   Chapter 4.5, Problem 24E ### 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 − x

To determine

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

Explanation

Given:

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

Calculation:

Rewrite the equation (1).

y=f(x)=x2+xxy=f(x)=x(x+1)x

(a)

Calculate the domain.

Assume the value inside the radical is greater than or equal to zero.

x2+x=0x(x+1)=0x=0x+1=0x=1

Therefore, the given equation is an upward opening parabola. So, the domain is (,1) and (0,)

(b)

Calculate the intercepts.

Calculate the value of x-intercept.

Take y=0 then,

y=x2+xx0=x2+xxx2+x=xx2+x=x2x2+xx2=0x=0

Therefore, the only x -intercept is same at (0,0)

(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)=x2+xxf(1)=(12)+(1)(1)=1

Substitute +1 for x in the equation (1)

f(x)=x2+xxf(1)=(12)+1(1)=0.414

Hence, f(x)f(x) so, there is no symmetry.

(d)

Calculate asymptotes.

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

limxf(x)=limx(x2+xx)x2+x+xx2+x+x=limxx2+xx2x2+x+x=limxxx2+x+x=limxxxx2+x+xx=limx1x2+xx2+xx=limx11+1x+1=11+1+1=11+0+1=12

Therefore, the asymptote is y=12 which is in horizontal axis. So, no vertical axis exists.

(e)

Calculate the intervals.

Differentiate the equation (1).

f(x)=(x2+x)12xf'(x)=12(x2+x)12.(2x+1)1f'(x)=2x+12(x2+x)1f'(x)=2x+12(x2+x)2(x2+x)>0

Consider the numerator and equate f'(x) is equal to zero.

2x+12(x2+x)=02(x2+x)=2x+1Take square on both sides,(2(x2+x))2=(2x+1)24(x2+x)=4x2+1+4x4x2+4x4x214x=01=0

Therefore, numerator is equal to zero.

Consider the denominator and equate f'(x) is equal to zero

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### 64x8y36

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

#### limx(lnx)1/x= a) 0 b) 1 c) e d)

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

#### Define the concept of internal validity and a threat to internal validity.

Research Methods for the Behavioral Sciences (MindTap Course List) 