PbAf 499
University of Washington
Homework Assignment #5
Find the derivative of the function at the indicated point(s).
1. y = 5 + 2x, x=2
2. y = 5 - 2x, x=2
y'=-2
3. y = -2 + 6x, x=2
y'=6
4. y = x2, x=3
y'=2x
x=3,y'=6
2
5. y = 1 - x , x=1
y'=-2x
x=1,y'=-2
6. y = 1 - 2x2, x=1
y'=-4x
x=1,y'=-4
2
7. y = 5 + x , x=3
y'=2x
x=3,y'=6
8. y = x2 - 2, x=1,2
y'=2x
x=1,y'=2
x=2,y'=4
2
9. y = x - 3, x=4
y'=2x
x=4,y'=8
10. y = x2 + 2, x=4
y'=2x
x=4,y'=8
11. y = 4x - x2 + 3, x=1,2,3
12. y = 4x - x2, x=1,2,3
y'=4-2x
x=1,y'=2 x=2,y'=0 x=3,y'=-2
13. y = 4x - x2 - 4, x=1,2,3
y'=4-2x
x=1,y'=2 x=2,y'=0 x=3,y'=-2
2
14. y = x - 4x, x=1,2,3
y'=2x-4
x=1,y'=-2 x=2,y'=0 x=3,y'=2
15. y = x2 - 4x + 4, x=1,2,3
y'=2x-4
x=1,y'=-2 x=2,y'=0 x=3,y'=2
3
2
16. y = x , x=-1,0,1
y'=3x
x=-1,y'=3 x=0,y'=0 x=1,y'=3
17. y = x3 + 3, x=-1,0,1
y'=3x2
x=-1,y'=3 x=0,y'=0 x=1,y'=3
2
18. y = x + 2x + 1, x=-1,0
y'=2x+2
x=-1,y'=0
x=0,y'=2
19. y = (4-x)2, x=3,4
y'=-8+2x
x=3,y'=-2
x=4,y'=0
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
y'=2, there is no value of x for which y'=0
21. y = 5 - 2x
y'=-2, there is no value of x for which y'=0
22. y = -2 + 6x
y'=6, there is no value of x for which y'=0
23. y = x2
y'=2x, y'=0 at x=0, this is a minimum
2
24. y = 1 - x
25. y = 1 - 2x2
y'=-4x, y'=0 at x=0, this is a maximum
2
26. y = 5 + x
y'=2x, y'=0 at x=0, this is a minimum
27. y = x2 - 2
y'=2x, y'=0 at x=0, this is a minimum
28. y = x2 - 3
y'=2x, y'=0 at x=0, this is a minimum
2
29. y = x + 2
y'=2x, y'=0 at x=0, this is a minimum
30. y = 4x - x2 + 3 y'=4-2x, y'=0 at x=2, this is a maximum
31. y = 4x - x2
y'=4-2x, y'=0 at x=2, this is a maximum
32. y = 4x - x2 - 4
y'=4-2x, y'=0 at x=2, this is a maximum
2
33. y = x - 4x
y'=2x-4, y'=0 at x=2, this is a minimum
34. y = x2 - 4x + 4
35. y = x3
y'=3x2, y'=0 at x=0, this is neither a max nor a min
36. y = x3 + 3
y'=3x2, y'=0 at x=0, this is neither a max nor a min
2
37. y = x + 2x + 1 y'=2x+2, y'=0 at x=-1, this is a minimum
38. y = (4-x)2
y'=-8+2x, y'=0 at x=4, this is a minimum
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
39.
y
 5  7x
w
y
 6  7w
x
40.
y
 10w  7x2
w
y
 18x2  14x w
x
41.
42.
y
1

w x
y  w
 2
x
x
43.
y
12x3
7
 10w 
 3
3
w
w
x
y 18x2 21w

 4
x
w2
x
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?
The profit maximizing quantity doesn’t change if the fixed costs change.
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?
AC(q) = (100+q2)/q = 100/q + q2/q = 100q-1 + q
dAC/dq = -1*100*q-2 + 1 = 0
-100/q2 = -1
q2 = 100
q=10
48. If Bob's fixed costs are 400, at what quantity will average cost be minimized?
AC(q) = (400+q2)/q = 400/q + q2/q = 400q-1 + q
dAC/dq = -1*400*q-2 + 1 = 0
-400/q2 = -1
q2 = 400
q=20
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?
Profit = price*q – TC(q)
Profit = (100-q)*q – 20 – 2q – q2
dP/dq = 100 – 2q – 2 – 2q = 0
98=4q
98/4 = 24.5 = q*
Pp= 100 – 24.5 = 75.5.
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?
A change in fixed costs doesn’t change the profit maximizing price and quantity.
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.
Profit = price*q – TC(q)
Profit = (120-q/2)*q – 20 – 2q – q2
dP/dq = 120 – q – 2 – 2q = 0
118=3q
118/3 = 39.333 = q*
Their profit maximizing quantity increases when demand increases.
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
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
For all these questions, the two equations are:
MUA
2B 10 PA



MUB
A
20 PB
or
2B 1

or A  4B
A
2
10A  20B  600
Substituting the first into the second yields:
10(4B)  20B  600
40B  20B  600
60B  600
B  10, A  40
58. How do the answers change as the specification of the utility function changes?
They don't.
59. PA=10, PB=20, M=600, U(A,B)=AB
MUA
B 10 PA



MUB
A 20 PB
10A  20B  600
10(2B)  20B  600
20B  20B  600
40B  600
B  15, A  30
or
B 1

or A  2B
A 2
60. PA=10, PB=20, M=600, U(A,B)=A1/2B1/2
1
1
1 2 2
A B
MUA
B
10 PA
 2 1 1 


MUB
A
20 PB
1 2 2
A B
2
10A  20B  600
or
B
1

or A  2B
A
2
10(2B)  20B  600
20B  20B  600
40B  600
B  15, A  30
61. PA=10, PB=20, M=600, U(A,B)=A1/3B1/3
1
1
1 3 3
A B
MUA
B
10 PA
 3 1 1 


MUB
A
20 PB
1 3 3
A B
3
10A  20B  600
or
B
1

or A  2B
A
2
10(2B)  20B  600
20B  20B  600
40B  600
B  15, A  30
62. How do the answers change as the specification of the utility function changes?
Again, they don't change.
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?
MUA

MUB
10 
3
A
4
1 1
4 B4
3 3
A 4B 4
1
4
10A  PBB  720
10 

3B 10 PA


A
PB
PB
or
3BPB
3B 10

or A 
A
PB
10
Substituting yie lds
3BPB
10
 PBB  720
10
3BPB  PBB  720
PB  5
PB  10
3B  5  5B  720
15B  5B  720
20B  720
3  36  5
B  36, A 
 54
10
3B  10  10B  720
30B  10B  720
40B  720
3  18  10
B  18, A 
 54
10
PB  20
PB  40
3B  20  20B  720
60B  20B  720
80B  720
3  9  20
B  9, A 
 54
10
3B  40  40B  720
120B  40B  720
160B  720
3  4.5  40
B  4.5, A 
 54
10
Price
Demand for Good B
45
40
35
30
25
20
15
10
5
0
P
0
10
20
Quantity
30
40
The demand curve is not linear.
Because the total amount spent on good B ($180) does not change when the price changes, the
price elasticity of demand is -1. This may be a good model over some range of prices for good
B, but it is almost certain that if the price rose enough people would simply stop buying it.