Review Problems
Midterm 2
December 8, 2007
1. Problem 1 - Find the indicated derivative of the following functions.
(a) Find
dy
dx
if
y
Answer:y 0 =
(b) If
dy
dx
=
= p3 3xx
2x
p (32 x
x
1
1
1)4=3
4, nd
.
d3 y
dx3 .
4
Answer:y 000 = (x2 4)
3= 2
t2 ).
if
y = ln( p
(c) Find dy
dt
5 2t
Answer:y 0 = 2t (5 12t)
3t
2
(d) Find dy
dt if y = [ln( t 1 )] .
Answer:y 0 =
t(t
3t
2
ln
1)
(t 1)
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
y
y
x
=
2
(x
3)2
6x
(x 3)3
12x + 18
(x 3)4
0 =
00 =
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
( )=
C x
40; 000
x
+ 600 + 8x
nd the maximum prot.
Answer: Prot P (x) = 6400x
the prot; P (64) = 208; 490:67;
18x2
2
x3
3
40; 000;
x
= 64 maximizes
4. Problem 4 - Problem 21 page 740
Answer: Dimension of the box: length = 8, width = 4, height = 8.
5. Problem 5 - Problem 9 page 729
Answer: 40 people will maximize the revenue.
6. Problem 6 - Use second derivative test to nd the relative maxima,
relative minima, and point of inection, and sketch the graph of
y
= x4
8x3 + 16x2
Answer: relative min: (0; 0) and (4; 0); relative max: (2; 16)); POI: (2
p23 ; 7:111) and (2 + p23 ; 7:111)
3