# Mathematics

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22.–/1 pointsLarCalc9 5.1.042.

Find the limit. (If you need to use or – , enter INFINITY or –INFINITY, respectively. Round your answer to four decimal places.)

23.–/1 pointsLarCalc9 5.1.043.MI.

Find an equation of the tangent line to the graph of the logarithmic function at the point (1, 0).

y = ln x3

y =

24.–/11 pointsLarCalc9 5.1.043.MI.SA.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise Find an equation of the tangent line to the graph of the logarithmic function at the point (1, 0).

y = ln x6

lnlim x

x − 7x→8+

6/10/16, 11:46 PMSec 5.1 HW Assignment

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25.–/1 pointsLarCalc9 5.1.047.

Find the derivative of the function.

f(x) = ln(8x) f ‘(x) =

26.–/1 pointsLarCalc9 5.1.052.MI.

Find the derivative of the function.

f ‘(x)=

27.–/11 pointsLarCalc9 5.1.052.MI.SA.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise Find the derivative of the function.

f(x) = 8×2 ln 8x

f(x) = 5×2 ln 5x

6/10/16, 11:46 PMSec 5.1 HW Assignment

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28.–/1 pointsLarCalc9 5.1.055.

Find the derivative of the function.

y = ln(x x2 – 9) y’ =

29.–/1 pointsLarCalc9 5.1.058.MI.

Find the derivative of the function.

f ‘(x) =

30.–/9 pointsLarCalc9 5.1.058.MI.SA.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise Find the derivative of the function.

f(x) = ln 6x x + 9

f(x) = ln 3x x + 6

6/10/16, 11:46 PMSec 5.1 HW Assignment

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31.–/1 pointsLarCalc9 5.1.063.

Find the derivative of the function.

y’ =

32.–/1 pointsLarCalc9 5.1.064.MI.

Find the derivative of the function.

y’ =

y = ln x + 5 x − 5

y = ln 4 x − 1

x + 1

6/10/16, 11:46 PMSec 5.1 HW Assignment

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33.–/14 pointsLarCalc9 5.1.064.MI.SA.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise Find the derivative of the function.

34.–/1 pointsLarCalc9 5.1.066.

Find the derivative of the function.

f ‘(x) =

35.–/1 pointsLarCalc9 5.1.070.

Find the derivative of the function.

y'(x) =

y = ln 9 x − 6 x + 6

f(x) = ln(x + )3 + x2

y = ln(|csc(3x)|)

6/10/16, 11:46 PMSec 5.1 HW Assignment

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36.–/1 pointsLarCalc9 5.1.074.MI.

Find the derivative of the function.

f ‘(x) =

37.–/2 pointsLarCalc9 5.1.077.

Consider the following.

(a) Find an equation of the tangent line to the graph of f at the given point. y =

(b) Use a graphing utility to graph the function and its tangent line at the point.

f(x) = ln 5 + cos2 x

y = 8×2 − ln(x), (1, 8)

6/10/16, 11:46 PMSec 5.1 HW Assignment

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(c) Use the derivative feature of a graphing utility to confirm your results.

38.–/1 pointsLarCalc9 5.1.084.MI.

Use implicit differentiation to find

=

.dy dx

ln xy + 6x = 45 dy dx

6/10/16, 11:46 PMSec 5.1 HW Assignment

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39.–/8 pointsLarCalc9 5.1.084.MI.SA.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise

Use implicit differentiation to find

40.–/1 pointsLarCalc9 5.1.087.

Use implicit differentiation to find an equation of the tangent line to the graph at the given point.

y =

.dy dx

ln xy + 5x = 20

x + y − 1 = ln(x4 + y10), (1, 0)

6/10/16, 11:46 PMSec 5.1 HW Assignment

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41.–/2 pointsLarCalc9 5.1.096.MI.

Locate the relative extremum and inflection points. Use a graphing utility to confirm your results.

relative extremum

(x, y) = (

)

inflection point

(x, y) = (

)

42.–/1 pointsLarCalc9 5.1.101.

Use logarithmic differentiation to find dy/dx.

y = 3×2 ln x 4

y = x , x > 0x2 + 42

=dy dx

6/10/16, 11:46 PMSec 5.1 HW Assignment

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43.–/1 pointsLarCalc9 5.1.106.MI.

Use logarithmic differentiation to find dy/dx.

44.–/7 pointsLarCalc9 5.1.118.

The atmospheric pressure decreases with increasing altitude. At sea level, the average air pressure is one atmosphere (1.033227 kilograms per square centimeter). The table shows the pressures p (in atmospheres) at selected altitudes h (in kilometers).

h 0 5 10 15 20 25

p 1 0.55 0.25 0.12 0.06 0.02

(a) Use a graphic utility to find a model of the form (If an answer does not exist, enter DNE. Round your coefficients to four decimal places.) p =

(b) Use a graphing utility to find the logarithmic model for the data. (If an answer does not exist, enter DNE. Round your coefficients to four decimal places.) h =

km

(c) Use the model to estimate the altitude when p = 0.65. (Round your answer to two decimal places.)

y = , x > 2(x + 1)(x − 2) (x − 1)(x + 2)

=dy dx

p = a + b ln(h) for the data.

h = a + b ln(p)

6/10/16, 11:46 PMSec 5.1 HW Assignment

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h = km

(d) Use the model to estimate the pressure when h = 9. (Round your answer to two decimal places.) p = atm

(e) Use the model to find the rates of change of pressure when h = 4 and h = 20. Interpret the results. (Round your answers to four decimal places.)

As the altitude increases, the rate of change of pressure —Select— .

45.–/3 pointsLarCalc9 5.1.119.

A person walking along a dock drags a boat by a 10-meter rope. The boat travels along a path known as tractrix (see figure). The equation of this path is given below.

(a) What are the slopes of this path when x = 5 and x = 7? (Round your answers to three decimal places.) y'(5) = y'(7) =

(b) What does the slope of the path approach as x → 10?

h = 4:

h = 20:

y = 10 ln − 10 +

x 100 − x2

100 − x2

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