Fall 2014 © Drost
MATH 142
pg 1
Exam 2 REVIEW
Fall 2014, Mrs. Drost
Section 4.1 Derivatives
1) Find the derivative of f ( x)   2 x 2 
2) Find the derivative of f  x  
3) Find
1
x  3  2 ln x .
2
x 3/ x
.
x
dy
for y   x  ln x 2
dx
4) Find the equation of the tangent to f  x   x 2  e x  1 at x  0 .
5) Estimate the cost of the 35th item where the cost function is C  x   x  x  12   50 .
Section 4.2
Products and Quotients
6) Find f ' ( x) for f  x    x3  3ln x  2e x  3x 
(3x  4) 2
. Do not simplify.
x 3  x 2  3x
7) Find f ' ( x) for f ( x ) 
1
, where x is the
1  x2
number of items sold and p is the price in dollars. Find the marginal revenue.
8) Suppose the demand equation for a product is given by p 
9) Find
d
 f  g  h  if f , g , and h are differentiable.
dx
10) Given the cost function: C ( x)  2 x3  4 x 2  10 x  50 ; find the marginal average cost.
Section 4.3
Chain Rule
 8x

11) Find:
d
dx
12) Find:
d
15 5 2 x 3  3
dx
13) Find:
d 5x
e  [ln x]2 

dx
8
7
 10


Fall 2014 © Drost
MATH 142
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14) The volume, V, in board feet of certain trees is given by V  10  .007  d  5  for
3
d  10 , where d is the diameter in inches. Find the rate of change of V with respect to
d when d  12.
15) Find
ex  1
dy
where y 
3
dx
 2 x  5
Section 4.4 Derivatives of Exponentials and Logarithmic Functions
d 3x
16)
3 
dx
17)

d x2
e ln x
dx

18) Find the derivative of y   ln  5 x  
3
19) Suppose the price and demand are related by p ( x)  e 4 x . Find where the marginal
revenue is zero.
 (2 x  5) 2  e  x 
20) Find the derivative of y  ln 

42 x 1


Section 4.5 Elasticity of Demand
21) Given: x  4 p  20 , find the elasticity of demand at $3 , and state whether the
demand is elastic, inelastic, or unit elastic.
22) Given the price elasticity of air transportation is 1.10 , if the price is decreased by
10% , what will be the approximate change in demand?
23) Given x  30  10 p , find the price which maximizes revenue, and find the maximum
revenue.
24) The demand function for beef consumption per capita in pounds, x , and p is the
price of beef divided by disposable income per capita, then x  126.5  1800 p . Find the
elasticity of demand at p  0.05 .
25) A restaurant owner sells 100 dinner specials for $10 each. After raising the price to
$11 , she noticed that only 90 specials were sold. What is the elasticity of demand, and
what price maximizes revenue?
Section 5.1 The 1st Derivative
Fall 2014 © Drost
MATH 142
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26) Find the critical values and determine the intervals where f(x) is increasing and f(x)
3 2
is decreasing if f ( x )  1   2 .
x x
27) f   x   2  x  1  x  1  x  2  , f  1  2, f 1  3, f  2   0 Sketch the
graph.
2
3
Find the domain, critical values, intervals where the function is increasing/decreasing,
and all relative extrema.
28)
f  x   4 x 6  6 x 4  5
29)
f  x   e x  e x
30)
f  x   2 x  ln 2 x  4
Section 5.2 The 2nd Derivative
31) Find all inflection points for f ( x )  x 4  10 x 3  24 x 2  3x  5 .
32) Find the second derivative of f  x  
1
 x
x
33) Find the inflection points for f   x   12 x3  48 x 2 .
34) Find the intervals where f ( x) is concave up or down and any inflection points.
given f   x   x  x  2   x  3 . The domain is all reals.
2
35) Find the intervals where f ( x) is concave up or down and any inflection points.
given f  x    x  4  / x . The domain is all reals, except x  0 .
2
Section 5.3 Limits at Infinity
5 x 2  3x  1
x  6 x 2  x  7
36) Find: lim
37) Find: lim 4 
x 
1
x
Fall 2014 © Drost
MATH 142
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38) Find: lim  5  2e x 
x  
39) Find: lim
x 
x 1
x5
x4  x2  x  1
x  
x3  1
40) Find: lim
Section 5.4 Additional Curve Sketching
3 x  12
.
x  3x  4
2 x 2  5x  3
42) Sketch the graph of f ( x ) 
. Include a sketch of all asymptotes.
x2  9
41) Find vertical asymptotes for f ( x) 
43) Sketch the graph of f ( x) 
2
x
.
x 1
2
44) Sketch the graph of f ( x)  x 
25
.
x
45) Sketch a graph of the function:
f   0 and f   x   0 on  , 0 
f   x   0 and f   x   0 on  0,  
f  0   0 , lim
x  
f  x   1 , lim f ( x)  4
x 
Section 5.5 Absolute Extrema
Locate the value(s) where each function has an absolute maximum, an absolute
minimum, if they exist, over the given intervals.
46)
f  x   x3  3x 2  2 on  0, 4
47)
f  x   4  2 x  x 2 on  ,  
48)
f  x   x 2  1 on  3, 4
49) Find two numbers whose sum is 42, and for which the sum of their squares is a
minimum.
Fall 2014 © Drost
MATH 142
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50) Find two non-negative numbers x and y with x  y  60 for which the term x 2 y is
maximized.
Section 5.6 Optimization
51) A trailer rental agency rents 10 trailers per day at a rate of $30 per day. For each $5
increase in rate, one less trailer is rented. At what rate should the trailers be rented to
produce the maximum income? How many trailers will be rented?
52) A rancher wants to build a fence to enclose a rectangular area of 1,800 m 2 for cattle.
The fence a long one side is to be made of heavy-duty material that costs $15 per meter.
The material along the remaining three sides costs only $5 per meter. Find the
dimensions and total cost of fencing for the field that is least expensive to fence.
53) A company is planning to manufacture a new blender. After conducting extensive
market surveys, the research department estimates a weekly demand of 600 blenders at a
price of $50 per blender and a weekly demand of 800 blenders at a price of $40 per
blender. Assuming the demand equation is linear, use the research department's estimates
to find the price that maximizes revenue.
54) A box is to be made out of a piece of cardboard that measures 12 inches by 12
inches. Squares of equal size will be cut out of each corner and then the ends and sides
will be folded up to form a rectangular box. What size square should be cut from each
corner to obtain a maximum volume?
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