2.1 A Preview of Calculus
65
2.1 A Preview of Calculus
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Decide whether the following problem can be solved using precalculus, or
whether calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the distance traveled in 16 seconds by an object traveling at a constant velocity of 20
feet per second.
calculus, 320 ft
calculus, 340 ft
precalculus, 320 ft
calculus, 640 ft
precalculus, 640 ft
____ 2. Decide whether the following problem can be solved using precalculus, or
whether calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the distance traveled in 20 seconds by an object moving with a velocity of
feet per second.
calculus, 162.4485 ft
precalculus, 163.7985 ft
calculus, 165.4777 ft
precalculus, 165.4777 ft
precalculus, 162.4485 ft
66
Chapter 2: Limits and Their Properties
____
3.
Decide whether the following problem can be solved using precalculus, or
whether calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
A cyclist is riding on a path whose elevation is modeled by the function
where x and are measured in miles. Find the rate of change of elevation when x = 4.
precalculus, 0.08
calculus, 0.2
calculus, 0.64
calculus, 0.08
precalculus, 0.2
____ 4. Decide whether the following problem can be solved using precalculus, or
whether calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
A cyclist is riding on a path whose elevation is modeled by the function where x
and are measured in miles. Find the rate of change of elevation when x = 5.
2.1 A Preview of Calculus
67
calculus, 2
precalculus, 0.2
calculus, 0.2
precalculus, 2
precalculus, 0.45
____ 5. Decide whether the following problem can be solved using precalculus, or
whether calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the area of the shaded region bounded by the triangle with vertices (0,0), (8,9), (17,0).
precalculus , 153
calculus , 229.5
precalculus , 76.5
precalculus , 229.5
calculus , 153
68
Chapter 2: Limits and Their Properties
____
6.
Decide whether the following problem can be solved using precalculus, or
whether calculus is required. If the problem can be solved using precalculus, solve it. If the
problem seems to require calculus, use a graphical or numerical approach to estimate the
solution.
Find the area of the shaded region.
calculus , 11
precalculus , 11
precalculus , 13
calculus , 16
precalculus , 16
____ 7.
Consider the function
and the point
on the graph of f.
Graph f and the secant line passing through
and
for
.
a.
d.
2.1 A Preview of Calculus
69
b. e.
c.
____ 8.
Consider the function
and the point
on the graph of f.
Find the slope of the secant line passing through
and
for
. Round
your answer to four decimal places.
m=0.1000
m=0.0122
m=0.0122
m=0.3133
m=0.1000
____ 9.
Consider the function
and the point
on the graph of f.
Estimate the slope m of the tangent line of f at
. Round your answer to four decimal
places.
m=0.1667
m=0.0832
m=0.3800
m=0.0556
m=0.0833
70
Chapter 2: Limits and Their Properties
____
10.
Consider the function
and the point
on the graph of f.
Graph f and the secant line passing through
and
for
.
a.
d.
b. e.
c.
2.1 A Preview of Calculus
71
____ 11.
Consider the function
and the point
on the graph of
f. Find the slope of the secant line passing through and for . Round
your answer to one decimal place.
3.5
2.0
3.0
4.5
9.0
____ 12.
Consider the function
and the point
on the
graph of f. Estimate the slope of the tangent line of f at
.
a.
10
b.
3
c.
8
d.
2
e.
9
____ 13. Use the rectangles in the following graph to approximate the area of the region
bounded by .
3.9082
2.6055
1.9541
1.4656
0.9770
72 Chapter 2: Limits and Their Properties
____ 14. Use the rectangles in the following graph to approximate the area of the region
bounded by .
0.7850
1.5700
3.1400
1.1775
1.0519
____ 15. Use the rectangles in the graph given below to approximate the area of the region bounded by
Round your answer to three decimal places.
2.481 units2
6.371 units2
3.585 units2
6.872 units2
6.903 units2
2.1 A Preview of Calculus
73
____ 16. Consider the length of the graph of
Approximate the length of the curve by finding the sum of the lengths of four line segments, as
shown in following figure. Round your answer to two decimal places.
6.11
8.12
5.66
8.49
7.11
74 Chapter 2: Limits and Their Properties
2.1 A Preview of Calculus
Answer Section
2.2 Finding Limits Graphically and Numerically
75
2.2 Finding Limits Graphically and Numerically
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1.
Complete the table and use the result to estimate the limit.
x
2.9
2.99
2.999
3.001
3.01
3.1
0.525000
0.275000
–0.100000
0.400000
–0.475000
____ 2. Complete the table and use the result to estimate the limit.
x
6.9
6.99
6.999
7.001
7.01
7.1
–0.062500
0.067500
–0.192500
0.047500
–0.172500
76 Chapter 2: Limits and Their Properties
____ 3. Complete the table and use the result to estimate the limit.
x
–10.1
–10.01
–10.001
–9.999
–9.99
–9.9
0.974745
–1.099745
–1.224745
1.058078
1.224745
____ 4. Complete the table and use the result to estimate the limit.
x
–0.1
–0.01
–0.001
0.001
0.01
0.1
0
1
0.5
____ 5. Complete the table and use the result to estimate the limit.
x
–0.1
–0.01
–0.001
0.001
0.01
0.1
0
0.5
1
2.2
Finding Limits Graphically and Numerically
77
____ 6.
Determine the following limit. (Hint:
Use the graph to calculate the limit.)
6
1
5
4
does not exist
____ 7. Determine the following limit. (Hint: Use the graph to calculate the limit.)
5
1
0
4
does not exist
78 Chapter 2: Limits and Their Properties
____ 8. Let
Determine the following limit. (Hint: Use the graph to calculate the limit.)
5
4
3
0
does not exist
2.2 Finding Limits Graphically and Numerically
79
____ 9. Let
Determine the following limit. (Hint: Use the graph to calculate the limit.)
6
25
1
5
does not exist.
____ 10. Determine the following limit. (Hint: Use the graph to calculate the limit.)
80 Chapter 2: Limits and Their Properties
–2
0
–4
2
does not exist
____ 11. A ring has a inner circumference of 10 centimeters. What is the radius of the ring?
Round your answer to four decimal places.
0.7958 centimeter
3.1831 centimeters
1.5915 centimeters
1.7841 centimeters
10.1321 centimeters
____ 12. A ring has a inner circumference of 9 centimeters. If the ring’s inner circumference can
vary between 8 centimeters and 10 centimeters how can the radius vary? Round your answer to
five decimal places.
Radius can vary between 6.48456 centimeters and 10.13212 centimeters.
Radius can vary between 1.59577 centimeters and 1.78412 centimeters.
Radius can vary between 1.27324 centimeters and 1.59155 centimeters.
Radius can vary between 2.54648 centimeters and 3.18310 centimeters.
Radius can vary between 0.43239 centimeter and 2.43239 centimeters.
____ 13. A sphere has a volume of 4.76 cubic inches. What is the radius of the sphere? Round your
answer to four decimal places.
1.0435 inches
1.6565 inches
1.0660 inches
2.1320 inches
1.9335 inches
____ 14. A sphere has a volume of 5.2 cubic inches. If the sphere’s volume can vary between 4.4
cubic inches and 6.1 cubic inches , how can the radius vary? Round your answer to five decimal
places.
Radius can vary between 1.01653 inches and 1.13348 inches.
Radius can vary between 1.61365 inches and 1.79929 inches.
Radius can vary between 0.27474 inch and 1.97474 inches.
Radius can vary between 1.85897 inches and 2.18882 inches.
Radius can vary between 1.02490 inches and 1.20676 inches.
2.2 Finding Limits Graphically and Numerically
81
2.2 Finding Limits Graphically and Numerically
Answer Section
2.
5.
9.
11.
14.
82 Chapter 2: Limits and Their Properties
2.3 Evaluating Limits Analytically
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the limit.
a.
b.
c.
d.
e.
____ 2. Find the limit.
a.
b.
3
22
3
7
3
10
____ 3. Find the limit.
a.
–3
2.3 Evaluating Limits Analytically
83
____ 4. Find the limit.
a.
b.
c.
d.
e.
____ 5. Find the limit.
a.
b.
c.
d.
e. 0
____ 6. Find the limit.
a.
b.
c.
d.
e.
84 Chapter 2: Limits and Their Properties
____ 7. Find the lmit.
a.
b.
c.
d.
e.
____ 8. Let and . Find the limit.
–18
25
21
8
9
____ 9. Let and . Find the limit.
a. 2
b.
8
–8
____ 10. Let and . Find the limit.
6
14
11
10
2