PBAF 499 HW#5
PbAf 499, Summer 2011
University of Washington
Homework Assignment #5, Due 7/19
On this homework assignment, I will be grading the starred questions.
Find the derivative of the function at the indicated point(s).
1.* y = 5 + 2x, x=2
2. y = 5 - 2x, x=2
3. y = -2 + 6x, x=2
4. y = x2, x=3
5. y = 1 - x2, x=1
6. y = 1 - 2x2, x=1
7. y = 5 + x2, x=3
8. y = x2 - 2, x=1,2
9. y = x2 - 3, x=4
10. y = x2 + 2, x=4
11.* y = 4x - x2 + 3, x=1,2,3
12. y = 4x - x2, x=1,2,3
13. y = 4x - x2 - 4, x=1,2,3
14. y = x2 - 4x, x=1,2,3
15. y = x2 - 4x + 4, x=1,2,3
16. y = x3, x=-1,0,1
17. y = x3 + 3, x=-1,0,1
18. y = x2 + 2x + 1, x=-1,0
19. y = (4-x)2, x=3,4 (Note: (4-x)2 = (4-x)(4-x) = 42 -4x-4x+x2 = 16 - 8x + x2)
Find the value(s) of x for which the derivatives of the following functions equals zero. Is each a
minimum, a maximum or neither?
20. y = 5 + 2x
21. y = 5 - 2x
22. y = -2 + 6x
23. y = x2
24.* y = 1 - x2
25. y = 1 - 2x2
26. y = 5 + x2
27. y = x2 - 2
28. y = x2 - 3
29. y = x2 + 2
30. y = 4x - x2 + 3
31. y = 4x - x2
32. y = 4x - x2 - 4
33. y = x2 - 4x
34.* y = x2 - 4x + 4
35. y = x3
PBAF 499 HW#5
36. y = x3 + 3
37. y = x2 + 2x + 1
38. y = (4-x)2
Find the partial derivatives with respect to each of the explanatory
variables of each of the following functions.
39. y = 5w + 6x + 7xw
40. y = 5w2 + 6x3 + 7x2w
41.* y = 5w2 + 6x2 + 7xw2
42. y = w/x
43. y = 5w2 + 6x3/w2 + 7w/x3
That was sure fun. Now let's try some story problems.
44.* Bob's Balls makes and sells balls. Balls are priced at $20 each, and they can't change that
because the price is set in the world ball market. Their costs can be expressed as a function of
the quantity they produce: TC(q)=100 + q2. As this function indicates. Bob's fixed costs are
100. Bob's profit is equal to revenue minus cost. What is their profit maximizing quantity?
What is their profit at this quantity?
45.* How does the profit maximizing quantity change if Bob's fixed costs double to 200?
46. How does the profit maximizing quantity change if Bob's fixed costs are 50?
47. Average cost is defined as AC(q)=TC(q)/q, or total cost divided by quantity. At what
quantity is Bob's average cost minimized? What is the average cost at that quantity?
48. If Bob's fixed costs are 400, at what quantity will average cost be minimized?
49. American Big Tobacco (ABT) makes and sells chewing tobacco laced with fiberglass.
Because they're the only company making and selling this exact product, the market price
depends on the quantity they sell. The price is given by p(q)=100-q and their costs are given by
TC(q)=20+2q+q2. Calculate their profit maximizing quantity. What is the market price at this
quantity?
50. Because of a legal ruling, ABT has to give $40 to Whittier Elementary School. The result is
that their fixed costs increase from 20 to 60. How does this affect their profit maximizing
quantity and price?
51. A new report emphasizes the health benefits of chewing fiberglass, and ABT's product is a
good source of nutritional fiberglass. As a result, demand for ABT's product rises to p(q)=120(q/2). Now solve for their profit maximizing quantity.
Solve the following utility maximization problems.
52. PA=10, PB=20, M=600, U(A,B)=100A2B
53. PA=10, PB=20, M=600, U(A,B)=10A2B
PBAF 499 HW#5
54.* PA=10, PB=20, M=600, U(A,B)=A2B
55. PA=10, PB=20, M=600, U(A,B)=100AB1/2
56. PA=10, PB=20, M=600, U(A,B)=10AB1/2
57. PA=10, PB=20, M=600, U(A,B)=10A2/3B1/3
58. How do the answers change as the specification of the utility function changes?
59. PA=10, PB=20, M=600, U(A,B)=AB
60. PA=10, PB=20, M=600, U(A,B)=A1/2B1/2
61. PA=10, PB=20, M=600, U(A,B)=A1/3B1/3
62. How do the answers change as the specification of the utility function changes?
63. Calculate the utility maximizing quantities for PA=10, M=720, U(A,B)=10A3/4B1/4 and
PB=5,10,20,40. On a graph, plot the quantities of good B along the horizontal axis and the
associated prices on the vertical axis to get a demand curve. Is it linear? What is the price
elasticity of demand for good B? Does this seem like a good model?