280 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
5.1 The Natural Logarithmic Function: Differentiation
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. What is the domain of the function ?
a.
b.
c.
d.
e.
____ 2. What is the domain of the function ?
a.
b.
c.
d.
e.
____ 3. Write the following expression as a logarithm of a single quantity.
a.
b.
c.
d.
e.
5.1 The Natural Logarithmic Function: Differentiation 281
____ 4. Write the following expression as a logarithm of a single quantity.
a.
b.
c.
d.
e.
____ 5. Find the following limit if it exists: . Use when appropriate.
a.
b.
c.
d.
e.
____ 6. Find the following limit if it exists: . Use when appropriate.
a.
b.
c.
d.
e.
____ 7. Find an equation of the tangent line to the graph of at the point (1,0).
a.
b.
c.
d.
e.
282 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
____ 8. Differentiate the function .
a.
b.
c.
d.
e.
____ 9. Differentiate the function .
a.
b.
c.
d.
e.
____ 10. Find the derivative of the function .
a.
b.
c.
d.
e.
5.1 The Natural Logarithmic Function: Differentiation 283
____ 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 function .
a.
b.
c.
d.
e.
284 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
____ 14. Find the slope-intercept equation of the line tangent to the graph of when
.
a.
b.
c.
d.
e.
____ 15. Use implicit differentiation to find .
a.
b.
c.
d.
e.
____ 16. Use implicit differentiation to find .
a.
b.
c.
5.1 The Natural Logarithmic Function: Differentiation 285
d.
e.
____ 17. Use implicit differentiation to find at the point .
a.
b.
c.
d.
e.
____ 18. Locate any relative extrema and inflection points of the function . Use
a graphing utility to confirm your results.
a. relative minimum value: ; no inflection points
b. relative minimum value: inflection point:
c. relative maximum value: inflection point:
d. relative minimum value: no inflection points
e. relative maximum value: ; no inflection points
____ 19. Determine the x-coordinate(s) of any relative extrema and inflection points of the
function .
a.
relative minimum: ; no inflection points
b.
relative maximum: ; inflection point:
c.
no relative maximum or minimum; inflection point:
286 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
d. no relative extrema or inflection points.
e.
relative minimum: ; inflection point:
____ 20. Determine the x-coordinate(s) of any relative extrema and inflection points of the
function .
a.
relative minimum: ; inflection point:
b.
no relative extrema; inflection point:
c.
relative minimum: ; no inflection points
d.
no relative extrema; inflection point:
e.
relative maximum: inflection point:
____ 21. Use logarithmic differentiation to find the derivative of .
a.
b.
c.
d.
e.
5.1 The Natural Logarithmic Function: Differentiation 287
a. $
b. $
c. $
d. $
e. $
____ 23. Suppose the term of a mortgage t (in years) can be
approximated by where x is the monthly payment in dollars. Find
the instantaneous rate of change of t with respect to x when . Round your answer to four
decimal places.
a. 0.0044
b. –0.1619
c. 0.0619
d. –0.0619
e. –0.0044
____ 24. The relationship between the number of decibels and the intensity I of sound in
watts per centimeter squared is . Determine the number of decibels of a sound
with an intensity of watts per square centimeter.
a. 250 decibels
b. 65 decibels
c. 70 decibels
d. decibels
e. 75 decibels
____ 25. Suppose the table below shows the temperature at which a given liquid boils
at selected pressures p (pounds per square inch). A model that approximates the data is
. Find the rate of change of T with respect to p when P = 20. Round
your answer to two decimal places.
P 5 10 14.696
(1 atm)
20 30 40 60 80 100
T 234.68 288.88 320.55 346.89 383.19 410.28 450.7 481.26 506.22
____ 22. Suppose the term of a mortgage t (in years) can be
approximated by where x is the monthly payment in dollars. Use
the model to approximate the term of a home mortgage for which the monthly payment is $ .
What is the total amount paid? Round your answer to the nearest cent. Your answer may be slightly
different depending on the number of digits your technology uses in computations.
288 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
c.
d.
e.
____ 26. Suppose the table below shows the temperature at which a given liquid boils at
selected pressures p (pounds per square inch). A model that approximates the data is
. Use a graphing utility to graph .
P 5 10 14.696(1 atm) 20 30 40 60 80 100
T
234.68
288.88
320.55
346.89
383.19
410.28
450.7
481.26
506.22
a. d.
b. e.
a.
b.
5.1 The Natural Logarithmic Function: Differentiation 289
c.
____ 27. Suppose the table below shows the temperature at which a given liquid boils at
selected pressures p (pounds per square inch). A model that approximates the data is
. Find .
P 5 10 14.696
(1 atm)
20 30 40 60 80 100
T
204.16
236.64
256.08
272.55
295.68
313.29
340.12
360.86
378.07
a. 1
b. 0
c.
d.
e.
290 Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions
5.1 The Natural Logarithmic Function: Differentiation
Answer Section
5.1 The Natural Logarithmic Function: Differentiation 291