Exam 2 Review
Zarestky’s Math 142, Spring 2011
Sections 3.7, 4.1-4.4, 4.7, 5.1-5.2, 5.4
These problems are meant to be a highlight of the material covered on the exam and not a comprehensive
review! Every type of problem will not be on this review, and you should work lots of other problems
from the text and the Week in Review.
1. Given the profit function P( x ) = x (1000 ! x ) !5000 , where P is in dollars and x is the number of
items produced and sold:
A. Where is profit increasing and decreasing?
B. What is the total profit from producing 576 items?
C. What is the rate of change of profit at 576 items?
D. Use marginal profit to estimate the profit from producing 577 items.
E. What is the average profit per item when 576 items are produced?
F. Use marginal average profit to estimate average profit per item when 577 items are produced.
"$ 4 2
%
$$ x ! +10''''
x
&
2. Write the function y = 5$#
(
as a composition of functions y = g (u) and u = h( x ) .
)
Then find the derivative of g h( x ) .
3. Compute the following derivatives.
d
A.
(5x 2 +1) x 6 !10x
dx
d "$$ x 2 +1 %''
B.
'
$
dx $$# log8 (!x )''&
(
)
C.
d "$ (
$4
dx $$#
D.
3
*
d' !
))ln #(1+ 2x ) 4 x 5 + 6 $& ,,
dx ( "
% ,+
(
50!x 36
5
) %''
) ''&
4. Find the value(s) of x where the function g ( x ) = x 3e x has a horizontal tangent line.
5. Use the graph of y = f (x) to identify the following:
A. The intervals where f ! is positive and negative
B. The intervals where f ! is increasing and decreasing
C. The x-coordinate(s) where f has inflection point(s)
6. A rational function h(x) has domain x ≠ ±3 and h!( x ) =
32x
( x "9)
2
2
.
A. Find the intervals where h is increasing and decreasing. Identify any local extrema.
B. Find h′′(x) and simplify the numerator.
C. Find the intervals where h is concave up and down. Identify any inflection points.
D. Sketch a possible graph of h(x).
7. Assume y = f (x) is continuous on (−∞, ∞) and use the information below to sketch a possible graph of f.
f (−3) = 0, f (−1) = 0, f (0) = −2, f (4) = 0;
f ! (−2) = 0, f ! (0) DNE, and f ! (4) = 0;
f !! (−5) = 0, f !! (0) DNE, and f !! (4) = 0;
8. Let the price-demand equation be x =
10
!2 .
p
A. Find the elasticity function E(p).
B. Classify the elasticity at the price p = 9.
C. Should price be raised or lowered to increase revenue?
D. If price is lowered by 3%, what is the corresponding change in demand?
9.
Assuming that f is continuous on its domain and that all intercepts are included in the table of
values, use the given information to sketch a possible graph of f.
Domain: All real x, except x = −1, 4, where there are vertical asymptotes;
lim f ( x ) = "3
x!"#