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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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Chapter
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 = x4 − 4x

To determine

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

Explanation

Given:

The Cartesian equation is as below.

y=x44x (1)

Calculation:

a)

Calculate the domain.

From the equation (1), the function is defined for all real values of x (there is no restriction on the value of x). Hence, the domain of f(x) is all real values of x .

For the polynomial, the domain is (,) .

b)

Calculate the intercepts.

Calculate the value of x-intercept.

Substitute 0 for y in the equation (1).

x44x=0x(x34)=0x34=0x3=4x=1.587

Hence, x -intercept is (0,0) and (1.587,0) .

Calculate the y-intercept.

Substitute 0 for x in the equation (1).

y=(0)44(0)=0

Therefore, the y -intercept is (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(1)=(1)44(1)=1+4=5

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

f(1)=(1)44(1)=14=3

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

d)

Calculate asymptotes.

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

limxy=

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

limxy=

Here, the value of limit gets infinity; this implies there is no horizontal asymptote or vertical asymptote.

e)

Calculate the intervals.

Differentiate the equation (1) with respect to x .

f'(x)=4x34 (2)

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

4x34=04x3=4x3=1x=1

Take the interval as (,1) .

Substitute 0 for x in the equation (2).

f'(0)=4(0)34=4

Here, the condition f'(0)<0 is true. Thus, function f is decreasing in the interval (,1) .

Take the interval as (1,) .

Substitute 2 for x in the equation (12)

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