# 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

ππ‘

β

(π) =

π

ππ‘

(π₯ 2 π¦ 3 )

ππ

π

π

= π₯ 2 β
(π¦ 3 ) + (π₯ 2 ) β
π¦ 3

ππ‘

ππ‘

ππ‘

ππ¦

ππ₯

= π₯ 2 β
3π¦ 2 β
ππ‘ + 2π₯ β
ππ‘ β
π¦ 3

ππ¦

ππ₯

= 3π₯ 2 π¦ 2 β ππ‘ + 2π₯π¦ 3 β ππ‘

(c) From allot (b), the reprove of alter of the profit (after a while i-elation to season) is

ππ

2 2 ππ¦

3 ππ₯

=

3π₯

π¦

β

+

2π₯π¦

β
ππ‘ .

ππ‘

ππ‘

Then,

ππ¦

ππ‘

ππ₯

ππ‘

= β3 (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

ππ

= 3 β
(10)2 β
(2)2 β
5 + 2 β
(10) β
(2)3 β
β3 = 300 β
20 + 20 β
β24

ππ‘

= 6000 β 480 = 5520

Question 2:

(a) β« π₯ π ππ₯

Using the sway government of integration:

=

π₯ π+1

π+1

+ πΆ; whither π β β1

(b) β« π₯ β1 ππ₯

Writing any unimaginative look to the sway of β1, is the selfselfsame as initiative that

expressionβs reciprocal:

1

= β« ππ₯

π₯

This unimpaired is similar to the dishonorable unimpaired involving logarithms:

= ln|π₯| + πΆ

(c) β« π ππ₯ ππ₯

For this unimpaired, the ...

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