Name: ________________________ Class: ___________________ Date: __________
ID: A
Test 3 Review
Short Answer
1. Find the value of the derivative (if it exists) of
f (x )  (x  2)
4 /5
7. A rectangle is bounded by the x- and y-axes and the
(5  x )
(see figure). What length and
graph of y 
2
width should the rectangle have so that its area is a
maximum?
at the indicated extremum.
2. Find the differential dy of the function
y  x 2  3x  2.
8. A Norman window is constructed by adjoining a
semicircle to the top of an ordinary rectangular
window (see figure). Find the dimensions of a
Norman window of maximum area if the total
perimeter is 38 feet.
3. Find the differential dy of the function
y  x cos (7x) .
4. Find the differential dy of the function y  2x 3 / 7 .
5. Find the equation of the tangent line T to the graph
ÊÁ 19 ˆ˜
19
of f (x )  2 at the given point ÁÁÁÁ 2, ˜˜˜˜ .
x
Ë 4 ¯
6. Assume that the amount of money deposited in a
bank is proportional to the square of the interest
rate the bank pays on this money. Furthermore, the
bank can reinvest this money at 36%. Find the
interest rate the bank should pay to maximize
profit. (Use the simple interest formula.)
1
Name: ________________________
ID: A
9. The graph of f is shown below. For which values of
x is f  (x ) zero?
13. Analyze and sketch a graph of the function
x 2  2x  9
f (x ) 
.
x
14. Analyze and sketch a graph of the function
x
f (x ) 
.
x 1
15. Determine the slant asymptote of the graph of
5x 2  9x  5
f (x ) 
.
x1
16. The graph of a function f is is shown below. Sketch
the graph of the derivative f .
10. The graph of f is shown below. For which value of
x is f  (x ) zero?
11. A rectangular page is to contain 144 square inches
of print. The margins on each side are 1 inch. Find
the dimensions of the page such that the least
amount of paper is used.
12. Determine the slant asymptote of the graph of
f (x ) 
x 2  8x  14
.
x6
2
Name: ________________________
ID: A
17. The graph of a function f is is shown below. Sketch
the graph of the derivative f .
21. Find the limit.
lim
x
8x  2
5 x 2  4
22. Find the limit.
ÊÁ
3 ˆ˜
lim ÁÁÁÁ 5 2 ˜˜˜˜
x ¯
xË
23. Find all critical numbers of the function
g (x )  x 4  4x 2 .
24. Find the limit.
lim
x
18. The graph of a function f is is shown below. Sketch
the graph of the derivative f .
6x
64 x 2  5
25. A business has a cost of C  1.5x  500 for
producing x units. The average cost per unit is
C
C  . Find the limit of C as x approaches
x
infinity.
26. Match the function f (x ) 
2sinx
with one of the
x2  1
following graphs.
27. Find the cubic function of the form
f (x )  ax 3  bx 2  cx  d, where a  0 and the
coefficients a,b,c,d are real numbers, which
satisfies the conditions given below.
19. Sketch the graph of the relation xy 2  2 using any
extrema, intercepts, symmetry, and asymptotes.
Relative maximum: ÊÁË 3,0ˆ˜¯
Relative minimum: ÊÁË 5,2 ˆ˜¯
Inflection point: ÊÁË 4,1 ˆ˜¯
x2
x2  1
using any extrema, intercepts, symmetry, and
asymptotes.
20. Sketch the graph of the function f (x ) 
28. Find all relative extrema of the function
f (x )  x 2/3  6 . Use the Second Derivative Test
where applicable.
3
Name: ________________________
ID: A
29. The graph of f is shown. Graph f, f' and f'' on the
same set of coordinate axes.
36. The graph of f is shown in the figure. Sketch a
graph of the derivative of f.
30. Determine the open intervals on which the graph of
y  6x 3  8x 2  6x  5 is concave downward or
concave upward.
37. Find the relative extremum of
f (x )  9x 2  54x  2 by applying the First
Derivative Test.
31. Find the points of inflection and discuss the
concavity of the function f (x )  x x  16 .
38. Find the open interval(s) on which
f (x )  2x 2  12x  8 is increasing or decreasing.
32. Find all points of inflection on the graph of the
1
function f (x )  x 4  2x 3 .
2
39. For the function f (x )  (x  1 ) 3 :
2
(a) Find the critical numbers of f (if any);
(b) Find the open intervals where the function is
increasing or decreasing; and
(c) Apply the First Derivative Test to identify all
relative extrema.
33. Find all relative extrema of the function
f (x )  4x 2  32x  62 . Use the Second Derivative
Test where applicable.
Use a graphing utility to confirm your results.
34. Find the points of inflection and discuss the
concavity of the function f (x )  sin x  cos x on
the interval ÊÁË 0,2 ˆ˜¯ .
35. Determine the open intervals on which the graph of
f (x )  3x 2  7x  3 is concave downward or
concave upward.
4
Name: ________________________
ID: A
45. Determine whether Rolle's Theorem can be applied
˘
È
to f (x )  x 2  10x on the closed interval ÍÍÎ 0,10 ˙˙˚ .
If Rolle's Theorem can be applied, find all values of
c in the open interval ÁÊË 0, 10˜ˆ¯ such that f  (c)  0.
x3
 3x given
4
below to estimate the open intervals on which the
function is increasing or decreasing.
40. Use the graph of the function y 
46. Find the critical number of the function
f (x )  6x 2  108x  7 .
47. Find a function f that has derivative
f  (x )  12x  6 and with graph passing through the
point (5,6).
48. The graph of f is shown in the figure. Sketch a
graph of the derivative of f.
41. Determine whether the Mean Value Theorem
can be applied to the function f (x )  x 2 on the
closed interval [3,9]. If the Mean Value
Theorem can be applied, find all numbers c in
the open interval (3,9) such that
f  (c) 
f (9)  f (3)
.
9  (3)
42. Which of the following functions passes through
the point ÊÁË 0,10 ˆ˜¯ and satisfies f  (x )  12?
49. The height of an object t seconds after it is dropped
from a height of 250 meters is s (t )  4.9t 2  250.
Find the time during the first 8 seconds of fall at
which the instantaneous velocity equals the average
velocity.
43. Sketch the graph of the function
ÏÔÔ
Ô 10x  25 0  x  5
f(x)  ÌÔ
ÔÔ 2
5x8
Óx
and locate the absolute extrema of the function on
˘
È
the interval ÍÍÎ 0, 8 ˙˙˚ .
50. Locate the absolute extrema of the function
f(x)  3x 2  12x  3 on the closed interval [4, 4].
44. Identify the open intervals where the function
f (x )  6x 2  6x  4 is increasing or decreasing.
51. Determine whether Rolle's Theorem can be applied
to the function f (x )  (x  5 ) (x  6 ) (x  7 ) on the
È ˘
closed interval ÍÍÎ 5,7 ˙˙˚ . If Rolle's Theorem can be
applied, find all numbers c in the open interval
ÁÊ 5,7˜ˆ such that f  (c)  0.
Ë ¯
5
ID: A
Test 3 Review
Answer Section
SHORT ANSWER
1. ANS:
f (2) is undefined.
PTS: 1
DIF: Easy
REF: 3.1.5
OBJ: Understand the relationship between the value of the derivative and the extremum of a function
MSC: Skill
NOT: Section 3.1
2. ANS:
(2x  3)dx
PTS: 1
DIF: Medium
REF: 3.9.11
OBJ: Calculate the differential of y for a given function
NOT: Section 3.9
3. ANS:
cos (7x)  7x sin (7x)dx
PTS: 1
DIF: Medium
REF: 3.9.18
OBJ: Calculate the differential of y for a given function
NOT: Section 3.9
4. ANS:
6 4 / 7
x
dx
7
PTS: 1
DIF: Medium
REF: 3.9.12
OBJ: Calculate the differential of y for a given function
NOT: Section 3.9
5. ANS:
19x 57
y

4
4
MSC: Skill
MSC: Skill
MSC: Skill
PTS: 1
DIF: Easy
REF: 3.9.2
OBJ: Write an equation of a line tangent to the graph of a function at a specified point
MSC: Skill
NOT: Section 3.9
6. ANS:
24 %
PTS: 1
DIF: Difficult
REF: 3.7.58
OBJ: Apply calculus techniques to solve a minimum/maximum problem involving interest rates
MSC: Application NOT: Section 3.7
1
ID: A
7. ANS:
x  2.5; y  1.25
PTS: 1
DIF: Difficult
REF: 3.7.26
OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a rectangle
bounded beneath a line
MSC: Application NOT: Section 3.7
8. ANS:
76
38
x
feet; y 
feet
4
4
PTS: 1
DIF: Medium
REF: 3.7.25
OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the area of a Norman window
MSC: Application NOT: Section 3.7
9. ANS:
x  0; x  4
PTS: 1
DIF: Easy
REF: 3.6.71a
OBJ: Identify properties of the derivative of a function given the graph of the function
MSC: Skill
NOT: Section 3.6
10. ANS:
x  2
PTS: 1
DIF: Easy
REF: 3.6.71b
OBJ: Identify properties of the second derivative of a function given the graph of the function
MSC: Skill
NOT: Section 3.6
11. ANS:
14,14
PTS: 1
DIF: Medium
REF: 3.7.17
OBJ: Apply calculus techniques to solve a minimum/maximum problem involving the print area on a page
MSC: Application NOT: Section 3.7
12. ANS:
y  x2
PTS: 1
DIF: Medium
REF: 3.6.15
OBJ: Identify the slant asymptote of the graph of a function
NOT: Section 3.6
13. ANS:
MSC: Skill
none of the above
PTS: 1
DIF: Medium
REF: 3.6.15
OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes
MSC: Skill
NOT: Section 3.6
2
ID: A
14. ANS:
PTS: 1
DIF: Medium
REF: 3.6.14
OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes
MSC: Skill
NOT: Section 3.6
15. ANS:
y  5x  4
PTS: 1
DIF: Medium
REF: 3.6.16
OBJ: Identify the slant asymptote of the graph of a function
NOT: Section 3.6
16. ANS:
MSC: Skill
PTS: 1
DIF: Medium
REF: 3.6.4
OBJ: Graph a function's derivative given the graph of the function
MSC: Skill
NOT: Section 3.6
3
ID: A
17. ANS:
PTS: 1
DIF: Medium
REF: 3.6.2
OBJ: Graph a function's derivative given the graph of the function
MSC: Skill
NOT: Section 3.6
18. ANS:
PTS: 1
DIF: Medium
REF: 3.6.3
OBJ: Graph a function's derivative given the graph of the function
MSC: Skill
NOT: Section 3.6
4
ID: A
19. ANS:
PTS: 1
DIF: Medium
REF: 3.5.67
OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes
MSC: Skill
NOT: Section 3.5
20. ANS:
PTS: 1
DIF: Medium
REF: 3.5.64
OBJ: Graph a function using extrema, intercepts, symmetry, and asymptotes
MSC: Skill
NOT: Section 3.5
21. ANS:
0
PTS: 1
DIF: Medium
REF: 3.5.23
OBJ: Evaluate the limit of a function at infinity
NOT: Section 3.5
5
MSC: Skill
ID: A
22. ANS:
5
PTS: 1
DIF: Medium
REF: 3.5.19
OBJ: Evaluate the limit of a function at infinity
NOT: Section 3.5
23. ANS:
critical numbers: x  0, x  2 , x   2
PTS: 1
DIF: Easy
REF: 3.1.12
OBJ: Identify the critical numbers of a function
NOT: Section 3.1
24. ANS:
3

4
PTS: 1
DIF: Medium
REF: 3.5.28
OBJ: Evaluate the limit of a function at infinity
NOT: Section 3.5
25. ANS:
1.5
PTS: 1
MSC: Application
26. ANS:
DIF: Easy
NOT: Section 3.5
REF: 3.5.88
PTS: 1
DIF: Medium
REF: 3.5.5
OBJ: Identify the graph that matches the given function
NOT: Section 3.5
6
MSC: Skill
MSC: Skill
MSC: Skill
OBJ: Evaluate limits at infinity in applications
MSC: Skill
ID: A
27. ANS:
f (x ) 
1 3
45
x  6x 2 
x  27
2
2
PTS: 1
DIF: Medium
REF: 3.4.73
OBJ: Construct a function that has the specified relative extrema and inflection points
MSC: Skill
NOT: Section 3.4
28. ANS:
relative minimum: ÁÊË 0,6 ˜ˆ¯
PTS: 1
DIF: Medium
REF: 3.4.47
OBJ: Identify all relative extrema for a function using the Second Derivative Test
MSC: Skill
NOT: Section 3.4
29. ANS:
PTS: 1
DIF: Medium
REF: 3.4.62
OBJ: Graph a function's derivative and second derivative given the graph of the function
MSC: Skill
NOT: Section 3.4
30. ANS:
ÊÁ
ÊÁ 4 ˆ˜
4 ˆ˜
concave upward on ÁÁÁÁ , ˜˜˜˜ ; concave downward on ÁÁÁÁ , ˜˜˜˜
9¯
Ë
Ë 9 ¯
PTS: 1
DIF: Medium
REF: 3.4.9
OBJ: Identify the intervals on which a function is concave up or concave down
MSC: Skill
NOT: Section 3.4
31. ANS:
no inflection points; concave up on ÊÁË 16, ˆ˜¯
PTS: 1
DIF: Medium
REF: 3.4.27
OBJ: Identify all points of inflection for a function and discuss the concavity
MSC: Skill
NOT: Section 3.4
7
ID: A
32. ANS:
ÊÁ 0,0 ˆ˜ ÊÁ 2,8 ˆ˜
Ë ¯Ë
¯
PTS: 1
DIF: Easy
REF: 3.4.19
OBJ: Identify all points of inflection for a function
NOT: Section 3.4
33. ANS:
relative min: f (4)  2 ; no relative max
MSC: Skill
PTS: 1
DIF: Medium
REF: 3.4.40
OBJ: Identify all relative extrema for a function using the Second Derivative Test
MSC: Skill
NOT: Section 3.4
34. ANS:
none of the above
PTS: 1
DIF: Medium
REF: 3.4.34
OBJ: Identify all points of inflection for a function and discuss the concavity
MSC: Skill
NOT: Section 3.4
35. ANS:
concave upward on ÊÁË , ˆ˜¯
PTS: 1
DIF: Medium
REF: 3.4.5
OBJ: Identify the intervals on which a function is concave up or concave down
MSC: Skill
NOT: Section 3.4
36. ANS:
PTS: 1
DIF: Medium
REF: 3.3.62
OBJ: Graph a function's derivative given the graph of the function
MSC: Skill
NOT: Section 3.3
8
ID: A
37. ANS:
relative maximum: ÊÁË 3,83 ˆ˜¯
PTS: 1
DIF: Easy
REF: 3.3.19c
OBJ: Identify the relative extrema of a function by applying the First Derivative Test
MSC: Skill
NOT: Section 3.3
38. ANS:
increasing on ÊÁË ,3ˆ˜¯ ; decreasing on ÊÁË 3, ˆ˜¯
PTS: 1
DIF: Easy
REF: 3.3.19b
OBJ: Identify the intervals on which the function is increasing or decreasing
MSC: Skill
NOT: Section 3.3
39. ANS:
(a) x  1
(b) decreasing: ÊÁË ,1ˆ˜¯ ; increasing: ÊÁË 1, ˆ˜¯
(c) relative min: f (1)  0
PTS: 1
DIF: Medium
REF: 3.3.29c
OBJ: Identify the intervals on which the function is increasing or decreasing; Identify the relative extrema of a
function by applying the First Derivative Test
MSC: Skill
NOT: Section 3.3
40. ANS:
increasing on ÊÁË ,2ˆ˜¯ and ÊÁË 2, ˆ˜¯ ; decreasing on ÊÁË 2,2 ˆ˜¯
PTS: 1
DIF: Easy
REF: 3.3.5
OBJ: Estimate the intervals where a function is increasing and decreasing from a graph
MSC: Skill
NOT: Section 3.3
41. ANS:
MVT applies; c = 6
PTS: 1
DIF: Medium
REF: 3.2.39
OBJ: Identify all values of c guaranteed by the Mean Value Theorem
MSC: Skill
NOT: Section 3.2
42. ANS:
f (x )  12x  10
PTS: 1
DIF: Easy
REF: 3.2.74
OBJ: Construct a function that has a given derivative and passes through a given point
MSC: Skill
NOT: Section 3.2
43. ANS:
left endpoint: ÊÁË 0,  25 ˆ˜¯ absolute minimum
right endpoint: ÊÁË 8, 64ˆ˜¯ absolute maximum
PTS: 1
DIF: Medium
REF: 3.1.41
OBJ: Graph a function and locate the absolute extrema on a given closed interval
MSC: Skill
NOT: Section 3.1
9
ID: A
44. ANS:
ÊÁ
ÊÁ 1 ˆ˜
1 ˆ˜
decreasing: ÁÁÁ , ˜˜˜˜ ; increasing: ÁÁÁÁ , ˜˜˜˜
ÁË
2¯
Ë 2 ¯
PTS: 1
DIF: Medium
REF: 3.3.9
OBJ: Identify the intervals on which a function is increasing or decreasing
MSC: Skill
NOT: Section 3.3
45. ANS:
Rolle's Theorem applies; c = 5
PTS: 1
DIF: Easy
REF: 3.2.11
OBJ: Identify all values of c guaranteed by Rolle's Theorem
NOT: Section 3.2
46. ANS:
x9
PTS: 1
DIF: Easy
REF: 3.3.19a
OBJ: Identify the critical numbers of a function
NOT: Section 3.3
47. ANS:
f (x )  6x 2  6x  114
MSC: Skill
MSC: Skill
PTS: 1
DIF: Medium
REF: 3.2.76
OBJ: Construct a function that has a given derivative and passes through a given point
MSC: Skill
NOT: Section 3.2
48. ANS:
PTS: 1
DIF: Medium
REF: 3.3.60
OBJ: Graph a function's derivative given the graph of the function
MSC: Skill
NOT: Section 3.3
10
ID: A
49. ANS:
4 seconds
PTS: 1
DIF: Easy
REF: 3.2.53b
OBJ: Identify all values of c guaranteed by the Mean Value Theorem in applications
MSC: Application NOT: Section 3.2
50. ANS:
absolute max: ÊÁË 4, 99ˆ˜¯ ; absolute min: ÊÁË 2,  9 ˆ˜¯
PTS: 1
DIF: Medium
REF: 3.1.20
OBJ: Locate the absolute extrema of a function on a given closed interval
MSC: Skill
NOT: Section 3.1
51. ANS:
Rolle's Theorem applies; c  6 
3
3
, 6
3
3
PTS: 1
DIF: Medium
REF: 3.2.13
OBJ: Identify all values of c guaranteed by Rolle's Theorem
NOT: Section 3.2
11
MSC: Skill