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Math
CalculusSingle Variable Calculus: Early Transcendentals, Volume I8th Edition

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Chapter 4.6, Problem 34E
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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 9E
Problem 10E
Problem 13E
Problem 14E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 33E
Problem 34E
Problem 35E
Problem 38E
Problem 39E
Problem 40E
BuyFindarrow_forward

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)cosx−sinx(−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)2−sinx(c2+cos2x+2ccosx)=sinx(ccosx−c2+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=cos−1(−c) and x=2π−cos−1(−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=cos−1(−1c)

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

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Chapter 4.6,
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In Review Exercise 27 to 30, supply the missing coordinates for the vertices, using as few variables as possibl...

Elementary Geometry For College Students, 7e

Calculate the following trade discounts and trade discount rates. Round answers to the nearest tenth of a perce...

Contemporary Mathematics for Business & Consumers

Label each of the following statements as either true or false. Every decimal representation of a real number t...

Elements Of Modern Algebra

Each of the following problems refers to triangle ABC. In each case, find the area of the triangle. Round to th...

Trigonometry (MindTap Course List)

Solve each equation using the quadratic formula (when necessary, round result to three significant digits): 2x2...

Elementary Technical Mathematics

Radians and Degrees The fundamental limit limxsinxx=1 assumes that x is measured in radians. Suppose you assume...

Calculus: Early Transcendental Functions (MindTap Course List)

Using Arc Length The graph of y=f(x) passes through the origin. The arc length of the curve from (0,0) to (x,f(...

Calculus of a Single Variable

In Exercises 1 to 8, find the perimeter of each polygon. ABCD with ABBC d1=4md2=10m

Elementary Geometry for College Students

TRAFFIC CONTROL The accompanying figure shows the flow of downtown traffic during the rush hours on a typical w...

Finite Mathematics for the Managerial, Life, and Social Sciences

Using the Second Fundamental Theorem of Calculus In Exercises 75-80, use the Second Fundamental Theorem of Calc...

Calculus

The area of the shaded region is given by:

Study Guide for Stewart's Multivariable Calculus, 8th

Finding a General Solution Using Separation of Variables In Exercises 5-18, find the general solution of the di...

Calculus (MindTap Course List)

In Exercises 3338, use a double integral in polar coordinates to find the volume of the solid bounded by the gr...

Multivariable Calculus

True or False: If a number p is in the interval of convergence of then so is the number .

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

Applications Comparison of compounding methods An initial investment of 30000 grows at an annual rate of 8% for...

College Algebra (MindTap Course List)

Describe the purpose of a phase in a single-case design and explain how patterns within a phase are defined, in...

Research Methods for the Behavioral Sciences (MindTap Course List)

Suppose that you arc an inspector for the Fish and Game Department and that you are given the task of determini...

Introduction To Statistics And Data Analysis

Marvel Studios motion picture Guardians of the Galaxy opened over the first two days of the 2014 Labor Day week...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Find x.

Mathematics For Machine Technology

Describe the relationship between a sample and the population (both target and accessible) in a research study ...

Research Methods for the Behavioral Sciences (MindTap Course List)

A regional director responsible for business development in the state of Pennsylvania isconcerned about the num...

Statistics for Business & Economics, Revised (MindTap Course List)

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