The natural log function and integration homework answers

Calculus Homework

Use a graphing utility to evaluate the logarithm by the methods given below. (Round your answers to four decimal places.)

ln(24)

(a) Use the natural logarithm key. 3.1780

(b) Use the integration capabilities to evaluate the integral .

3.1780

2. –/2 pointsLarCalc9 5.1.006.

Use a graphing utility to evaluate the logarithm by the methods given below. (Round your answers to four decimal places.)

ln(0.30)

(a) Use the natural logarithm key.

(b) Use the integration capabilities to evaluate the integral .

Sec 5.1 HW Assignment (Homework)

ahmed alarbash MATH 21414, section 13539, Summer 1 2016 Instructor: Mahmoud Basharat

WebAssign

dt 24 1

t1

dt 0.30 1

t1

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3. 1/1 points | Previous AnswersLarCalc9 5.1.008.

Match the function with its graph.

f(x) = –ln(x/6)

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4. 1/1 points | Previous AnswersLarCalc9 5.1.010.

Match the function with its graph.

f(x) = –ln(x – 9)

5. 2/2 points | Previous AnswersLarCalc9 5.1.011.

Sketch the graph of the function.

f(x) = 8 ln(x)

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

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State the domain of the function.

−∞ < x ≤ 0

−∞ < x < 0

0 ≤ x ≤ ∞

0 < x < ∞

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6. 2/2 points | Previous AnswersLarCalc9 5.1.014.

Sketch the graph of the function.

State the domain of the function.

f(x) = ln(|x|)

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

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7. 2/2 points | Previous AnswersLarCalc9 5.1.016.

Sketch the graph of the function.

f(x) = 4 ln(x) + 2

x < 0

all reals

x ≠ 0

x > 0

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

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State the domain of the function.

−∞ < x < 0

0 ≤ x < ∞

x > 0

x > -1

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8. 3/4 points | Previous AnswersLarCalc9 5.1.020.MI.

Use the properties of logarithms, given that ln 2 ≈ 0.6931 and ln 3 ≈ 1.0986, to approximate the logarithm. Use a calculator to confirm your approximations.

(a) ln 0.25 ≈ -1.3862

(b) ln 24 ≈ 3.1780

(c) ≈ 0.8283

(d) ≈ -4.2766

9. 0/1 points | Previous AnswersLarCalc9 5.1.021.

Use the properties of logarithms to expand the logarithmic expression.

1x

ln 12 3

ln 1 72

ln x 49

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10.–/1 pointsLarCalc9 5.1.023.

Use the properties of logarithms to expand the logarithmic expression.

11.–/1 pointsLarCalc9 5.1.026.MI.

Use the properties of logarithms to expand the logarithmic expression.

12.–/3 pointsLarCalc9 5.1.026.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 the properties of logarithms to expand the logarithmic expression.

ln xa w

ln a − 1

ln a − 26

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13.–/1 pointsLarCalc9 5.1.027.

Use the properties of logarithms to expand the logarithmic expression.

14.–/1 pointsLarCalc9 5.1.031.

Write the expression as a logarithm of a single quantity.

15.–/1 pointsLarCalc9 5.1.033.

Write the expression as the logarithm of a single quantity.

ln x − 78

x

ln(x − 88) − ln(x + 86)

4 ln(x + 7) + ln(x) − ln(x2 − 8) 1 3

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16.–/1 pointsLarCalc9 5.1.034.MI.

Write the expression as a logarithm of a single quantity.

17.–/10 pointsLarCalc9 5.1.034.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 Write the expression as a logarithm of a single quantity.

18.–/1 pointsLarCalc9 5.1.035.

Write the expression as the logarithm of a single quantity.

2 [ln x − ln(x + 3) − ln(x − 3)]

7 [ln x − ln(x + 4) − ln(x − 4)]

3 ln(2) − ln(x2 + 2)1 2

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19.–/1 pointsLarCalc9 5.1.036.

Write the expression as a logarithm of a single quantity.

20.–/1 pointsLarCalc9 5.1.039.

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

21.–/1 pointsLarCalc9 5.1.041.

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

ln(x2 + 5) − ln(x + 8) − ln(x − 8) 7 2

ln(x − 9)lim x→9+

ln x2(6 − x)lim x→5−

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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+

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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) = 8x2 ln 8x

f(x) = 5x2 ln 5x

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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

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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

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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)|)

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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 = 8x2 − ln(x), (1, 8)

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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

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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)

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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 = 3x2 ln x 4

y = x , x > 0x2 + 42

=dy dx

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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)

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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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