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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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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 = 1 x 2 4

To determine

To find: sketch the curve using the equation y=1x24 .

Explanation

Given:

The equation of the curve is,

y=1x24 (1).

Calculation:

Rewrite the equation (1).

f(x)=1x24f(x)=1(x2)2

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

A)

Compute the domain.

Considering the denominator equation (1)

x24=0x=±2

Therefore, domains are (,2) , (2,2) and (2,) .

B)

Compute y intercepts.

Substitute the value 0 for x in equation (1).

y=1024y=14

Compute x intercepts.

Substitute the value 0 for f(x) in the equation.

x=0

Therefore, y intercepts and x intercepts are (0,14) .

C)

Compute the symmetric:

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

Substitute the 1 for x .

f(x)=1x24f(1)=1(1)24f(1)=13

Substitute the 1 for x .

f(x)=xx24f(1)=1(1)24f(1)=13

f(x)=f(x)

so, the function is even.

Therefore, the curve is symmetric about the y axis .

D)

Compute the horizontal asymptote:

limx±xx24=0

Therefore, the horizontal asymptote y is 0 .

Compute the vertical asymptote:

limx2xx24=

limx2+xx24=

limx2+xx24=

limx2xx24=

Therefore, the vertical asymptote x is ±2 .

E)

Compute the interval of increase or decrease.

Apply UV method to differentiate the equation (2)

f'(x)=vu'uv'v2f'(x)=(x24)(0)(1)(2x)(x24)2f'(x)=2x(x24)2<0

The equation has negative sign before, so, it is always negative.

Using reciprocal rule

The inequality got reversed.

f'(x)=2x(x24)2>0

Then, the function is decreasing on (0,2) and (2,) and increasing on (,2) and (2,0) .

F)

Find the local maximum and minimum values.

For maxima

If

f(x)0 then, the ,minima will occurs.

If

f(x)0 then, the maxima will occurs.

Substitute the 0 for f'(x) in equation (1)

f(0)=1024f(0)=14

In the curve there is no minima because the lowest limit are   and the maxima value is 14

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