   Chapter 4.6, Problem 34E ### 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

# Describe how the graph of f varies as c varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when c changes. You should also identify any transitional values of c at which the basic shape of the curve changes. f ( x ) = sin x c + cos x

To determine

To describe: How the graph of f varies as c varies.

Explanation

Given information:

The function is f(x)=sinxc+cosx

Calculation:

Show the function as follows:

f(x)=sinxc+cosx (1)

Differentiate both sides of the Equation (1).

f(x)=(c+cosx)cosxsinx(sinx)(c+cosx)2=ccosx+cos2x+sin2xc2+cos2x+2ccosx=1+ccosxc2+cos2x+2ccosx

Differentiate both sides of the Equation.

f(x)=(c2+cos2x+2ccosx)(csinx)(1+ccosx)(2cosx(sinx)+2c(sinx))(c2+cos2x+2ccosx)2=sinx(c3+ccos2x+2c2cosx)+(1+ccosx)(2cosxsinx+2csinx)(c2+cos2x+2ccosx)2=(1+ccosx)(2cosxsinx+2csinx)(c2+cos2x+2ccosx)2sinx(c2+cos2x+2ccosx)=sinx(ccosxc2+2)(c2+cos2x+2ccosx)3

Consider the function is an odd function on the interval (0,2π) . So, the graph is symmetric about the origin.

Sketch the function for |c|1 as shown in Figure 1.

Refer to Figure 1.

The function f has VAs when the denominator is zero, that is at x=cos1(c) and x=2πcos1(c) .

If the value of c=1 , then there are VAs at x=0 and x=2π .

If the value of c increases, it move closer to x=π , which is the single VA when c=1 .

If c=0 , then f(x)=tanx . There are no extreme points and inflection points occur at multiples of π .

Sketch the function for c>1 as shown in Figure 2.

Equating the derivative of the function is equal to zero.

f(x)=01+ccosxc2+cos2x+2ccosx=01+ccosx=0x=cos1(1c)

Now the VA disappears and there is a local maximum and a local minimum

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