Name: ________________________ Class: ___________________ Date: __________ ID: A

Final Exam Calculus I Review

1. A business has a cost of C  0.5x  800 for Ê ˆ6

producing x units. The average cost per unit is Ë 
5. Find the indefinite integral t 3 ÁÁÁ 2  t 4 ˜˜˜ dt .
¯
C
C  . Find the limit of C as x approaches
x
6. A rectangular page is to contain 81 square inches of
infinity.
print. The margins on each side are 1 inch. Find the
dimensions of the page such that the least amount
2. Suppose the position function for a free-falling of paper is used.
object on a certain planet is given by
s (t )  17t 5  v 0 t  s 0 . A silver coin is dropped 7. Use sigma notation to write the sum
6 6 6 6
from the top of a building that is 1378 feet tall.    .
11 12 13 1  13
Determine the velocity function for the coin.

ÁÊÁ ˜ˆ˜
n

8. Write the limit lim  ÁÁÁÁ 2 ˜˜˜˜ x i , as a definite

3. Test for symmetry with respect to each axis and to Á 4 ˜
the origin.
   0 i  1 Á ÁË c i ˜˜¯
ÈÍ ˘
x2  4 integral on the interval ÍÎ 8, 10 ˙˙˚ , where ci is any
y
x point in the i th subinterval.

4. An open box of maximum volume is to be made

from a square piece of material 20 centimeters on a 9. Find the limit of s (n) as n  .
side by cutting equal squares from the corners and
ÈÍ ˘
turning up the sides (see figure). Write the volume ÍÍ n 4 (n  1) 2 ˙˙˙
5 ÍÍ ˙˙
V as a function of x, the length of the corner s (n )  4 ÍÍ
ÍÍ
˙˙
˙˙
squares. n ÍÍÎ 10 ˙˙˚

È ˘
10. Determine all values of x in the interval ÍÍÎ 1,3˙˙˚ for
Ê ˆ
5 ÁÁ x 2  1 ˜˜
Ë ¯
which the function f (x )  2
equals its
x
20
average value .
3

11. Find all critical numbers of the function

g (x )  x 4  22x 2 .

a. V  x(20  2x) 2 b. V  x(20  2x) 12. Evaluate the definite integral of the trig function.
c. V  x 2  (20  2x) d. V  x 2 (20  2x) 7
e. V  x  (20  x) 2 0 (5sins  2cos s)ds

Use a graphing utility to verify your results.

1
Name: ________________________ ID: A

13. Find the limit (if it exists). ÊÁ ˆ˜

n
ÁÁ 9 ˜˜
19. Write the limit lim  ÁÁÁÁ 2 ˜˜˜˜ x i , as a definite
x  22  5    0 i  1 Á ÁË c i ˜˜¯
lim È ˘
x3 x3 integral on the interval ÍÍÎ 5, 7 ˙˙˚ , where ci is any point
in the i th subinterval.
14. Let f(x)  5  3x 2 and g (x )  x  6 . Find the
limit.
20. Use implicit differentiation to find an equation of
lim g ÊÁË f (x ) ˆ˜¯ x2 y2
x3
the tangent line to the ellipse   1 at
2 338
ÊÁ 1,13 ˆ˜ .
Ë ¯
a. 2 5 b. 4 2 c. 38 d. 11 e. 3
21. Find the indefinite integral of the following
15. Let f (x)  5  5x 2 and g (x )  x  3 . Find the function and check the result by differentiation.
limit.

lim g ÊÁË f (x ) ˆ˜¯  (5  z ) z dz

x3
22. Determine whether the Mean Value Theorem
a. 53 b. 23 c. 2 2 d. 3 e. 5 2 can be applied to the function f (x )  x 2 on the
closed interval [2,10]. If the Mean Value
16. Find a function f that has derivative f  (x )  8x  3 Theorem can be applied, find all numbers c in
and with graph passing through the point (5,8). the open interval (2,10) such that
f (10)  f (2)
f  (c)  .
17. Determine all values of x , (if any), at which the 10  (2)
graph of the function has a horizontal tangent.

6
y (x ) 
x9

dy
18. Find by implicit differentiation given that
dx
2xy  5.

dy dy dy
a.  5xy b.  5xy c.  xy
dx dx dx
dy 5y dy y
d.  e. 
dx x dx x

2
Name: ________________________ ID: A

23. A rectangle is bounded by the x- and y-axes and the 29. Find the constant a such that the function
(5  x )
graph of y  (see figure). What length and ÏÔ
2 ÔÔ 5, x  3
width should the rectangle have so that its area is a ÔÔ
f (x )  ÌÔ ax  b, 3  x  7
maximum? ÔÔ
ÔÔ 5, x7
Ó

is continuous on the entire real line.

30. Use the Quotient Rule to differentiate the

5 x
function f  (x )  .
2
x  10

31. Determine the slant asymptote of the graph of

6x 2  2x  5
f (x )  .
x1

32. Solve the differential equation.

24. Find the derivative of the function f (x )  7x  4
using the limiting process. df
 20s 3  4, f (1)  9
ds
Ê ˆ
2 ÁÁ x 2  1 ˜˜
Ë ¯
25. Find the average value of f (x )  2
on the 33. Find an equation of the line that passes through the
x points (33,  18) and (22, 27).
È ˘
interval ÍÍÎ 1,3˙˙˚ .
9 9 9
a. y   x b. y   x  9 c. y   x  9
26. Find an equation of the line through the points of 11 11 11
intersection of y  x 2 and y  16x  x 2 . d. y 
9
x  9 e. y 
9
x9
11 11
27. Evaluate the following definite integral by the limit
definition. 34. Find all relative extrema of the function
f (x )  x 2/3  2 . Use the Second Derivative Test
10 where applicable.
Ê ˆ
 ÁÁÁË 3s 2  2˜˜˜¯ ds 35. Determine the open intervals on which the graph of
4
f (x )  7x 2  4x  5 is concave downward or
Ê ˆ concave upward.
2 ÁÁ x 2  4 ˜˜
Ë ¯
28. Find the average value of f (x )  2
on the
x
È ˘
interval ÍÍÎ 1,3˙˙˚ .

3
Name: ________________________ ID: A

36. Find the indefinite integral and check the result by 41. Evaluate the following definite integral.
differentiation.
 Ê ˆ
0
6
t sinÁÁÁ 6t 2 ˜˜˜ dt
7x  6x  6
2

 x4
dx Ë ¯

Use a graphing utility to check your answer.

37. Assume that the amount of money deposited in a
bank is proportional to the square of the interest 42. Determine the limit (if it exists).
rate the bank pays on this money. Furthermore, the
bank can reinvest this money at 24%. Find the 4 (1  cos x)
interest rate the bank should pay to maximize lim
x0 x2
profit. (Use the simple interest formula.)

38. Sketch the graph of the function 43. Evaluate the following definite integral.
ÏÔÔ
Ô 40x  100 0  x  5
f(x)  ÌÔ  Ê ˆ
0
6
ÔÔ 2 t cos ÁÁÁ 4t 2 ˜˜˜ dt
Ó 4x 5x8 Ë ¯
and locate the absolute extrema of the function on
È ˘
the interval ÍÎÍ 0, 8 ˙˚˙ . Use a graphing utility to check your answer.

39. Assume that x and y are both differentiable 44. Write the following limit as a definite integral on
È ˘
dx dy the interval ÍÍÎ 3, 9 ˙˙˚ where ci is any point in the i th
functions of t. Find when x = 6 and =  4 for
dt dt subinterval.
the equation xy  42.
n
40. Find the points of inflection and discuss the lim  3c 2i  5c i x i
concavity of the function f (x )  sin x  cos x on  x  0 i  1

the interval ÊÁË 0,2 ˆ˜¯ .

x  1 
45. Find the x-values (if any) at which f (x )  is not continuous.
x1

a. f (x ) is not continuous at x  1 and the discontinuity is removable.

b. f (x ) is not continuous at x  1 and the discontinuity is nonremovable.
c. f (x ) is not continuous at x  0 and the discontinuity is removable. d. f (x ) is continuous for all real x.
e. f (x ) is not continuous at x  0 ,  1 and x  0 is a removable discontinuity.

4
Name: ________________________ ID: A

46. A man 6 feet tall walks at a rate of 5 feet per second 49. Evaluate the integral.
away from a light that is 15 feet above the ground
(see figure). When he is 8 feet from the base of the 6
Ê ˆ
light, at what rate is the tip of his shadow moving?
 ÁÁÁË 12z 2  5˜˜˜¯ d z
5

given,

 x 3 dx 
671
,
4
5

 x 2 dx 
91
,
3
5

 x dx 
11
,
2
5

 dx 1 .
47. Determine whether Rolle's Theorem can be applied 5
È ˘
to f (x )  x 2  30x on the closed interval ÍÍÎ 0,30 ˙˙˚ .
If Rolle's Theorem can be applied, find all values of 50. Find the derivative of the function.
c in the open interval ÊÁË 0, 30ˆ˜¯ such that f  (c)  0.
Ê ˆ
y  cos ÁÁÁ 3x 6  7 ˜˜˜
Ë ¯
48. Identify the open intervals where the function
f (x )  5x 2  2x  1 is increasing or decreasing. 51. Use the quotient rule to differentiate the following
3x
function f (x )  and evaluate f  (1) .
3
x 7

52. Find the derivative of the algebraic function

Ê ˆ3
f (s)  ÁÁÁ s 6  3 ˜˜˜ .
Ë ¯

53. Determine the slant asymptote of the graph of

3x 2  x  5
f (x )  .
x1

5
Name: ________________________ ID: A

54. Use the summation formulas to rewrite the

n

expression  30k (k4  1) without the summation

k1 n
notation.

55. Find a function f that has derivative f  (x )  4x  4

and with graph passing through the point (–4,–2).

56. Find the limit (if it exists).

9x
lim
x9
 x 2  81

57. Evaluate the definite integral of the algebraic

function.

2
0 2  z  1dz

Use a graphing utility to verify your results.

58. Suppose the position function for a free-falling

object on a certain planet is given by
s (t )  14t 2  v 0 t  s 0 . A silver coin is dropped
from the top of a building that is 1374 feet tall. Find
the instantaneous velocity of the coin when t  2.

59. Determine the limit (if it exists).

sin 3 x
lim
x0 x3

60. Suppose the position function for a free-falling

object on a certain planet is given by
s (t )  13t 2  v 0 t  s 0 . A silver coin is dropped
from the top of a building that is 1366 feet tall. Find
the instantaneous velocity of the coin when t  4.

6
 ID: A

Final Exam Calculus I Review

Answer Section
1. ANS:
0.5

PTS: 1 DIF: Easy REF: 3.5.88 OBJ: Evaluate limits at infinity in applications
MSC: Application NOT: Section 3.5
2. ANS:
v (t )  85t 4

PTS: 1 DIF: Medium REF: 2.2.97a

OBJ: Write the velocity function for a specified position function
MSC: Application NOT: Section 2.2
3. ANS:
symmetric with respect to the origin

PTS: 1 DIF: Easy REF: 0.1.37

OBJ: Identify the type of symmetry of the graph of an equation
MSC: Skill NOT: Section 0.1
4. ANS: A PTS: 1 DIF: Medium REF: 0.3.97a
OBJ: Create functions in applications MSC: Application NOT: Section 0.3
5. ANS:
ÊÁ ˆ7
ÁÁ 2  t 4 ˜˜˜
Ë ¯
C
28

PTS: 1 DIF: Easy REF: 4.5.15

OBJ: Evaluate the indefinite integral of a function using substitution
MSC: Skill NOT: Section 4.5
6. ANS:
11,11

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
7. ANS:
13

 1 6 j
j1

PTS: 1 DIF: Easy REF: 4.2.8 OBJ: Write a sum in sigma notation
MSC: Skill NOT: Section 4.2

1
 ID: A

8. ANS:
10

 x 2 dx
4

PTS: 1 DIF: Easy REF: 4.3.12

OBJ: Write a limit as a definite integral on an interval MSC: Skill
NOT: Section 4.3
9. ANS:
unbounded

PTS: 1 DIF: Medium REF: 4.2.37 OBJ: Evaluate limits at infinity

MSC: Skill NOT: Section 4.2
10. ANS:
x 3

PTS: 1 DIF: Easy REF: 4.4.52b

OBJ: Identify the points where a function equals its average value over a given interval
MSC: Skill NOT: Section 4.4
11. ANS:
critical numbers: x  0, x  11, x   11

PTS: 1 DIF: Easy REF: 3.1.12

OBJ: Identify the critical numbers of a function MSC: Skill
NOT: Section 3.1
12. ANS:
–10

PTS: 1 DIF: Easy REF: 4.4.27

OBJ: Evaluate the definite integral of a function MSC: Skill
NOT: Section 4.4
13. ANS:
1
10

PTS: 1 DIF: Medium REF: 1.3.55

OBJ: Evaluate the limit of a function analytically MSC: Skill
NOT: Section 1.3
14. ANS: C PTS: 1 DIF: Medium REF: 1.3.25c
OBJ: Evaluate the limit of composite functions MSC: Skill
NOT: Section 1.3
15. ANS: A PTS: 1 DIF: Medium REF: 1.3.25c
OBJ: Evaluate the limit of composite functions MSC: Skill
NOT: Section 1.3

2
 ID: A

16. ANS:
f (x )  4x 2  3x  77

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
17. ANS:
The graph has no horizontal tangents.

PTS: 1 DIF: Difficult REF: 2.2.61

OBJ: Calculate the values for which the slope of a function is zero
MSC: Skill NOT: Section 2.2
18. ANS: E PTS: 1 DIF: Easy REF: 2.5.6
OBJ: Differentiate an equation using implicit differentiation MSC: Skill
NOT: Section 2.5
19. ANS:
7

 x 2 dx
9

PTS: 1 DIF: Easy REF: 4.3.12

OBJ: Write a limit as a definite integral on an interval MSC: Skill
NOT: Section 4.3
20. ANS:
y  2x  15

PTS: 1 DIF: Easy REF: 2.5.41a

OBJ: Write an equation of a line tangent to the graph of an ellipse at a specified point.
MSC: Skill NOT: Section 2.5
21. ANS:
3 5
10 2 2 2
z  z C
3 5

PTS: 1 DIF: Easy REF: 4.5.37

OBJ: Evaluate the indefinite integral of a function MSC: Skill
NOT: Section 4.5
22. 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

3
 ID: A

23. 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
24. ANS:
7
f  (x ) 
2 7x  4

PTS: 1 DIF: Medium REF: 2.1.23

OBJ: Calculate the derivative of a function by the limit process MSC: Skill
NOT: Section 2.1
25. ANS:
8
3

PTS: 1 DIF: Easy REF: 4.4.52a

OBJ: Calculate the average value of a function over a given interval
MSC: Skill NOT: Section 4.4
26. ANS:
y  8x

PTS: 1 DIF: Medium REF: 0.2.71

OBJ: Write an equation of a line through the points of intersection of quadratic equations
MSC: Skill NOT: Section 0.2
27. ANS:
–924

PTS: 1 DIF: Easy REF: 4.3.8

OBJ: Evaluate a definite integral by the limit definition MSC: Skill
NOT: Section 4.3
28. ANS:
14
3

PTS: 1 DIF: Easy REF: 4.4.52a

OBJ: Calculate the average value of a function over a given interval
MSC: Skill NOT: Section 4.4
29. ANS:
a  1 , b  2

PTS: 1 DIF: Medium REF: 1.4.67

OBJ: Identify the value of a parameter to ensure a function is continuous
MSC: Skill NOT: Section 1.4

4
 ID: A

30. ANS:
ÊÁ ˆ
ÁÁ 10  10x  x 2 ˜˜˜
Ë ¯
f  (x ) 
ÊÁ 2 ˆ 2
ÁÁ x  10 ˜˜˜
Ë ¯

PTS: 1 DIF: Difficult REF: 2.3.8

OBJ: Differentiate a function using the quotient rule MSC: Skill
NOT: Section 2.3
31. ANS:
y  6x  4

PTS: 1 DIF: Medium REF: 3.6.16

OBJ: Identify the slant asymptote of the graph of a function MSC: Skill
NOT: Section 3.6
32. ANS:
f (s)  5s 4  4s  10

PTS: 1 DIF: Medium REF: 4.1.59 OBJ: Solve a differential equation

MSC: Skill NOT: Section 4.1
33. ANS: B PTS: 1 DIF: Easy REF: 0.2.40
OBJ: Write an equation of a line given two points on the line MSC: Skill
NOT: Section 0.2
34. ANS:
relative minimum: ÊÁË 0,2 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
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:
7 3 2
  2  3 C
x x x

PTS: 1 DIF: Medium REF: 4.1.28

OBJ: Evaluate the indefinite integral of a function MSC: Skill
NOT: Section 4.1

5
 ID: A

37. ANS:
16 %

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
38. ANS:
left endpoint: ÊÁË 0,  100 ˆ˜¯ absolute minimum
right endpoint: ÁÊË 8, 256 ˜ˆ¯ 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
39. ANS:
dx 24

dt 7

PTS: 1 DIF: Easy REF: 2.6.3b

OBJ: Calculate the value of an implicit derivative from given information
MSC: Skill NOT: Section 2.6
40. 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
41. ANS:
0

PTS: 1 DIF: Medium REF: 4.5.85

OBJ: Evaluate the definite integral of a function using substitution
MSC: Skill NOT: Section 4.5
42. ANS:
–2

PTS: 1 DIF: Medium REF: 1.3.66

OBJ: Evaluate the limit of a function analytically MSC: Skill
NOT: Section 1.3
43. ANS:
0

PTS: 1 DIF: Medium REF: 4.5.85

OBJ: Evaluate the definite integral of a function using substitution
MSC: Skill NOT: Section 4.5

6
 ID: A

44. ANS:
9

 3x 2  5x dx
3

PTS: 1 DIF: Easy REF: 4.3.11

OBJ: Write a limit as a definite integral on an interval MSC: Skill
NOT: Section 4.3
45. ANS: B PTS: 1 DIF: Medium REF: 1.4.49
OBJ: Identify the removable discontinuities of a function MSC: Skill
NOT: Section 1.4
46. ANS:
25
ft/sec
3

PTS: 1 DIF: Difficult REF: 2.6.33a

OBJ: Solve a related rate problem involving a man walking away from a light source
MSC: Application NOT: Section 2.6
47. ANS:
Rolle's Theorem applies; c = 15

PTS: 1 DIF: Easy REF: 3.2.11

OBJ: Identify all values of c guaranteed by Rolle's Theorem MSC: Skill
NOT: Section 3.2
48. ANS:
ÊÁ 0 ˆ˜ ÊÁ 0 ˆ˜
increasing: ÁÁÁÁ , ˜˜˜˜ ; decreasing: ÁÁÁÁ , ˜˜˜˜
Ë 0¯ Ë 0 ¯

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
49. ANS:
359

PTS: 1 DIF: Easy REF: 4.3.38

OBJ: Evaluate the definite integral of a function MSC: Skill
NOT: Section 4.3
50. ANS:
Ê ˆ
y   18x 5 sin ÁÁÁ 3x 6  7 ˜˜˜
Ë ¯

PTS: 1 DIF: Medium REF: 2.4.50

OBJ: Differentiate a trigonometric function using the chain rule
MSC: Skill NOT: Section 2.4

7
 ID: A

51. ANS:
15
f  (1) 
64

PTS: 1 DIF: Difficult REF: 2.3.15

OBJ: Differentiate a function using the quotient rule and evaluate the derivative
MSC: Skill NOT: Section 2.3
52. ANS:
Ê ˆ2
f  (s)  18s 5 ÁÁÁ s 6  3 ˜˜˜
Ë ¯

PTS: 1 DIF: Difficult REF: 2.4.7

OBJ: Differentiate a function using the chain rule MSC: Skill
NOT: Section 2.4
53. ANS:
y  3x  4

PTS: 1 DIF: Medium REF: 3.6.16

OBJ: Identify the slant asymptote of the graph of a function MSC: Skill
NOT: Section 3.6
54. ANS:
10 10

n n3

PTS: 1 DIF: Medium REF: 4.2.47

OBJ: Rewrite a sum without summation notation MSC: Skill
NOT: Section 4.2
55. ANS:
f (x )  2x 2  4x  50

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
56. ANS:
1

18

PTS: 1 DIF: Easy REF: 1.4.10 OBJ: Evaluate one-sided limits

MSC: Skill NOT: Section 1.4
57. ANS:
4

PTS: 1 DIF: Medium REF: 4.4.23

OBJ: Evaluate the definite integral of a function MSC: Skill
NOT: Section 4.4

8
 ID: A

58. ANS:
–56 ft/sec

PTS: 1 DIF: Medium REF: 2.2.97c OBJ: Interpret a derivative as a rate of change
MSC: Application NOT: Section 2.2
59. ANS:
1

PTS: 1 DIF: Medium REF: 1.3.69

OBJ: Evaluate the limit of a function analytically MSC: Skill
NOT: Section 1.3
60. ANS:
–104 ft/sec

PTS: 1 DIF: Medium REF: 2.2.97c OBJ: Interpret a derivative as a rate of change
MSC: Application NOT: Section 2.2