   Chapter 4.5, Problem 53E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Use the guidelines of this section to sketch the curve.y = earctan x

To determine

To Find: The curve by plotting x and y coordinates using guideline.

Explanation

Given:

The given curve is as below.

y=earctanx (1)

Calculation:

(a)

Calculate the domain.

The function contain real numbers, therefore the domain for the function f(x) is (,) .

(b)

Calculate the intercepts.

Substitute 0 for x in the equation (1).

y=earctanxy=earctan(0)y=1

Therefore, there is no intercept of x and y -intercept is y=1 .

(c)

Calculate the symmetry.

Apply negative and positive values for x in the equation (1).

Substitute -1 for x in the equation (1).

f(x)=earctanxf(1)=earctan(1)=0.456

Substitute +1 for x in the equation (1).

f(x)=earctanxf(1)=earctan(1)=2.193

Therefore, the conditions for symmetry are not satisfied.

f(x)f(x)

f(x)-f(x)

There is no symmetry about the y axis and the origin.

(d)

Find the asymptote equations.

Substitute ± for x using limit in the equation (1).

limxf(x)=limxearctanx=earctan()=eπ2=4.81limxf(x)=limxearctanx=earctan()=eπ2=0.208

Therefore, the horizontal asymptotes are y=eπ2 and y=eπ2 .

(e)

Calculate the intervals.

Differentiate the equation (1).

f(x)=earctanxf'(x)=earctanx(11+x2)>0

Therefore, the function will get positive values for all values of x, and hence the function f will keep on increasing on domain (,) .

(f)

Calculate the local minimum and maximum values.

The local maxima will not occur for the function inside the interval of (,) . Therefore, there is no local extrema.

(g)

Calculate the concavity and point of inflection.

Differentiate the equation f'(x)=earctanx.(11+x2) with respect to x by applying U/V method

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