2.3 Product and Quotient Rules and Higher-Order Derivatives 101
2.3 Product and Quotient Rules and Higher-Order Derivatives
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Find the derivative of the algebraic function .
a.
b.
c.
d.
e.
____ 2. Use the Product Rule to differentiate .
a.
b.
c.
d.
e.
____ 3. Use the Product Rule to differentiate .
a.
b.
c.
d.
e.
102 Chapter 2: Differentiation
____ 4. Use the Product Rule to differentiate.
a.
b.
c.
d.
e.
____ 5. Use the Product Rule to differentiate .
a.
b.
c.
d.
e.
____ 6. Use the Quotient Rule to differentiate the function .
a.
b.
2.3 Product and Quotient Rules and Higher-Order Derivatives 103
c.
d.
e.
____ 7. Use the Quotient Rule to differentiate the function .
a.
b.
c.
104 Chapter 2: Differentiation
d.
e.
____ 8. Use the Quotient Rule to differentiate the function .
a.
b.
c.
d.
e.
2.3 Product and Quotient Rules and Higher-Order Derivatives 105
____ 9. Use the Quotient Rule to differentiate the following function and
evaluate .
a.
b.
c.
d.
e.
____ 10. Find the derivative of the algebraic function .
a.
b.
c.
d.
e.
106 Chapter 2: Differentiation
____ 11. Find the derivative of the function .
a.
b.
c.
d.
e.
____ 12. Find the derivative of the function.
.
a.
b.
c.
d.
e.
____ 13. Find the derivative of the trigonometric function .
a.
b.
c.
d.
e.
____ 14. Find the derivative of the function.
a.
b.
c.
d.
e.
2.3 Product and Quotient Rules and Higher-Order Derivatives 107
____ 15. Find an equation of the tangent line to the graph of f at the given point.
a.
b.
c.
d.
e.
____ 16. Determine all values of x, (if any), at which the graph of the function has a horizontal
tangent.
a.
b.
c.
d.
e. The graph has no horizontal tangents.
____ 17. The length of a rectangle is and its height is , where t is time in seconds and
the dimensions are in inches. Find the rate of change of the area, A, with respect to time.
a. square inches/second
b. square inches/second
c. square inches/second
d. square inches/second
e. square inches/second
____ 18. The radius of a right circular cylinder is and its height is , where t is time
in seconds and the dimensions are in inches. Find the rate of change of the volume of the cylinder, V,
with respect to time.
a. cubic inches per second
b. cubic inches per second
108 Chapter 2: Differentiation
c. cubic inches per second
d. cubic inches per second
e. cubic inches per second
____ 19. The ordering and transportation cost C for the components used in manufacturing a
product is where C is measured in thousands of dollars and x is the
order size in hundreds. Find the rate of change of C with respect to x for x = 24. Round your answer
to two decimal places.
a. –6.44 thousand dollars per hundred
b. 8.04 thousand dollars per hundred
c. 3.28 thousand dollars per hundred
d. –4.92 thousand dollars per hundred
e. –7.96 thousand dollars per hundred
____ 20. A population of 620 bacteria is introduced into a culture and grows in number
according to the equation where t is measured in hours. Find the rate at
which the population is growing when t = 2. Round your answer to two decimal places.
a. 226.7 bacteria per hour
b. 68.89 bacteria per hour
c. 65.26 bacteria per hour
d. 51.52 bacteria per hour
e. 61.23 bacteria per hour
____ 21. When satellites observe Earth, they can scan only part of Earth’s surface. Some
satellites have sensors that can measure the angle shown in the figure. Let h represent the satellite’s
distance from Earth’s surface and let r represent Earth’s radius. Find the rate at which h is changing
with respect to when (Assume r = 4460 miles.) Round your answer to the nearest unit.
2.3 Product and Quotient Rules and Higher-Order Derivatives 109
a. –2973 mi/radian
b. –5150 mi/radian
c. 5150 mi/radian
d. –8920 mi/radian
e. 2973 mi/radian
____ 22. Find the second derivative of the function .
a.
b.
c.
d.
e.
____ 23. Find the second derivative of the function .
a.
b.
110 Chapter 2: Differentiation
c.
d.
e.
____ 24. Find the second derivative of the function .
a.
b.
c.
d.
e.
____ 25. Given the derivative below find the requested higher-order derivative.
.
a.
b.
c.
d.
e.
____ 26. Suppose that an automobile’s velocity starting from rest is where v is
measured in feet per second. Find the acceleration at 9 seconds. Round your answer to one decimal
place.
a. 1.9 ft/sec2
b. 0.9 ft/sec2
c. 0.6 ft/sec2
d. 0.2 ft/sec2
e. 8.3 ft/sec2
2.3 Product and Quotient Rules and Higher-Order Derivatives 111
2.3 Product and Quotient Rules and Higher-Order Derivatives
Answer Section
112 Chapter 2: Differentiation