2.4 The Chain Rule 113
2.4 The Chain Rule
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. Find the derivative of the function.
a.
b.
c.
d.
e.
114 Chapter 2: Differentiation
____ 3. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 4. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 5. Find the derivative of the function.
a.
b.
c.
d.
e.
2.4 The Chain Rule 115
____ 6. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 7. Find the derivative of the function .
a.
b.
c.
d.
e.
____ 8. Find the derivative of the function .
a.
b.
c.
d.
e.
116 Chapter 2: Differentiation
____ 9. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 10. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 11. Find the derivative of the function.
a.
b.
c.
d.
e.
2.4 The Chain Rule 117
____ 12. Find the derivative of the function.
a.
b.
c.
d.
e.
____ 13. Evaluate the derivative of the function at the point .
a.
b.
c.
d.
e.
____ 14. Evaluate the derivative of the function at the given point.
,
a.
b.
c.
d.
e.
118 Chapter 2: Differentiation
____ 15. Evaluate the derivative of the function at the point .
a.
b.
c.
d.
e.
____ 16. Evaluate the derivative of the function at the point .
a.
b.
c.
d.
e.
____ 17. Find an equation to the tangent line for the graph of f at the given point.
,
a.
b.
c.
d.
e.
____ 18. Find an equation to the tangent line to the graph of the function at the
point . The coefficients below are given to two decimal places.
a.
b.
c.
d.
e.
2.4 The Chain Rule 119
____ 19. Find the second derivative of the function.
a.
b.
c.
d.
e.
____ 20. Find the second derivative of the function .
a.
b.
c.
d.
e.
____ 21. The displacement from equilibrium of an object in harmonic motion on the end of a
spring is where y is measured in feet and t is the time in seconds. Determine
the position of the object when . Round your answer to two decimal places.
a. 0.12 feet
b. 0.21 feet
c. 17.53 feet
d. 0.46 feet
e. 0.11 feet
____ 22. The displacement from equilibrium of an object in harmonic motion on the end of a
spring is where y is measured in feet and t is the time in seconds. Determine
the velocity of the object when . Round your answer to two decimal places.
a. 8.00 ft/sec
b. 4.68 ft/sec
c. 6.00 ft/sec
d. 1.00 ft/sec
e. 5.00 ft/sec
120 Chapter 2: Differentiation
____ 23. Suppose a 15-centimeter pendulum moves according to the equation
where is the angular displacement from the vertical in radians and t is the time in seconds.
Determine the rate of change of when t = 7 seconds. Round your answer to four decimal places.
a. 2.5034 radians per second
b. 3.6185 radians per second
c. 0.3129 radians per second
d. 3.1535 radians per second
e. 4.1724 radians per second
____ 24. A buoy oscillates in simple harmonic motion as waves move past it. The
buoy moves a total of 3.5 feet (vertically) between its low point and its high point. It returns to its
high point every 14 seconds. Write an equation describing the motion of the buoy if it is at its high
point at t = 0.
a.
b.
c.
d.
e.
____ 25. A buoy oscillates in simple harmonic motion as waves move past it. The
buoy moves a total of 13.5 feet (vertically) between its low point and its high point. It returns to its
high point every 10 seconds. Determine the velocity of the buoy as a function of t.
a.
b.
c.
d.
e.
2.4 The Chain Rule 121
2.4 The Chain Rule
Answer Section
122 Chapter 2: Differentiation