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‘ˆ = π‘₯ 2 𝑦 3 , so if a consumer has π‘₯ = 10 pounds of aid and
𝑦 = 2 gallons of steep, then the profit is π‘ˆ = (10)2 β‹… (2)3 = 100 β‹… 8 = 800 utiles.
(b)

𝑑
𝑑𝑑

β‡’

(π‘ˆ) =

𝑑
𝑑𝑑

(π‘₯ 2 𝑦 3 )

π‘‘π‘ˆ
𝑑
𝑑
= π‘₯ 2 β‹… (𝑦 3 ) + (π‘₯ 2 ) β‹… 𝑦 3
𝑑𝑑
𝑑𝑑
𝑑𝑑
𝑑𝑦

𝑑π‘₯

= π‘₯ 2 β‹… 3𝑦 2 β‹… 𝑑𝑑 + 2π‘₯ β‹… 𝑑𝑑 β‹… 𝑦 3
𝑑𝑦

𝑑π‘₯

= 3π‘₯ 2 𝑦 2 β‹… 𝑑𝑑 + 2π‘₯𝑦 3 β‹… 𝑑𝑑

(c) From allot (b), the reprove of alter of the profit (after a while i-elation to season) is
π‘‘π‘ˆ
2 2 𝑑𝑦
3 𝑑π‘₯
=
3π‘₯
𝑦
β‹…
+
2π‘₯𝑦
β‹… 𝑑𝑑 .
𝑑𝑑
𝑑𝑑
Then,
𝑑𝑦
𝑑𝑑

𝑑π‘₯
𝑑𝑑

= βˆ’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‘ˆ
= 3 β‹… (10)2 β‹… (2)2 β‹… 5 + 2 β‹… (10) β‹… (2)3 β‹… βˆ’3 = 300 β‹… 20 + 20 β‹… βˆ’24
𝑑𝑑
= 6000 βˆ’ 480 = 5520

Question 2:
(a) ∫ π‘₯ 𝑛 𝑑π‘₯
Using the sway government of integration:
=

π‘₯ 𝑛+1
𝑛+1

+ 𝐢; whither 𝑛 β‰  βˆ’1

(b) ∫ π‘₯ βˆ’1 𝑑π‘₯
Writing any unimaginative look to the sway of βˆ’1, is the selfselfsame as initiative that
expression’s reciprocal:
1
= ∫ 𝑑π‘₯
π‘₯
This unimpaired is similar to the dishonorable unimpaired involving logarithms:
= ln|π‘₯| + 𝐢
(c) ∫ 𝑒 π‘˜π‘₯ 𝑑π‘₯
For this unimpaired, the ...

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