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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 − 8x2 + 8

To determine

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

Explanation

Given:

The Cartesian equation is as below.

y=x48x2+8 (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).

x48x2+8=0

Substitute u for x2 in the above equation.

u28u+8=0

Calculate the value of u using the quadratic formula.

u=b±b24ac2a

Substitute 1 for a , -8 for b and 8 for c in the above equation.

u=(8)±(8)24(1)(8)2(1)=8±64322=8±5.6572=8+5.6572,85.6572u=6.828,1.172

Substitute x2 for u in the above equation.

x2=6.828,1.172x=2.613,1.082

Hence, x -intercept are (2.613,0) and (1.082,0) .

Calculate the y-intercept.

Substitute 0 for x in the equation (1).

f(0)=(0)48(0)2+8=8

Therefore, the y -intercept is (0,8) .

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)48(1)2+8=18+8=1

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

f(1)=(1)48(1)2+8=18+8=1

Here, the function f(x)=f(x) is true, it is an even function. So it has y-axis 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)=4x316x (2)

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

4x316x=04x(x24)=0x24=0x2=4x=±2,0

Take the interval as (,2) .

Substitute -3 for x in the equation (2).

f'(3)=4(3)316(3)=108+48=60

Here f'(3)<0 , thus function f is decreasing in the interval (,2) .

Take the interval as (2,0) .

Substitute -1 for x in the equation (2).

f'(1)=4(1)316(1)=4+16=12

Here f'(1)>0 , thus the function f is increasing in the interval (2,0) .

Take the interval as (0,2) .

Substitute 1 for x in the equation (2).

f'(1)=4(1)316(1)=416=12

Here f'(1)<0 , thus the function f is decreasing in the interval (0,2) .

Take the interval as (2,)

f'(3)=4(3)316(3)=10848=60

Here f'(3)>0 , thus the function f is increasing in the interval (2,)

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