MP CHAPTER 12 SOLUTIONS
MP SECTION 12.1
3h + h
2
1. lim
 =
lim
(3 + h) = 3
h
→
0
h
h
→
0
2.
25x
0
≤
x
≤
100
f(x) =
2500 + 20(x  100) 100
≤
x
≤
200
25(100) + 20(100) + 15(x  200)
x
≥
200
This function is continuous for all nonnegative x. However,
f(x) has no derivative at x = 100 and x = 200 (the slope of f(x)
abruptly changes at these points).
3a.
x(e
x
) + e
x
(x
2
+ 1)(2x) x
2
(2x)
3b.

(x
2
+ 1)
2
3c. 3e
3x
3d. 6/(3x + 2)
3
3e. 3x
2
/x
3
= 3/x
4.
∂
f/
∂
x
1
= 2x
1
exp(x
2
)
∂
f/
∂
x
2
= x
1
2
exp(x
2
)
∂
f
2
/
∂
x
1
∂
x
2
=
∂
f
2
/
∂
x
2
∂
x
1
= 2x
1
exp(x
2
) =
∂
2
f/
∂
2
x
1
= 2exp(x
2
),
∂
2
f/
∂
2
x
2
= x
1
2
exp(x
2
)
5a. f'(p)<0 if a price increase lowers demand. Thus we expect to
find f'(p)<0.
5b. Let r(p) = pf(p). If r'(p)<0, then price decrease increases
revenue. Now
r'(p) = pf'(p) +f(p)<0 if
dq
p 
< q
or
dp
p
dq

 < 1 or E<1.
q
dp
5c.
If 1<E<0, then r'(p)>0, so a price cut will decrease
revenue.
6a. lim
k( 1  e
cx
) = k
x
→∞
1
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View Full Document6b. The maximum size of the market as measured in terms of sales
per year.
6c. If f(x) = k(1 e
cx
) is sales response from $x of advertising,
then f'(x) is (approximately) the sales response due to increasing
advertising from x to x + 1. Note that
f'(x) = kce
cx
= c times (part of market not buying product).
7a. Cost of producing x'th unit is c'(x) = k(1  b)x
b
which is a
decreasing function of x.
7b. If total amount produced is x, then production cost per unit
is k(1  b)x
b
. If total amount produced is 2x, then production
cost per unit is k(1  b)(2x)
b
. Thus doubling the amount
produced reduces cost per unit to 100(2
b
)% of what it was
previously.
.
8. Let total output = f(m,w) = 3m
1/3
w
2/3
. Then
∂
f/
∂
w
= 2m
1/3
w
1/3
,
∂
f/
∂
m
= m
2/3
w
2/3
.
Thus
∂
f/
∂
m(216, 1000) = 1/36(100) = 100/36
∂
f/
∂
w(216, 1000) = 2(6)(1/10) = 1.2
One extra hour of machine time increases output by
approximately 1(100/36) = 100/36 units while two hours of labor
increases output by approximately 2(1.2) = 2.4 units. Thus one
hour of machine time is a better buy than two hours of labor.
MP SECTION 12.2
1a. Let
S = soap opera ads and F = football ads. Then we wish to
min z = 50S + 100F
st
5S
1/2
+ 17F
1/2
≥
40 (men)
20S
1/2
+ 7F
1/2
≥
60 (women)
S
≥
0, F
≥
0
1b. Since doubling
S does not double the contribution of S to
each constraint,we are violating the proportionality assumption.
Additivity is not violated.
1c. This accounts for the fact that an extra soap opera ad yields
a benefit which is a decreasing function of the number of football
ads. This accounts for the fact that we may not want to double
count people who see both types of ads.
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 Fall '07
 HOCHBAUM

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