✹✺✮✱ ➓ ✸✳✷✭✹✼✮

P P❡r♠✉t❛t✐♦♥s✱ ➓ ✶✵✳✶✭✶✹✺✮✱ ➓ ✶✵✳✷✭✶✹✼✮✱

❋✐♥❛♥❝❡✱ ➓ ✷✳✶✭✷✾✮✱ ➓ ✷✳✷✭✸✺✮

➓ ✶✵✳✸✭✶✹✾✮

❋✉♥❝t✐♦♥s✱ ➓ ✹✳✶✭✺✸✮✱ ➓ ✹✳✷✭✺✹✮

❙ ❙❡q✉❡♥❝❡s✱ ➓ ✶✳✶✭✶✶✮✱ ➓ ✶✳✷✭✶✼✮

● ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s✱ ✶✸

❙❡r✐❡s✱ ➓ ✶✳✶✭✶✶✮✱ ➓ ✶✳✷✭✶✼✮

●❡♦♠❡tr②✱ ➓ ✼✳✶✭✾✸✮✱ ➓ ✼✳✷✭✶✵✸✮

❙♦✉t❤ ❆❢r✐❝❛✱ ➓ ✭✶✮✱ ➓ ✶✳✶✭✶✶✮✱ ➓ ✶✳✷✭✶✼✮✱

●r❛❞❡ ✶✷✱ ➓ ✭✶✮✱ ➓ ✶✳✶✭✶✶✮✱ ➓ ✶✳✷✭✶✼✮✱ ➓ ✷✳✶✭✷✾✮✱

➓ ✷✳✶✭✷✾✮✱ ➓ ✷✳✷✭✸✺✮✱ ➓ ✸✳✶✭✹✺✮✱ ➓ ✸✳✷✭✹✼✮✱

➓ ✷✳✷✭✸✺✮✱ ➓ ✸✳✶✭✹✺✮✱ ➓ ✸✳✷✭✹✼✮✱ ➓ ✹✳✶✭✺✸✮✱

➓ ✹✳✶✭✺✸✮✱ ➓ ✹✳✷✭✺✹✮✱ ➓ ✺✳✶✭✻✶✮✱ ➓ ✺✳✷✭✻✼✮✱

➓ ✹✳✷✭✺✹✮✱ ➓ ✺✳✶✭✻✶✮✱ ➓ ✺✳✷✭✻✼✮✱ ➓ ✺✳✸✭✼✷✮✱

➓ ✺✳✸✭✼✷✮✱ ➓ ✻✳✶✭✽✺✮✱ ➓ ✻✳✷✭✽✽✮✱ ➓ ✼✳✶✭✾✸✮✱

➓ ✻✳✶✭✽✺✮✱ ➓ ✻✳✷✭✽✽✮✱ ➓ ✼✳✶✭✾✸✮✱ ➓ ✼✳✷✭✶✵✸✮✱

➓ ✼✳✷✭✶✵✸✮✱ ➓ ✽✳✶✭✶✶✺✮✱ ➓ ✽✳✷✭✶✶✾✮✱ ➓ ✾✳✶✭✶✷✾✮✱

➓ ✽✳✶✭✶✶✺✮✱ ➓ ✽✳✷✭✶✶✾✮✱ ➓ ✾✳✶✭✶✷✾✮✱ ➓ ✾✳✷✭✶✸✹✮✱

➓ ✾✳✷✭✶✸✹✮✱ ➓ ✶✵✳✶✭✶✹✺✮✱ ➓ ✶✵✳✷✭✶✹✼✮✱ ➓ ✶✵✳✸✭✶✹✾✮

➓ ✶✵✳✶✭✶✹✺✮✱ ➓ ✶✵✳✷✭✶✹✼✮✱ ➓ ✶✵✳✸✭✶✹✾✮

❙t❛t✐st✐❝s✱ ➓ ✾✳✶✭✶✷✾✮✱ ➓ ✾✳✷✭✶✸✹✮

●r❛♣❤s✱ ➓ ✹✳✶✭✺✸✮✱ ➓ ✹✳✷✭✺✹✮

❚ ❚r✐❣♦♥♦♠❡tr②✱ ➓ ✽✳✶✭✶✶✺✮✱ ➓ ✽✳✷✭✶✶✾✮

■ ■♥✈❡rs❡✱ ➓ ✹✳✷✭✺✹✮

❆❚❚❘■❇❯❚■❖◆❙

✶✺✼

❆ttr✐❜✉t✐♦♥s

❈♦❧❧❡❝t✐♦♥✿ ❙✐②❛✈✉❧❛ t❡①t❜♦♦❦s✿ ●r❛❞❡ ✶✷ ▼❛t❤s

❊❞✐t❡❞ ❜②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴❝♦❧✶✶✷✹✷✴✶✳✷✴

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧▲♦❣❛r✐t❤♠s✧

❇②✿ ❘♦r② ❆❞❛♠s✱ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t✱ ▼❛r❦ ❍♦r♥❡r✱ ❍❡❛t❤❡r ❲✐❧❧✐❛♠s

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✶✽✽✸✴✶✳✹✴

P❛❣❡s✿ ✶✲✶✵

❈♦♣②r✐❣❤t✿ ❘♦r② ❆❞❛♠s✱ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t✱ ▼❛r❦ ❍♦r♥❡r✱ ❍❡❛t❤❡r ❲✐❧❧✐❛♠s

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❙❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s✿ ❆r✐t❤♠❡t✐❝ ✫ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s✱ ❘❡❝✉rs✐✈❡ ❋♦r♠✉❧❛❡ ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❆r✐t❤♠❡t✐❝ ✫ ●❡♦♠❡tr✐❝ ❙❡q✉❡♥❝❡s✱ ❘❡❝✉rs✐✈❡ ❋♦r♠✉❧❛❡✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✵✷✴✶✳✶✴

P❛❣❡s✿ ✶✶✲✶✻

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❙❡q✉❡♥❝❡s ❛♥❞ s❡r✐❡s✿ ❙✐❣♠❛ ♥♦t❛t✐♦♥✱ ❋✐♥✐t❡ ✫ ■♥✜♥✐t❡ ❙❡r✐❡s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❙✐❣♠❛ ♥♦t❛t✐♦♥✱ ❋✐♥✐t❡ ✫ ■♥✜♥✐t❡ ❙❡r✐❡s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✵✶✴✶✳✶✴

P❛❣❡s✿ ✶✼✲✷✽

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋✐♥❛♥❝❡✿ ■♥tr♦❞✉❝t✐♦♥✱ ❙❡q✉❡♥❝❡s ✫ ❙❡r✐❡s✱ ❋✉t✉r❡ ❱❛❧✉❡ ♦❢ P❛②♠❡♥ts ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥✱ ❙❡q✉❡♥❝❡s ✫ ❙❡r✐❡s✱ ❋✉t✉r❡ ❱❛❧✉❡ ♦❢ P❛②♠❡♥ts✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✾✻✴✶✳✶✴

P❛❣❡s✿ ✷✾✲✸✺

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋✐♥❛♥❝❡✿ ■♥✈❡st♠❡♥ts ✫ ▲♦❛♥s✱ ▲♦❛♥ ❙❝❤❡❞✉❧❡s✱ ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣✱ ❋♦r♠✉❧❛❡ ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥✈❡st♠❡♥ts ✫ ▲♦❛♥s✱ ▲♦❛♥ ❙❝❤❡❞✉❧❡s✱ ❈❛♣✐t❛❧ ❖✉tst❛♥❞✐♥❣✱ ❋♦r♠✉❧❛❡✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✾✹✴✶✳✶✴

P❛❣❡s✿ ✸✺✲✹✵

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋❛❝t♦r✐s✐♥❣ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s✿ ■♥tr♦❞✉❝t✐♦♥✱ ❋❛❝t♦r ❚❤❡♦r❡♠✱ ❋❛❝t♦r✐s✐♥❣ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥✱ ❋❛❝t♦r ❚❤❡♦r❡♠✱ ❋❛❝t♦r✐s✐♥❣ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✾✴✶✳✶✴

P❛❣❡s✿ ✹✺✲✹✼

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

✶✺✽

❆❚❚❘■❇❯❚■❖◆❙

▼♦❞✉❧❡✿ ✧❋❛❝t♦r✐s✐♥❣ ❈✉❜✐❝ P♦❧②♥♦♠✐❛❧s✿ ❋❛❝t♦r ❚❤❡♦r❡♠ ❊①❡r❝✐s❡s✱ ❙♦❧✈✐♥❣ ❈✉❜✐❝ ❊q✉❛t✐♦♥s✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❋❛❝t♦r ❚❤❡♦r❡♠ ❊①❡r❝✐s❡s✱ ❙♦❧✈✐♥❣ ❈✉❜✐❝ ❊q✉❛t✐♦♥s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✻✴✶✳✶✴

P❛❣❡s✿ ✹✼✲✹✽

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋✉♥❝t✐♦♥s ❛♥❞ ●r❛♣❤s✿ ■♥tr♦❞✉❝t✐♦♥✱ ❉❡✜♥✐t✐♦♥✱ ◆♦t❛t✐♦♥ ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥✱ ❉❡✜♥✐t✐♦♥✱ ◆♦t❛t✐♦♥✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✽✻✴✶✳✶✴

P❛❣❡s✿ ✺✸✲✺✹

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❋✉♥❝t✐♦♥s ❛♥❞ ●r❛♣❤s✿ ●r❛♣❤s ♦❢ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧●r❛♣❤s ♦❢ ■♥✈❡rs❡ ❋✉♥❝t✐♦♥s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✽✷✴✶✳✶✴

P❛❣❡s✿ ✺✹✲✻✵

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s✿ ■♥tr♦❞✉❝t✐♦♥✱ ▲✐♠✐ts✱ ❆✈❡r❛❣❡ ●r❛❞✐❡♥t ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥✱ ▲✐♠✐ts✱ ❆✈❡r❛❣❡ ●r❛❞✐❡♥t✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✵✴✶✳✶✴

P❛❣❡s✿ ✻✶✲✻✼

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s✿ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✭❋✐rst Pr✐♥❝✐♣❧❡s✱ ❘✉❧❡s✮ ❛♥❞ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❉✐✛❡r❡♥t✐❛t✐♦♥ ✭❋✐rst Pr✐♥❝✐♣❧❡s✱ ❘✉❧❡s✮ ❛♥❞ ❙❦❡t❝❤✐♥❣ ●r❛♣❤s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✶✸✴✶✳✶✴

P❛❣❡s✿ ✻✼✲✼✷

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❉✐✛❡r❡♥t✐❛❧ ❈❛❧❝✉❧✉s✿ ❙♦❧✈✐♥❣ Pr♦❜❧❡♠s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❙♦❧✈✐♥❣ Pr♦❜❧❡♠s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✼✸✴✶✳✶✴

P❛❣❡s✿ ✼✷✲✼✻

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

❆❚❚❘■❇❯❚■❖◆❙

✶✺✾

▼♦❞✉❧❡✿ ✧▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✿ ■♥tr♦❞✉❝t✐♦♥✱ ❚❡r♠✐♥♦❧♦❣②✱ ❚❤❡ ❋❡❛s✐❜❧❡ ❘❡❣✐♦♥ ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥✱ ❚❡r♠✐♥♦❧♦❣②✱ ❚❤❡ ❋❡❛s✐❜❧❡ ❘❡❣✐♦♥✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✾✸✴✶✳✶✴

P❛❣❡s✿ ✽✺✲✽✽

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣✿ ▼❡t❤♦❞✱ ❊①❡r❝✐s❡s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧▼❡t❤♦❞✱ ❊①❡r❝✐s❡s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✾✼✴✶✳✶✴

P❛❣❡s✿ ✽✽✲✾✵

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧●❡♦♠❡tr②✿ ■♥tr♦❞✉❝t✐♦♥✱ ❈✐r❝❧❡ ●❡♦♠❡tr② ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥✱ ❈✐r❝❧❡ ●❡♦♠❡tr②✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✷✼✴✶✳✶✴

P❛❣❡s✿ ✾✸✲✶✵✸

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧●❡♦♠❡tr②✿ ❈♦♦r❞✐♥❛t❡ ●❡♦♠❡tr②✱ ❊q✉❛t✐♦♥ ♦❢ ❚❛♥❣❡♥t✱ ❚r❛♥s❢♦r♠❛t✐♦♥s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❈♦♦r❞✐♥❛t❡ ●❡♦♠❡tr②✱ ❊q✉❛t✐♦♥ ♦❢ ❚❛♥❣❡♥t✱ ❚r❛♥s❢♦r♠❛t✐♦♥s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✷✽✽✴✶✳✶✴

P❛❣❡s✿ ✶✵✸✲✶✶✵

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❚r✐❣♦♥♦♠❡tr②✿ ❈♦♠♣♦✉♥❞ ■❞❡♥t✐t✐❡s✱ Pr♦❜❧❡♠ ❙♦❧✈✐♥❣ ❙tr❛t❡❣✐❡s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❈♦♠♣♦✉♥❞ ■❞❡♥t✐t✐❡s✱ Pr♦❜❧❡♠ ❙♦❧✈✐♥❣ ❙tr❛t❡❣✐❡s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✶✾✴✶✳✶✴

P❛❣❡s✿ ✶✶✺✲✶✶✾

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❚r✐❣♦♥♦♠❡tr②✿ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❚r✐❣ ❋✉♥❝t✐♦♥s ✭✷❉ ✫ ✸❉✮✱ ❖t❤❡r ●❡♦♠❡tr✐❡s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❚r✐❣ ❋✉♥❝t✐♦♥s ✭✷❉ ✫ ✸❉✮✱ ❖t❤❡r ●❡♦♠❡tr✐❡s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✶✵✴✶✳✶✴

P❛❣❡s✿ ✶✶✾✲✶✷✹

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

✶✻✵

❆❚❚❘■❇❯❚■❖◆❙

▼♦❞✉❧❡✿ ✧❙t❛t✐st✐❝s✿ ◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥✱ ❙❛♠♣❧✐♥❣✱ ❋✉♥❝t✐♦♥ ❋✐tt✐♥❣ ✫ ❘❡❣r❡ss✐♦♥ ❆♥❛❧②s✐s ✭●r❛❞❡ ✶✷✮✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧◆♦r♠❛❧ ❉✐str✐❜✉t✐♦♥✱ ❙❛♠♣❧✐♥❣✱ ❋✉♥❝t✐♦♥ ❋✐tt✐♥❣ ✫ ❘❡❣r❡ss✐♦♥ ❆♥❛❧②s✐s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✵✺✴✶✳✶✴

P❛❣❡s✿ ✶✷✾✲✶✸✹

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❙t❛t✐st✐❝s✿ ▲❡❛st ❙q✉❛r❡s✱ ❈❛❧❝✉❧❛t♦r ❲♦r❦✱ ❈♦rr❡❧❛t✐♦♥ ❈♦❡✣❝✐❡♥ts✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧▲❡❛st ❙q✉❛r❡s✱ ❈❛❧❝✉❧❛t♦r ❲♦r❦✱ ❈♦rr❡❧❛t✐♦♥ ❈♦❡✣❝✐❡♥ts✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✸✷✹✴✶✳✶✴

P❛❣❡s✿ ✶✸✹✲✶✹✵

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❈♦♠❜✐♥❛t✐♦♥s ❛♥❞ ♣❡r♠✉t❛t✐♦♥s✿ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ◆♦t❛t✐♦♥✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧■♥tr♦❞✉❝t✐♦♥ ❛♥❞ ◆♦t❛t✐♦♥✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✵✵✴✶✳✶✴

P❛❣❡s✿ ✶✹✺✲✶✹✻

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❈♦♠❜✐♥❛t✐♦♥s ❛♥❞ P❡r♠✉t❛t✐♦♥s✿ ❋✉♥❞❛♠❡♥t❛❧ ❈♦✉♥t✐♥❣ Pr✐♥❝✐♣❧❡ ❛♥❞ ❈♦♠❜✐♥❛t✐♦♥s✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧❋✉♥❞❛♠❡♥t❛❧ ❈♦✉♥t✐♥❣ Pr✐♥❝✐♣❧❡ ❛♥❞ ❈♦♠❜✐♥❛t✐♦♥s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✵✻✴✶✳✶✴

P❛❣❡s✿ ✶✹✼✲✶✹✽

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

▼♦❞✉❧❡✿ ✧❈♦♠❜✐♥❛t✐♦♥s ❛♥❞ P❡r♠✉t❛t✐♦♥s✿ P❡r♠✉t❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✧

❯s❡❞ ❤❡r❡ ❛s✿ ✧P❡r♠✉t❛t✐♦♥s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s✧

❇②✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

❯❘▲✿ ❤tt♣✿✴✴s✐②❛✈✉❧❛✳❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✸✾✶✵✸✴✶✳✶✴

P❛❣❡s✿ ✶✹✾✲✶✺✷

❈♦♣②r✐❣❤t✿ ❋r❡❡ ❍✐❣❤ ❙❝❤♦♦❧ ❙❝✐❡♥❝❡ ❚❡①ts Pr♦❥❡❝t

▲✐❝❡♥s❡✿ ❤tt♣✿✴✴❝r❡❛t✐✈❡❝♦♠♠♦♥s✳♦r❣✴❧✐❝❡♥s❡s✴❜②✴✸✳✵✴

❆❜♦✉t ❈♦♥♥❡①✐♦♥s

❙✐♥❝❡ ✶✾✾✾✱ ❈♦♥♥❡①✐♦♥s ❤❛s ❜❡❡♥ ♣✐♦♥❡❡r✐♥❣ ❛ ❣❧♦❜❛❧ s②st❡♠ ✇❤❡r❡ ❛♥②♦♥❡ ❝❛♥ ❝r❡❛t❡ ❝♦✉rs❡ ♠❛t❡r✐❛❧s ❛♥❞

♠❛❦❡ t❤❡♠ ❢✉❧❧② ❛❝❝❡ss✐❜❧❡ ❛♥❞ ❡❛s✐❧② r❡✉s❛❜❧❡ ❢r❡❡ ♦❢ ❝❤❛r❣❡✳ ❲❡ ❛r❡ ❛ ❲❡❜✲❜❛s❡❞ ❛✉t❤♦r✐♥❣✱ t❡❛❝❤✐♥❣ ❛♥❞

❧❡❛r♥✐♥❣ ❡♥✈✐r♦♥♠❡♥t ♦♣❡♥ t♦ ❛♥②♦♥❡ ✐♥t❡r❡st❡❞ ✐♥ ❡❞✉❝❛t✐♦♥✱ ✐♥❝❧✉❞✐♥❣ st✉❞❡♥ts✱ t❡❛❝❤❡rs✱ ♣r♦❢❡ss♦rs ❛♥❞

❧✐❢❡❧♦♥❣ ❧❡❛r♥❡rs✳ ❲❡ ❝♦♥♥❡❝t ✐❞❡❛s ❛♥❞ ❢❛❝✐❧✐t❛t❡ ❡❞✉❝❛t✐♦♥❛❧ ❝♦♠♠✉♥✐t✐❡s✳

❈♦♥♥❡①✐♦♥s✬s ♠♦❞✉❧❛r✱ ✐♥t❡r❛❝t✐✈❡ ❝♦✉rs❡s ❛r❡ ✐♥ ✉s❡ ✇♦r❧❞✇✐❞❡ ❜② ✉♥✐✈❡rs✐t✐❡s✱ ❝♦♠♠✉♥✐t② ❝♦❧❧❡❣❡s✱ ❑✲✶✷

s❝❤♦♦❧s✱ ❞✐st❛♥❝❡ ❧❡❛r♥❡rs✱ ❛♥❞ ❧✐❢❡❧♦♥❣ ❧❡❛r♥❡rs✳ ❈♦♥♥❡①✐♦♥s ♠❛t❡r✐❛❧s ❛r❡ ✐♥ ♠❛♥② ❧❛♥❣✉❛❣❡s✱ ✐♥❝❧✉❞✐♥❣

❊♥❣❧✐s❤✱ ❙♣❛♥✐s❤✱ ❈❤✐♥❡s❡✱ ❏❛♣❛♥❡s❡✱ ■t❛❧✐❛♥✱ ❱✐❡t♥❛♠❡s❡✱ ❋r❡♥❝❤✱ P♦rt✉❣✉❡s❡✱ ❛♥❞ ❚❤❛✐✳ ❈♦♥♥❡①✐♦♥s ✐s ♣❛rt

♦❢ ❛♥ ❡①❝✐t✐♥❣ ♥❡✇ ✐♥❢♦r♠❛t✐♦♥ ❞✐str✐❜✉t✐♦♥ s②st❡♠ t❤❛t ❛❧❧♦✇s ❢♦r Pr✐♥t ♦♥ ❉❡♠❛♥❞ ❇♦♦❦s✳ ❈♦♥♥❡①✐♦♥s

❤❛s ♣❛rt♥❡r❡❞ ✇✐t❤ ✐♥♥♦✈❛t✐✈❡ ♦♥✲❞❡♠❛♥❞ ♣✉❜❧✐s❤❡r ◗❖❖P t♦ ❛❝❝❡❧❡r❛t❡ t❤❡ ❞❡❧✐✈❡r② ♦❢ ♣r✐♥t❡❞ ❝♦✉rs❡

♠❛t❡r✐❛❧s ❛♥❞ t❡①t❜♦♦❦s ✐♥t♦ ❝❧❛ssr♦♦♠s ✇♦r❧❞✇✐❞❡ ❛t ❧♦✇❡r ♣r✐❝❡s t❤❛♥ tr❛❞✐t✐♦♥❛❧ ❛❝❛❞❡♠✐❝ ♣✉❜❧✐s❤❡rs✳

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