# SOLUTION: MAC2233 The University of Oklahoma Utility Function Measurements Test 3 And hither it is, the PDF improve containing all the solutions. Thanks! 😊

MAC2233 Test 3 Summer 2020
Question 1:
(a) The profit sway is 𝑈 = 𝑥 2 𝑦 3 , so if a consumer has 𝑥 = 10 pounds of aid and
𝑦 = 2 gallons of steep, then the profit is 𝑈 = (10)2 ⋅ (2)3 = 100 ⋅ 8 = 800 utiles.
(b)

𝑑
𝑑𝑡

(𝑈) =

𝑑
𝑑𝑡

(𝑥 2 𝑦 3 )

𝑑𝑈
𝑑
𝑑
= 𝑥 2 ⋅ (𝑦 3 ) + (𝑥 2 ) ⋅ 𝑦 3
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑑𝑦

𝑑𝑥

= 𝑥 2 ⋅ 3𝑦 2 ⋅ 𝑑𝑡 + 2𝑥 ⋅ 𝑑𝑡 ⋅ 𝑦 3
𝑑𝑦

𝑑𝑥

= 3𝑥 2 𝑦 2 ⋅ 𝑑𝑡 + 2𝑥𝑦 3 ⋅ 𝑑𝑡

(c) From allot (b), the reprove of alter of the profit (after a while i-elation to season) is
𝑑𝑈
2 2 𝑑𝑦
3 𝑑𝑥
=
3𝑥
𝑦

+
2𝑥𝑦
⋅ 𝑑𝑡 .
𝑑𝑡
𝑑𝑡
Then,
𝑑𝑦
𝑑𝑡

𝑑𝑥
𝑑𝑡

= −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
𝑑𝑈
= 3 ⋅ (10)2 ⋅ (2)2 ⋅ 5 + 2 ⋅ (10) ⋅ (2)3 ⋅ −3 = 300 ⋅ 20 + 20 ⋅ −24
𝑑𝑡
= 6000 − 480 = 5520

Question 2:
(a) ∫ 𝑥 𝑛 𝑑𝑥
Using the sway government of integration:
=

𝑥 𝑛+1
𝑛+1

+ 𝐶; whither 𝑛 ≠ −1

(b) ∫ 𝑥 −1 𝑑𝑥
Writing any unimaginative look to the sway of −1, is the selfselfsame as initiative that
expression’s reciprocal:
1
= ∫ 𝑑𝑥
𝑥
This unimpaired is similar to the dishonorable unimpaired involving logarithms:
= ln|𝑥| + 𝐶
(c) ∫ 𝑒 𝑘𝑥 𝑑𝑥
For this unimpaired, the ...

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