Stewart – Calculus 8e ET Chapter 2 Form A
1. Find the limit.
2. Evaluate the limit.
3. Find the limit.
4. How close to do we have to take x so that
5. How close to 2 do we have to take x so that is within a distance of 0.01 from 13?
6. If f and g are continuous functions with
7. Find the limit.
8. Find the limit.
9. Find an equation of the tangent line to the curve.
Stewart – Calculus 8e ET Chapter 2 Form A
10. Find the limit if
11. Use the graph of the function to find each limit.
12345–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
, ,
12. Complete the table by computing at the given values of x, accurate to five decimal places.
Use the results to guess at the indicated limit, if it exists, to three decimal places.
x 3.9 3.99 3.999 4.001 4.01 4.1
13. Use the precise definition of a limit to prove that
14. Find the limit .
15. Find the numbers, if any, where the function is discontinuous.
Stewart – Calculus 8e ET Chapter 2 Form A
16. Find the interval(s) where is continuous.
17. You are given the graph of f. Find the horizontal and vertical asymptotes of the graph of f.
1234567–1–2–3–4–5–6–7 x
1
2
3
4
5
6
7
–1
–2
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–6
–7
y
18. The graph shows the percentage of households in a certain city watching television during a 24-hr period on
a weekday ( corresponds to 6 a.m.). By computing the slope of the respective tangent line, estimate the
rate of change of the percentage of households watching television at a-12 p.m.
Note that
3
dy
2 4 6 8 10 12 14 16 18 20 t (hr)
0.075
0.15
0.225
y (%)
19. Find the derivative of the function and evaluate at the given value of x.
Stewart – Calculus 8e ET Chapter 2 Form A
20. Let .
a. Sketch the graph of f.
b. For what values of x is f differentiable?
c. Find a formula for .
Stewart – Calculus 8e ET Chapter 2 Form A
Answer Key
Stewart – Calculus 8e ET Chapter 2 Form A
Stewart – Calculus 8e ET Chapter 2 Form B
1. The position of a car is given by the values in the following table.
Estimate the instantaneous velocity when by averaging the velocities for the periods
.
2. Find the limit.
3. Evaluate the limit.
4. Evaluate the limit.
5. Find the limit.
6. Consider the following function.
Determine the values of a for which exists.
Stewart – Calculus 8e ET Chapter 2 Form B
7. How close to do we have to take x so that
8. Use a graph to find a number such that whenever .
Round your answer to the nearest thousandth.
9. Use the definition of the limit to find values of that corresponds to .
Round your answer to the nearest thousandth.
10. Use a graph to find a number such that whenever .
11. How close to 2 do we have to take x so that is within a distance of 0.01 from 13?
12. Use continuity to evaluate the limit.
13. For the function f whose graph is shown, state the following.
Stewart – Calculus 8e ET Chapter 2 Form B
14. If a cylindrical tank holds gallons of water, which can be drained from the bottom of the
tank in an hour, then Torricelli’s Law gives the volume of water remaining in the tank after t
minutes as
Find the rate at which the water is flowing out of the tank (the instantaneous rate of change of V
with respect to t) as a function of t.
15. Find an equation of the tangent line to the curve at the point (1, 1).
16. Use the graph of the function to find each limit.
12345–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
, ,
17. Use the precise definition of a limit to prove that
18. Find the limit.
Stewart – Calculus 8e ET Chapter 2 Form B
19. Sketch the graph of the derivative of the function f whose graph is given.
12345–1–2–3–4–5 x
1
2
3
4
5
6
y
20. Let .
a. Sketch the graph of f.
b. For what values of x is f differentiable?
c. Find a formula for .
Stewart – Calculus 8e ET Chapter 2 Form B
Answer Key
Stewart – Calculus 8e ET Chapter 2 Form B
Stewart – Calculus 8e ET Chapter 2 Form C
Select the correct answer for each question.
____ 1. Suppose the distance s (in feet) covered by a car moving along a straight road after t sec is given
by the function Calculate the (instantaneous) velocity of the car when
a. 223 ft/sec
b. 16 ft/sec
c. 560 ft/sec
d. 4130 ft/sec
____ 2. If a rock is thrown upward on the planet Mars with a velocity of , its height in meters t
seconds later is given by
.
Find the average velocity over the time interval .
a. m/s
b. m/s
c. m/s
d. m/s
e. m/s
____ 3. The point lies on the curve . If is the point Q , use your calculator to
find the slope of the secant line PQ (correct to six decimal places) for the value
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 2 Form C
____ 4. Use the graph of the function to state the value of if it exists.
a.
b.
c. does not exist
d. 1
e.
Stewart – Calculus 8e ET Chapter 2 Form C
____ 5. Sketch the graph of the function f and evaluate
a.
12345–1–2–3–4–5 x
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2
3
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5
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–2
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–5
y
1
c.
12345–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
3
b.
12345–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
3
d.
12345–1–2–3–4–5 x
1
2
3
4
5
–1
–2
–3
–4
–5
y
1
Stewart – Calculus 8e ET Chapter 2 Form C
____ 6. Evaluate the limit.
a.
b.
c.
d.
e.
____ 7. Find the limit , if it exists.
a. 1
b. 3
c. 2
d. Does not exist
____ 8. Find the limit by evaluating the derivative of a suitable function at an appropriate value of x.
(Hint: Use the definition of the derivative.)
a. 25
b. 75
c.
d.
____ 9. You are given that , , and . Find the limit
.
a. 17
b. –8
c. 0
d. 22
Stewart – Calculus 8e ET Chapter 2 Form C
____ 10. Find the limit.
a. 0
b.
c.
d.
e. does not exist
____ 11. A machinist is required to manufacture a circular metal disk with area 1000 . If the machinist is
allowed an error tolerance of in the area of the disk, how close to the ideal radius must the
machinist control the radius?
Round your answer to the nearest hundred thousandth.
a.
b.
c.
d.
e.
____ 12. If f and g are continuous functions with
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 2 Form C
____ 13. Use the graph to determine where the function is discontinuous.
12345–1–2–3–4–5 x
1
2
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y
a. At 0
b. On the interval
c. At
d. At 1
____ 14. Find the numbers, if any, where the function is discontinuous.
a.
b. 2
c. 1
d. 0
____ 15. Let
Find the value of k that will make f continuous on
a. 5
b. 46
c. –4
d. 3
Stewart – Calculus 8e ET Chapter 2 Form C
____ 16. Find the limit.
a.
b. 0
c.
d.
____ 17. Find the horizontal and vertical asymptotes of the graph of the function
.
a. HA , VA
b. HA , VA none
c. HA , VA none
d. HA , VA
Stewart – Calculus 8e ET Chapter 2 Form C
____ 18. Find the vertical asymptotes of the function.
a.
b.
c.
d.
e. None of these
____ 19. If use the definition of derivative to find
a.
b.
c.
d.
e. None of these