AP Calculus AB
Assignment 4.1
Unit 4 – Applications of Derivatives
For 1-3 find the critical numbers of f. Then use the critical number test to determine the absolute extreme
values of f.
1.) f  x   2x 3  3x2  12x  7 on 0,4
2.)
f  x   x2 
16
on 1,3
x
3.)
f  x   2x
 5x
5
3
4
3
on 0,27
For 4-5, find the critical numbers of each function.
x2
4.) f  x  
5.) f  x   x 2  3 2x  5
5x  4
1
Assignment 4.2
Determine if f satisfies the conditions of the mean value theorem for derivatives. If so, find all possible
values of c.
1
1.) f  x   5x2  3x  1 on [1, 3]
2.) f  x  
on [0, 2]
2
 x  1
4
on [1, 4]
x
3.)
f x  x
5.)
f  x   x3  2x2  x  3 on [-1, 1]
6.) f  x   x 3  1 on [-2, 4]
7.)
 
f  x   sin x on  0, 
 2
8.) f  x   0.2x2  sin x on [-1, 2]
2
3
on [-8, 8]
4.) f  x   x 
Hint:Use your calculator
2
Assignment 4.3
For 1-8, find the intervals over which f is increasing and decreasing. Give the values of x for which f has local
extreme values.
2
1.) f  x   2x 3  x2  20x  1
2.) f  x   10 x 3  x  1
3.)
f x  x
5.)
f  x   x  x2  9
4
3
 4x
1
3
4.) f  x   x 2  3 x 2  4
6.) f  x   x 2e
3
1
x2
7.)
1
f  x   x  sin x on 0,2 
2
8.) f  x   2cos x  sin2x on 0,2 
For 9-12, find the local extreme values.
9.)
f  x    x  2   x  1
3
4
1 
  
11.) f  x   sec  x  on   , 
 2 2
2 
10.) f  x  
x 3
x2
  
12.) f  x   2tan x  tan2 x on   , 
 3 3
4
Assignment 4.4
In 1 and 2, find the local extreme values using the 2nd derivative test.
1.)
f  x   3x 4  4 x3  6
2.) f  x    x 2  1
2
In 3-5, for each function, determine the intervals for which f is increasing and decreasing, local extreme
values, intervals for which f is concave up and concave down, inflection points and provide a sketch of the
graph.
3.)
f  x   8x
4.) f  x   x
1
f x  x
2
5.)
1
3
x
4
3
5
1
3
 3x  10 
5
4.1-4.4 Review
In 1 and 2, find the critical numbers of each function.
1.)
f  x   3 x2  x  2
f x 
2.)
x2
5x  4
1
3.) Use the critical number test to find the extema on [-1, 3] for f where f  x   x 3  2 x 2  5x  1 .
3
4.) Let f be the function given by y 
3
theorem on  ,
2
x 1
. Find all vaues of c that satisfy the conclusion of the mean value
x 1

5 .

5.) If f '  x   x  x  3  x  5  , at which values of x is f  x  a relative minimum value?
4
7
6
6.) Find the x-coordinates of all inflection points of f if f  x  
x8 9x6
.

56 30
7.) Given the function f defined by f  x   cos x  cos2 x for   x   .
a.) Find the x-intercepts of the graph of f.
b.) Find the x- and y-coordinates of all relative maximum points of f. Justify your answer.
c.) Find the intervals on which the graph of f is increasing.
d.) Using the information in parts a, b and c, sketch the graph of f.
7
8.) Let f be a function that is even and continuous on the closed interval [-3, 3]. The function f and its
derivatives have the properties indicated in the table below.
x
f x
0
0  x 1
1
Positive
1 x 2
0 Negative
1
2
2 x 3
-1
Negative
f ' x 
Undefined Negative 0 Negative Undefined
f ''  x 
Undefined
Positive
Positive
0 Negative Undefined Negative
a.) Find the x-coordinate of each point at which f attains an absolute maximum value or an absolute
minimum value. For each x-coordinate you give, state whether f attains an absolute maximum or an
absolute minimum.
b.) Find the x-coordinate of each point of inflection on the graph of f. Justify your answer.
c.) Sketch the graph of a function with all the given characteristics of f.
9.) The figure below shows the graph of f ' , the derivative of a function f. the domain of the function f is the
set of all x such that 3  x  3 .
-3 -2 -1
1
2 3
a.) For what values of x, 3  x  3 , does f have a relative maximum? A relative minimum? Justify your
answer.
b.) For what values of x is the graph of f concave up? Justify your answer.
c.) Use the information found in parts a and b and the fact that f  3  0 to sketch a possible graph of f.
8
6x
3x  5
a.) Find the equations of the horizontal and vertical asymptotes.
10.) g  x  
b.) Over what interval is g concave up?
c.) Sketch the graph of g.
11.) If y  2x 3  4ax 2  bx  3 has an inflection point at (2, -3), find the values of a and b.
12.) Let f be the function with derivative given by f '  x   2cos  x 3  on the interval 1.9  x  2.2 .
(Use your calculator.)
a.) How many points of inflection does f have on this interval?
b.) How many relative maxima does f have on this interval?
c.) To the nearest thousandth, find the largest value of x in the interval for which f has a local minimum.
d.) How many critical points does f have on the interval?
9
13.) The function f has a continuous 2nd derivative for all x. f  0  2 , f '  0   3 and f ''  0   0 . g is a
function whose derivative is g '  x   e2 x  3 f  x   2 f '  x   for all x. Show that
g '' x   e2 x  6 f  x   f ' x   2 f ''  x   and use the 2nd derivative test to determine if g has a local
maximum at x  0 .
Multiple Choice

3
x
14.) If f  x   sin   , then there exists a number c in the interval  x 
that satisfies the conclusion of
2
2
2
the Mean Value Theorem. Which of the following could be c?
(A)
2
3
(B)
3
4
(C)
5
6
(D) 
(E)
3
2
15.) Let f be a polynomial function with degree greater than 2. If a  b and f  a   f  b   1 , which of the
following must be true for at least one value of x between a and b?
I. f  x   0
II. f '  x   0
III. f ''  x   0
(A) None
(B) I only
(C) II only
10
(D) I and II only
(E) I, II and III
16.) The graph of the twice differentiable function f is shown. Which of the following is true?
3
(A) f '  3  f '' 3  f  3
(B) f  3  f ''  3  f ' 3
(D) f '  3  f  3  f ''  3
(E) f  3  f '  3  f '' 3
17.) The derivative of f  x  
(A) -1
(C) f ''  3  f  3  f '  3
x4 x5
attains its maximum value at x  ?

3 5
(B) 0
(C) 1
(D)
4
3
(E)
5
3
18.) Given the function defined by f  x   3x 5  20x 3 , find all values of x for which the graph of f is concave up.
(A) x  0
(D) x  2
(B)  2  x  0 or x  2
(E) 2  x  2
(C) 2  x  0 or x  2
19.) For what value of x does the function f  x    x  2  x  3 have a relative maximum?
2
(A) -3
(B) 
7
3
(C) 
11
5
2
(D)
7
3
(E)
5
2
20.) At what value of x does the graph of y 
(A) 0
(B) 1
1 1
have a point of inflection?

x2 x3
(C) 2
(D) 3
(E) At no value of x
21.) Let f and g be odd functions. If p, r, and s are nonzero functions defined as follows, which must be odd?
I. p  x   f  g  x  
II. r  x   f  x   g  x 
III. s  x   f  x  g  x 
(A) I only
(B) II only
(C) I and II only
(D) II and III only
(E) I, II and III
22.) Let f be a function given by f  x   cos 2x   ln 3x  . What is the least value of x at which the graph of f
changes concavity?
(A) 0.56
(B) 0.93
(C) 1.18
(D) 2.38
(E) 2.44
23.) Let f be a continuous function on the closed interval [1, 5]. Let f 1  3 and f  5  9 . Which of the
following is not necessarily true?
(A) There exists a number h on [1, 5] such that f  h  f  x  for all x in [1, 5].
(B) For all a and b in [1, 5], if a  b , then f  a   f  b .
(C) There exists a number h on (1, 5) such that f  h  3 .
(D) For all h in the open interval (1, 5), lim f  x   f  h  .
x h
(E) There exists a value h on the open interval (1, 5) such that f '  h  3 .
12
24.) For all x in the closed interval [3, 6], the function f has a positive first derivative and a negative second
derivative. Which of the following could be a table of values for f?
(A) x
3
4
5
6
f x
(B) x
3
4
5
6
16
18
21
25
f x
25
21
18
16
(C) x
3
4
5
6
f x
13
17
21
25
(D) x
3
4
5
6
f x
16
14
11
7
(E) x
3
4
5
6
f x
7
11
14
16
25.) The function f and its derivative f ' are strictly decreasing on [0, 4]. Which of the following could be a
value for f '  3 ?
x
f x
(A) -10
2.6 2.9 3.0 3.1
42.6 41.1 40 38.8
(B) -10.5
(C) -11
(D) -11.5
(E) -12
26.) The function f is twice differentiable and the graph of f has no inflection points. If f  6  3 , f '  6   
and f ''  6   2 , which could be a value for f  7  ?
(A) 2
(B) 2.5
(C) 2.9
13
(D) 3
(E) 4
1
2