Review Problems
Midterm 3
April 7, 2008
1. Problem 1 - Find the derivative of the following functions.
(a) g (x) = 3log5 (4x + 7).
Answer: y 0 =
(b) y =
3
ex +1
x
12
(ln5)(4x+7)
.
Answer: y 0 =
(c) y = x2 5x
ex
3 +1 (3x3
x2
1)
Answer: y 0 = 5x (2x + x2 ln5)
(d)Find dy=dx in the equation log2 y
Answer:
dy=dx
x
=5
= yln2
2. Problem 2 - The function and its rst and second derivatives are given.
Use these to nd any horizontal and vertical asymptotes, critical points,
relative maxima, relative minima, and point of inection. Then sketch the
graph of the function.
1
y
x
=
2
(x
3)2
6x
(x 3)3
00 = 12x + 18
y
(x 3)4
y
0 =
Answer: VA:x = 3; HA: y = 1; critical point: (0; 0); relative min: (0; 0);
POI: ( 3=2; 1=9)
3. Problem 3 - Prot
Suppose that in a monopolistic market, the demand function for a commodity is
p
= 7000
x
10x
2
3
where x is the number of units and p is the price (in dollars). If a company's average cost function for this commodity is
40; 000
+ 600 + 8x
C (x) =
x
nd the maximum prot. (Hint: Prot = Revenue - Cost. First, nd
Revenue = price * number of units. Second, nd Cost = average cost *
number of units. Third, nd the prot function. Then nd value of x that
maximize the prot.)
Answer: Prot P (x) = 6400x
18x2
2
x3
3
40; 000;
x
= 64 maximizes
the prot; P (64) = 208; 490:67;
4. Problem 4-Problem 33, page 778
Answer: (a)y 0 (1) = 0; (b)y = e
1
5. Problem 5 - Implicit Dierentiation
Problem 45 page 790
Answer:
y
=3
x
6. Problem 6 - Suppose that the demand for a product is given by pq + p =
100.
(a) Find the elasticity when p = $10, q = 9.
(b) Tell what type of elasticity this is : unitary, elastic, or inelastic.
(c) How would revenue be aected by a price decrease?
Answer: (a) 10=9; (b) elastic; (c) revenue will increase .
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7. Problem 7 - Problem 36 page 808
Answer: t = $880, T = $3520
8. Problem
R
p8 - Evaluate the following integrals
(a) (4 + x x12 ) dx
3= 2
Answer:
4x + 2x3 + x1 + C
p
R
(b) 7x3 x4 + 6 dx
Answer: 76 (x4 + 6)3=2 + C
9. Problem 9 - Revenue
Suppose that the marginal cost for a product is given by
MC
p
= 60
x
+1
The cost for produce 24 units is $5500. (Hint: this information helps you
to nd the value of the constant after taking the interal). Find the cost
when level of production, x, equals to 99.
Answer:
C
(99) = $40; 500
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