Stewart – Calculus 8e ET Chapter 3 Form A
1. Find the first and the second derivatives of the function.
2. Find an equation of the tangent line to the curve.
3. Find
4. Find all points at which the tangent line is horizontal on the graph of the function.
5. Calculate
.
6. Differentiate the function.
7. The position function of a particle is given by
When does the particle reach a velocity of m/s?
8. If a tank holds 5000 gallons of water, and that water can drain from the tank in 40 minutes, then
Torricelli’s Law gives the volume V of water remaining in the tank after t minutes as
.
Find the rate at which water is draining from the tank after minutes.
9. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with
a speed of ft/s. At what rate is his distance from second base decreasing when he is halfway to
first base? Round the result to the nearest hundredth.
Stewart – Calculus 8e ET Chapter 3 Form A
10. Evaluate
11. Let .
a. Find the derivative of f.
b. Find the point on the graph of f where the tangent line to the curve is horizontal.
c. Sketch the graph of f and the tangent line to the curve at the point found in part (b).
d. What is the rate of change of f at this point?
12. Find the derivative function.
13. A 18-ft ladder leaning against a wall begins to slide. How fast is the angle between the ladder and
the wall changing at the instant of time when the bottom of the ladder is 9 ft from the wall and
sliding away from the wall at the rate of 4 ft/sec? Round the answer to the nearest hundredth.
18 ft
(figure not to scale)
14. Find
by implicit differentiation.
15. Find the rate of change of y with respect to x at the given values of x and y.
; x = 4, y = 4
Stewart – Calculus 8e ET Chapter 3 Form A
16. Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of
intersection of the curves. Show that the curves of the given equations are orthogonal.
y –
7
4
x = x =
7
4
cos y
123–1–2–3 x
1
2
3
–1
–2
–3
y
17. Differentiate the function.
18. s(t) is the position of a body moving along a coordinate line; s(t) is measured in feet and t in
seconds, where . Find the position, velocity, and speed of the body at the indicated time.
; t = 1
19. Two chemicals react to form another chemical. Suppose that the amount of chemical formed in
time t (in hours) is given by
where is measured in pounds.
a. Find the rate at which the chemical is formed when Round to two decimal places.
b. How many pounds of the chemical are formed eventually?
20. A circle’s radius is increasing. Find the rate of change of the area of the circle with respect to the
radius r when
Stewart – Calculus 8e ET Chapter 3 Form A
Answer Key
Stewart – Calculus 8e ET Chapter 3 Form A
Stewart – Calculus 8e ET Chapter 3 Form B
1. Find in terms of .
2. If and , find
3. Differentiate.
4. Calculate y’.
5. Differentiate.
6. Differentiate.
7. Find the equation of the tangent line to the given curve at the specified point.
8. Differentiate.
9. Find
10. Calculate
.
Stewart – Calculus 8e ET Chapter 3 Form B
11. Use logarithmic differentiation to find the derivative of the function.
12. Find, correct to three decimal places the area of the region above the hyperbola , below
the x axis, and between the lines x = – 4 and x = – 3.
13. Evaluate the integral.
14. Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length 3 cm,
pressure difference and viscosity 0.028.
Find the velocity of the blood at radius r =
15. The volume of a cube is increasing at a rate of . How fast is the surface area increasing
when the length of an edge is .
16. Find the derivative of the function.
17. Differentiate the function.
Stewart – Calculus 8e ET Chapter 3 Form B
18. Two chemicals react to form another chemical. Suppose that the amount of chemical formed in
time t (in hours) is given by
where is measured in pounds.
a. Find the rate at which the chemical is formed when Round to two decimal places.
b. How many pounds of the chemical are formed eventually?
19. The volume of a right circular cone of radius r and height h is . Suppose that the radius
and height of the cone are changing with respect to time t.
a. Find a relationship between , , and .
b. At a certain instant of time, the radius and height of the cone are 12 in. and 13 in. and are
increasing at the rate of 0.2 in./sec and 0.5 in./sec, respectively. How fast is the volume of
the cone increasing?
20. Find the derivative of the function.
= sinh –1 7x
Stewart – Calculus 8e ET Chapter 3 Form B
Answer Key
Stewart – Calculus 8e ET Chapter 3 Form B
20.
Stewart – Calculus 8e ET Chapter 3 Form C
Select the correct answer for each question.
____ 1. Find the derivative of the function.
a.
b.
c.
d.
____ 2. Find the second derivative of the function.
a.
b.
c.
d.
____ 3. Find an equation of the line tangent to the graph of at the point where x = 0.
a.
b.
c.
d.
Stewart – Calculus 8e ET Chapter 3 Form C
____ 4. Find the given derivative by finding the first few derivatives and observing the pattern that occurs.
a.
b.
c.
d.
e. None of these
____ 5. Find the derivative of the function.
a.
b.
c.
d.
____ 6. Find
in terms of .
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 3 Form C
____ 7. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 8. Find the derivative of the function.
g(t) = tan(cos 2t)
a.
b.
c.
d.
____ 9. Find the tangent line to the ellipse at the point .
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 3 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 10. Find by implicit differentiation.
a.
b.
c.
d.
____ 11. Find an equation of the tangent line to the curve at the point
(4,1).
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 3 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 12. Find an equation of the tangent line to the given curve at the indicated point.
3–3
4
–4
x
y
a.
b.
c.
d.
____ 13. Differentiate the function.
a.
b.
c.
d.
Stewart – Calculus 8e ET Chapter 3 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 14. Use logarithmic differentiation to find the derivative of the function.
a.
b.
c.
d.
e.
____ 15. Use logarithmic differentiation to find the derivative of the function.
a.
b.
c.
d.
e.
____ 16. Suppose the daily total cost (in dollars) of manufacturing x televisions is
What is the marginal cost when x = 200? What is the actual cost incurred in manufacturing the
201st television?
a. $214.00, $215.07
b. $229.00, $228.93
c. $214.00, $214.18
d. $229.00, $230.15
Stewart – Calculus 8e ET Chapter 3 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 17. Water flows from a tank of constant cross-sectional area 50 through an orifice of constant
cross-sectional area located at the bottom of the tank. Initially, the height of the water in
the tank was 20 ft, and t sec later it was given by the equation
How fast was the height of the water decreasing when its height was 2 ft?
h
a. ft/sec
b. ft/sec
c.
2
25
ft/sec
d. ft/sec
____ 18. Find the instantaneous rate of change of the function when
a.
b. 3
c. 9
d.
____ 19. Compute and dy for the given values of x and .
a.
b.
c.
d.
e.
Stewart – Calculus 8e ET Chapter 3 Form C
© 2016 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
____ 20. Use the definitions of the hyperbolic functions to find the following limit.
a.
b.
c.
d.
e. None of these
Stewart – Calculus 8e ET Chapter 3 Form C
Answer Key
Stewart – Calculus 8e ET Chapter 3 Form D
Select the correct answer for each question.
____ 1. Use the definition of the derivative to find the derivative of the function.
a.
b.
c.
d.
____ 2. Find the derivative of the function.
a.
b.
c.
d.
____ 3. Differentiate:
a. d.
b. e.
c.
____ 4. Differentiate:
a. d.
b.
e.
c.