   Chapter 4.5, Problem 23E ### 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 = x 2 + x − 2

To determine

To sketch: The curve by plotting x and y coordinates using guidelines.

Explanation

Given:

The given curve is y=x2+x2 (1)

Calculation:

Rewrite equation (1) as follows.

y=f(x)=x2+x2=(x+2)(x1)

Calculate the domain of the function.

Consider the equation (x+2)(x1)=0 and simplify it as follows.

x+2=0x=2x1=0x=1

Therefore, the domain is (,2) and (1,).

Calculate the value of x-intercept.

Take the value of variable y=0.

y=(x+2)(x1)0=(x+2)(x1)(x+2)(x1)=0x=2 and 1

Therefore, the x-intercept is x=2 and x=1.

Calculate the y-intercept.

Take the value of variable x=0.

y=(0+2)(01)=(2)(1)=2

Since the value of y is y=2, the function has no y-intercept.

Calculate the symmetry.

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

Substitute 1 for x in the equation (1).

f(x)=x2+x2f(1)=(12)+(1)2=2

Substitute 1 for x in the equation (1).

f(x)=x2+x2f(1)=(12)+12=0

Hence, the condition f(x)f(x) is true and there is no symmetry.

Calculate asymptotes.

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

limx(x2+x2)12=(2+2)12=

Therefore, the value of limit when x tends to infinity is infinity.

So there is no horizontal asymptote.

The function f starts at a point and goes up to infinity and so there is no vertical asymptote.

Hence, there are no asymptotes for the given equation.

Differentiate equation (2) as below.

f(x)=(x2+x2)12f'(x)=12(x2+x2)12.(2x+1)f'(x)=2x+12(x2+x2)

If the condition f'(x)=0 is true when x=12, but the domain range value is not 12.

So, the condition f'(x)>0 is true, when the value of x>12 and f'(x)<0 when the value of x<12.

Therefore, the function f is increasing on range coordinate (1,) and f is decreasing on range coordinate (,2)

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 