M-273Q Midterm: MATH 273 Montana State OldExam1
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▼✷✼✸◗ ▼✉❧t✐✈❛r✐❛❜❧❡ ❈❛❧❝✉❧✉s ❖❧❞ ❊①❛♠ ✶
◆❛♠❡ ❛♥❞ s❡❝t✐♦♥✿
■♥str✉❝t♦r✬s ♥❛♠❡✿
■♥str✉❝t✐♦♥s✿ ❈❧♦s❡❞ ❜♦♦❦✳ ◆♦ ❝❛❧❝✉❧❛t♦r ❛❧❧♦✇❡❞✳ ❉♦✉❜❧❡✲s✐❞❡❞ ❡①❛♠✳ ◆❖ ❈❊▲▲ P❍❖◆❊❙✳
❙❤♦✇ ❛❧❧ ✇♦r❦ ❛♥❞ ✉s❡ ❝♦rr❡❝t ♥♦t❛t✐♦♥ t♦ r❡❝❡✐✈❡ ❢✉❧❧ ❝r❡❞✐t✦ ❲r✐t❡ ❧❡❣✐❜❧②✳
✔✭s✮ ❂
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❞❚
❞s
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☞✔✭①✮ ❂ ❥❢✵✵✭①✮❥
❬✶ ✰ ✭❢✵✭①✮✮✷❪✸❂✷
✔✭t✮ ❂ ❥❥❚✵✭t✮❥❥
❥❥r✵✭t✮❥❥ ✔✭t✮ ❂ ❥❥r✵✭t✮✂r✵✵✭t✮❥❥
❥❥r✵✭t✮❥❥✸
✶✳ ✭✶ ❝r❡❞✐t ✮ ❉❡❝✐❞❡ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t✐❡s ❛r❡ ✈❡❝t♦rs✱ s❝❛❧❛rs✱ ♦r ❛r❡ ♥♦♥s❡♥s✐❝❛❧ ✭t❤❛t ✐s✱
t❤❡ st❛t❡♠❡♥t ✐s ♥♦t ❞❡☞♥❡❞ ♦r ❞♦❡s ♥♦t ♠❛❦❡ s❡♥s❡✮
✌❱❡❝t♦r ✌❙❝❛❧❛r ✌◆♦♥s❡♥s❡ ✭✉✁✈✮✂✇
✌❱❡❝t♦r ✌❙❝❛❧❛r ✌◆♦♥s❡♥s❡ ❦✉✂✈❦
✌❱❡❝t♦r ✌❙❝❛❧❛r ✌◆♦♥s❡♥s❡ ✭✉✁✈✮✇
✌❱❡❝t♦r ✌❙❝❛❧❛r ✌◆♦♥s❡♥s❡ ✔✭t✮ ❂ ❥❥r✵✭t✮✂r✵✵ ✭t✮❥❥
❥❥r✵✭t✮❥❥✸
✷✳ ✭✶ ❝r❡❞✐t ✮ ❉❡t❡r♠✐♥❡ ✇❤❡t❤❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛t✐♦♥s ❞❡s❝r✐❜❡ ❛ ♣❧❛♥❡✱ ❛ ❧✐♥❡✱ ♦r ♥❡✐t❤❡r ✐♥ ❘✸✿
✌P❧❛♥❡ ✌▲✐♥❡ ✌◆❡✐t❤❡r r✭t✮ ❂❁✶❀✶❀✺❃✰t ❁ ✵❀✷❀✸❃
✌P❧❛♥❡ ✌▲✐♥❡ ✌◆❡✐t❤❡r ①✰②✰③❂ ✶
✌P❧❛♥❡ ✌▲✐♥❡ ✌◆❡✐t❤❡r ②❂ ✺ ✰ ③
✌P❧❛♥❡ ✌▲✐♥❡ ✌◆❡✐t❤❡r ①✭t✮ ❂ ✼ ✰ ✷t❀ ②✭t✮ ❂ ✶✶ ✺t❀ ③✭t✮ ❂ t
✙
✸✳ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ ❛ ✉♥✐t ✈❡❝t♦r ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧✐♥❡ r✭t✮ ❂❁ t ✰ ✹❀✷ ✰ ✷t❀ ✺✷t ❃ ✿
Pr♦❜❧❡♠ ✶ ✷ ✸ ❚♦t❛❧
❈r❡❞✐t ✶ ✶ ✶ ✸
●P❆ ❈r❡❞✐t P♦✐♥ts ❊❛r♥❡❞
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▼✷✼✸◗ ▼✉❧t✐✈❛r✐❛❜❧❡ ❈❛❧❝✉❧✉s ❖❧❞ ❊①❛♠ ✶ ✲ P❛❣❡ ✷ ♦❢ ✽
✹✳ ▲❡t ❛❂❁✶❀✷❀✶❃❛♥❞ ❜❂❁✷❀✶❀✸❃ ✿
✭❛✮ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ ❦❛❦✿
✭❜✮ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ ❜✂❛✿
✭❝✮ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ ❛✁❜✿
✭❞✮ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ ✸❛✷❜✿
✭❡✮ ✭✶ ❝r❡❞✐t ✮ ■s t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ ❛❛♥❞ ❜❛❝✉t❡ ✭❧❡ss t❤❛t ✙❂✷✮✱ ♦❜t✉s❡ ✭❣r❡❛t❡r t❤❛♥ ✙❂✷✮✱ ♦r
♥❡✐t❤❡r❄
Pr♦❜❧❡♠ ✹ ❚♦t❛❧
❈r❡❞✐t ✺ ✺
●P❆ ❈r❡❞✐t P♦✐♥ts ❊❛r♥❡❞
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▼✷✼✸◗ ▼✉❧t✐✈❛r✐❛❜❧❡ ❈❛❧❝✉❧✉s ❖❧❞ ❊①❛♠ ✶ ✲ P❛❣❡ ✸ ♦❢ ✽
✺✳ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ ❛ ✈❡❝t♦r ♦❢ ❧❡♥❣t❤ ✹ t❤❛t ✐s ♦rt❤♦❣♦♥❛❧ t♦ ❜♦t❤ ❛❂❁✶❀✷❀✶❃❛♥❞ ❜❂❁✷❀✶❀✸❃ ✿
✻✳ ❚❤✐s q✉❡st✐♦♥ ❤❛s t✇♦ ♣❛rts✳
✭❛✮ ✭✶ ❝r❡❞✐t ✮ ❋✐♥❞ t❤❡ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❧✐♥❡ t❤r♦✉❣❤ P✭✸❀✶❀✵✮ ❛♥❞ ◗✭✶❀✹❀✸✮✿
✭❜✮ ✭✶ ❝r❡❞✐t ✮ ❙❤♦✇ t❤❛t t❤❡ ❧✐♥❡ ②♦✉ ❢♦✉♥❞ ✐♥ ♣❛rt ✭❛✮ ✐s ♦rt❤♦❣♦♥❛❧ t♦ ①✭t✮ ❂ ✸t❀ ②✭t✮ ❂ ✸✰✽t❀ ③✭t✮ ❂
✼ ✰ ✻t✿
✼✳ ✭✷ ❝r❡❞✐t ✮ ❋✐♥❞ ❛♥ ❡q✉❛t✐♦♥ ❢♦r t❤❡ t❛♥❣❡♥t ❧✐♥❡ t♦ t❤❡ ❝✉r✈❡ r✭t✮ ❂❁ t❀ t✷❀ t✸❃❛t t❤❡ ♣♦✐♥t ✭✶❀✶❀✶✮✿
Pr♦❜❧❡♠ ✺ ✻ ✼ ❚♦t❛❧
❈r❡❞✐t ✶ ✷ ✷ ✺
●P❆ ❈r❡❞✐t P♦✐♥ts ❊❛r♥❡❞