310 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
5.4 Exponential Functions: Differentiation and Integration
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Solve the following equation for .
a.
b.
c.
d.
e.
____ 2. Solve the following equation for .
a.
b.
c.
d.
e.
____ 3. Solve the following equation for .
a.
b.
c.
d.
e.
5.4 Exponential Functions: Differentiation and Integration 311
____ 4. Solve the following equation for .
a. x
8
b.
c.
d.
e.
____ 5. Find an equation of the tangent line to the graph of at the point .
a.
b.
c.
d.
e.
____ 6. Differentiate the function .
a.
b.
c.
d.
e.
____ 7. Find if .
a.
b.
c.
d.
e.
312 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
____ 8. Find the derivative of the function .
a.
b.
c.
d.
e.
____ 9. Find
if .
a.
b.
c.
d.
e.
____ 10. Differentiate the function .
a.
b.
c.
d.
e.
5.4 Exponential Functions: Differentiation and Integration 313
____ 11. Find the derivative of the function . Simplify your answer.
a.
b.
c.
d.
e.
____ 12. Find the derivative of the function . Simplify your answer.
a.
b.
c.
d.
e.
____ 13. Find if
a.
b.
c.
d.
e.
314 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
____ 14. Use implicit differentiation to find
a.
b.
c.
d.
e.
____ 15. Find if .
a.
b.
c.
d.
e.
____ 16. Find all points of inflection on the graph of the function .
a.
b.
c.
d.
e.
5.4 Exponential Functions: Differentiation and Integration 315
____ 17. The value V (in dollars) of an item t years after it is purchased is V = 10,000e -0.5286t,
0 t 10. Find the rate of change of V with respect to t when . Round your answer to two
decimal places.
a. –3,115.72 dollars/year
b. –1,944.61 dollars/year
c. 3,115.72 dollars/year
d. –1,836.5 dollars/year
e. 1,944.61 dollars/year
____ 18. A meteorologist measures the atmospheric pressure P (in kilograms per square meter)
at altitude h (in kilometers). The data are shown below. Use the regression capabilities of the
graphing utility to find a linear model for the revised data points obtained by plotting the points
a.
b.
c.
d.
e.
____ 19. A meteorologist measures the atmospheric pressure P (in kilograms per square meter)
at altitude h (in kilometers). The data are shown below. Find the rate of change of the pressure with
respect to altitude when using the relation . Round your answer to
one decimal place.
a.
b.
c.
d.
e.
316 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
____ 20. The table lists the approximate value V of a mid-sized sedan for the years 2003
through 2009. The variable t represents the time in years, with corresponding to 2003. Use the
regression capabilities of a graphing utility to fit an exponential model to the data. Round all
numerical values in your answer to four decimal places. Your answer may be slightly different
depending on the number of digits your graphing utility uses in computations.
a.
b.
c.
d.
e.
____ 21. The table lists the approximate value V of a mid-sized sedan for the years 2003
through 2009. The variable t represents the time in years, with corresponding to 2003. Find the
rate of change in the value of the sedan when using the exponential model
. Round your answer to two decimal places.
a.
b.
c.
d.
e.
____ 22. Find the indefinite integral.
a.
b.
c.
d.
e.
5.4 Exponential Functions: Differentiation and Integration 317
____ 23. Find the indefinite integral.
a.
b.
c.
d.
e.
____ 24. Evaluate the definite integral.
a.
b.
c.
d.
e.
318 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
____ 25. Evaluate the definite integral.
a.
b.
c.
d.
e.
____ 26. Find the indefinite integral
a.
b.
1
32
c.
1
32
d.
e.
1
32
____ 27. A valve on a storage tank is opened for 4 hours to release a chemical in a
manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given by the linear
model . Write the linear model in exponential form.
a.
b.
c.
d.
e.
____ 28. A valve on a storage tank is opened for 3 hours to release a chemical in a
manufacturing process. The flow rate R (in liters per hour) at time t (in hours) is given by the
exponential model Use the definite integral to approximate the number of
liters of chemical released during the first 3 hours. Round your answer to two decimal places.
a. 1,594.86 liters
b. 1,222.68 liters
c. 1,887.64 liters
d. 1,732.19 liters
e. 1,340.70 liters
5.4 Exponential Functions: Differentiation and Integration 319
5.4 Exponential Functions: Differentiation and Integration
Answer Section
320 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions