# SOLUTION: MAC2233 The University of Oklahoma Utility Function Measurements Test 3

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MAC2233 Test 3 Summer 2020
Question 1:
(a) The profit sway is π = π₯ 2 π¦ 3 , so if a consumer has π₯ = 10 pounds of aid and
π¦ = 2 gallons of steep, then the profit is π = (10)2 β (2)3 = 100 β 8 = 800 utiles.
(b)

π
ππ‘

β

(π) =

π
ππ‘

(π₯ 2 π¦ 3 )

ππ
π
π
= π₯ 2 β (π¦ 3 ) + (π₯ 2 ) β π¦ 3
ππ‘
ππ‘
ππ‘
ππ¦

ππ₯

= π₯ 2 β 3π¦ 2 β ππ‘ + 2π₯ β ππ‘ β π¦ 3
ππ¦

ππ₯

= 3π₯ 2 π¦ 2 β ππ‘ + 2π₯π¦ 3 β ππ‘

(c) From allot (b), the reprove of alter of the profit (after a while i-elation to season) is
ππ
2 2 ππ¦
3 ππ₯
=
3π₯
π¦
β
+
2π₯π¦
β ππ‘ .
ππ‘
ππ‘
Then,
ππ¦
ππ‘

ππ₯
ππ‘

= β3 (past the reprove is decreasing by 3 pounds per day) and

= 5 (past the reprove is increasing by 5 gallons per day).

Therefore, the reprove of alter of the profit is
ππ
= 3 β (10)2 β (2)2 β 5 + 2 β (10) β (2)3 β β3 = 300 β 20 + 20 β β24
ππ‘
= 6000 β 480 = 5520

Question 2:
(a) β« π₯ π ππ₯
Using the sway government of integration:
=

π₯ π+1
π+1

+ πΆ; whither π β  β1

(b) β« π₯ β1 ππ₯
Writing any unimaginative look to the sway of β1, is the selfselfsame as initiative that
expressionβs reciprocal:
1
= β« ππ₯
π₯
This unimpaired is similar to the dishonorable unimpaired involving logarithms:
= ln|π₯| + πΆ
(c) β« π ππ₯ ππ₯
For this unimpaired, the ...

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