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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Section
BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Use the guidelines of this section to sketch the curve.

y = arctan(ex)

To determine

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

Explanation

Given:

The given curve is

f(x)=y=arctan(ex)

f(x)=y=tan1(ex) (1)

Calculation:

(a)

Calculate the domain.

For inverse of tangent is real numbers for the domain.

So the domain is (,)

(b)

Calculate the intercepts.

Calculate the y –intercept.

Substitute 0 for the value of x in equation (1).

f(x)=tan1(ex)

f(0)=tan1(e0)=π4

The y –intercept: y=π4

Calculate the value of x-intercept.

Substitute 0 for y in the equation (1).

tan1(ex)=0x=0

The x –intercept: x=0

(c)

Calculate the symmetry.

Substitute x for x in the equation (1).

f(x)=tan1(ex)f(x)f(x)

So the curve does not have symmetry.

(d)

Calculate asymptotes.

Apply limit of x tends to (x) in the equation (1).

limxf(x)=tan1(ex)=tan1(e)=

Apply limit of x tends to (x) in the equation (1).

limxf(x)=tan1(ex)=tan1(e)=

So there is no horizontal and vertical asymptote.

(e)

Calculate the intervals.

f(x)=tan1(ex)

Differentiate the equation (1) with respect to x .

ddxtan1(x)=11+x2

f'(x)=11+(ex)2×ex=11+e2x×exf'(x)=ex1+e2x

So equate f'(x) to zero

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