   Chapter 4.5, Problem 52E ### 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 = ln x x 2

To determine

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

Explanation

Given:

The given curve is y=lnxx2 (1)

Calculation:

(a)

Calculate the domain.

The domain of the real number should be greater than zero, so the value of x will be having only positive numbers. Therefore, the domain is (x:x>0) .

(b)

Calculate the intercepts.

Here, the function lnx cannot be zero, there is no y intercept.

Substitute 0 for y in the equation (1).

y=lnxx20=lnxx2

Here, the value of lnx is equal to zero, when the condition x=1 is true. Therefore, there is no intercept for y and x intercept at x=1 .

(c)

Calculate the symmetry.

Here, the function of lnxx2 cannot be neither even nor odd function.

(d)

Calculate asymptotes.

The variable y should have a vertical asymptote of x=0 because of x2 in the denominator. Therefore, the vertical asymptote is x=0 .

(e)

Calculate the intervals.

Differentiate the equation (1).

f(x)=lnxx2f'(x)=x2(1x)lnx(2x)(x2)2=x2xlnxx4=x(12lnx)x4

f'(x)=12lnxx3 (2)

Substitute 0 for f'(x) in the equation (2).

f'(x)=12lnxx30=12lnxx312lnx=02lnx=1lnx=12

Use log rule.

lnx=12logex=12x=e12x=e

The local maxima will occur at x=e .

Therefore, the function f'(x) is increasing on (0,e) and decreasing on (e,) .

(f)

Calculate the concavity.

Differentiate the equation (2) with respect to x by applying U/V method

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