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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 8, Problem 96RE
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Ron Larson
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Relative Extrema In Exercises 91-96, find the relative extrema of the trigonometric function in the interval ( 0 , 2 π ) . Use a graphing utility to confirm your results.

f ( x ) = 1 2 + sin x

To determine

To calculate: The relative extrema of the trigonometric function f(x)=12+sinx over the interval (0,2π).

Explanation

Given Information:

The provided trigonometric function is f(x)=12+sinx over the interval (0,2π).

Formula used:

Sine derivative Rule:

ddx[sinu]=cosududx

Quotient rule of differentiation:

ddx[a(x)b(x)]=b(x)a'(x)−a(x)b'(x)[b(x)]2

Condition for critical points:

dydx=f′(x)=0

If function f(x) is increasing, then f′(x)>0 and if function is decreasing, then f′(x)<0.

From first derivative Test,

if f′(x) changes positive to negative, then function f(x) has relative maxima and, if f′(x) changes negative to positive then function f(x) has relative minima.

Calculation:

Consider the provided trigonometric function:

f(x)=12+sinx

Apply the quotient rule of differentiation to find the derivative of the above function,

f′(x)=(2+sinx)ddt[1]−1ddt[2+sinx](2+sinx)2

Differentiate the above function with respect to x using Sine,

f′(x)=(2+sinx)(0)−1[0+cosx](2+sinx)2=0−cosx(2+sinx)2=−cosx(2+sinx)2

Apply f′(x)=0 in order to find critical points,

−cosx(2+sinx)2=0cosx=0

Since, the value of the cosθ is 0 at θ=π2, and θ=5π6. So,

cosx=0cosx=cosπ2x=π2

And,

cosx=0cosx=cos3π2x=3π2

Thus, critical points are x=π2 and x=3π2.

Now, rewrite the first derivative of the function.

f′(x)=−cosx(2+sinx)2

At x=π3,

f′(π3)=−cosπ3(2+sinπ3)2=−12(2+32)2=−0.06087=−ve<0

At x=π,

f′(π)=−cosπ(2+sinπ)2=−(−1)(2+0)2=0.25=+ve>0

At x=5π3,

f′(5π3)=−cos5π3(2+sin5π3)2=−0

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Sect-8.1 P-1CPSect-8.1 P-2CPSect-8.1 P-3CPSect-8.1 P-4CPSect-8.1 P-1SWUSect-8.1 P-2SWUSect-8.1 P-3SWUSect-8.1 P-4SWUSect-8.1 P-5SWUSect-8.1 P-6SWU
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Convert each expression in Exercises 25-50 into its technology formula equivalent as in the table in the text. ...

Finite Mathematics

In problems 23-36, use the rules of exponents to simplify so that only positive exponents remain. 30.

Mathematical Applications for the Management, Life, and Social Sciences

By how much must the sample size n be increased if the width of the CI (7.5) is to be halved? If the sample siz...

Probability and Statistics for Engineering and the Sciences

In Exercises 17 to 24, let U = {2. 6, 8, 10, 12, 14, 16, 18}, A = {2, 6, 10}, B = {6, 10, 16, 18}, and C = {14,...

Mathematical Excursions (MindTap Course List)

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. Objectives 2 5(x2y)...

Intermediate Algebra

Sketch the graph of the function g(x) = |x2 1 | |x2 4|.

Single Variable Calculus: Early Transcendentals

In Exercises 7 to12, use the angle provided to perform the constructions. Exercises 7-12 Construct an angle tha...

Elementary Geometry For College Students, 7e

Draw a graph for each inequality on the number line: b5

Elementary Technical Mathematics

Reword Definition 3.6 for a group with respect to addition. Definition 3.6 : Product Notation Let n be a positi...

Elements Of Modern Algebra

For Problems 49 through 56, use Figure 14, which shows the unit circle with increments of 0.1 units marked off ...

Trigonometry (MindTap Course List)

13. The finance charge ________ is the unearned portion of the finance charge that is returned to a borrower wh...

Contemporary Mathematics for Business & Consumers

Think About It The graph of the velocity function (see figure) represents the velocity in miles per hour during...

Calculus of a Single Variable

Finding the Limit of a Sequence In Exercises 25-28, use a graphing utility to graph the first 10 terms of the s...

Calculus: Early Transcendental Functions (MindTap Course List)

If MNPQin O, explain why MNPQ is an isosceles trapezoid. HINT: Draw a diagonal.

Elementary Geometry for College Students

INVESTMENT ANALYSIS Jorge has decided to invest at most 100,000 in securities in the form of corporate stocks. ...

Finite Mathematics for the Managerial, Life, and Social Sciences

Using a binomial series, the Maclaurin series for is:

Study Guide for Stewart's Multivariable Calculus, 8th

Projectile Motion In Exercises 41 and 42, use the model for projectile motion, assuming there is no air resista...

Multivariable Calculus

f(x)=x+2x(x5) is continuous for all x except: a) x = 2 b) x = 2, 0, 5 c) x = 0, 5 d) f is continuous for all x.

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

Precalculus or Calculus In Exercises 3-6.decide whether the problem can be solved using precalculus or whether ...

Calculus

Distinguish between qualitative and quantitative research and recognize examples of each.

Research Methods for the Behavioral Sciences (MindTap Course List)

Consider the following hypothesis test: H0: 80 Ha: 80 A sample of 100 is used and the population standard d...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Change each rational equation to quadratic form and solve it by the most efficient method. 5x=4x26

College Algebra (MindTap Course List)

Describe the disadvantages of a multiple-treatment design, compared to a two-treatment design, for a within-sub...

Research Methods for the Behavioral Sciences (MindTap Course List)

The regional transit authority for a major metropolitan area wants to determine whether there is any relationsh...

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

The article Economy Low, Generosity High (USA TODAY, July 28, 2009) noted that despite a weak economy in 2008, ...

Introduction To Statistics And Data Analysis

Using the Table of Block Thicknesses for a Metric Gage Block Set, determine a combination of gage blocks each o...

Mathematics For Machine Technology

Verifying Divergence In Exercises 11-18.verify that the infinite series diverges. n=14n+34n+1

Calculus (MindTap Course List)

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