Review Problems
Midterm 2
March 4, 2008
1. Problem 1 - Find the indicated derivative of the following functions.
(a) Find
dy
dx
if
y
Answer:y 0 =
(b) If
dy
dx
=
2x
p (32 x
x
Answer:y 000 =
(c) Find
= p3 3xx
d2 y
dt2
1
1
1)4=3
4, nd
(x2
4
4)3=2
if y = x5
.
d3 y
dx3 .
3x2 + p1x .
Answer:y 00 = 20x3 6 + 13 x 3=2
2
3
(d) Find dy
6x2 + 9)1=4 .
dt if y = 3 (x
Answer:y 0 = 12 (x3
6x2 + 9)
3=4
(x2
4x)
2. Problem 2 - The function and its rst and second derivatives are given.
Use these to nd critical points, relative maxima, relative minima, and
point of inection. Then sketch the graph of the function.
1
y
y
y
4 =3
=
(x 7)
7x (x 4)
3
28(x 1)
9x2=3
x
1=3
0 =
00 =
Answer: critical point: (0; 0) and (4; 24); relative max: (0; 0); relative
min: (4; 24); POI: (1; 6)
3. Problem 3 - Prot
A manufaturer estimates that its product can be produced at a total cost
of C (x) = 45; 000 + 100x + x3 dollars. If the manufacturer's total revenue
from the sale of x units is R(x) = 4900x dollars, determine the level of
production x that will maximize the prot. Find the maximum prot.
(Hint: First nd the prot function P(x) = R(x) -C(x). Then nd x to
maximize prot. Then nd the maximum prot.)
Answer:
x
= 40 maximizes the prot; P (40) = $83; 000;
4. Problem 4 - Problem 37 page 719
2
Answer: (a) t = 8 hrs; (b) t = 4 hrs.
5. Problem 5 - An agency charges $10 per person for a trip to a concert if
30 people travel in a group. But for each person above the 30, the charge
will be reduced by $0.20. How many people will maximize the total revenue for the agency if the trip is limited to at most 50 people.
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