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

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Chapter 8, Problem 91RE
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Calculus: An Applied Approach (Min...

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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.

y = sin π x 4

To determine

To calculate: The relative extrema of the trigonometric function y=sinπx4 over the interval (0,2π).

Explanation

Given Information:

The provided trigonometric function is y=sinπx4 over the interval (0,2π).

Formula used:

Derivative of sine function is;

ddx[sinu]=cosududx

General power derivative rule is;

ddxxn=nxn−1

Condition for critical points is;

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.

In order words, if function f(x) changes from increasing function to decreasing at any point critical, then function f(x) will have relative maxima at that point, and if function f(x) changes from decreasing function to increasing at any critical point, then function f(x) will have relative minima at that point.

Calculation:

Consider the provided trigonometric function:

y=sinπx4

Differentiate the above function with respect to x using Sine and Power derivative rules as;

dydx=ddx[sinπx4]y′(x)=(cosπx4)ddx[πx4]=(cosπx4)(π4)=π4cosπx4

Apply dydx=0 in order to find critical points as;

π4cosπx4=0cosπx4=0

Since, the value of the cosθ is zero at θ=π2 and θ=3π2 over the interval [0,2π].

So,

cosπx4=cosπ2πx4=π2x=2

And,

cosπx4=cos3π2πx4=3π2x=6

Thus, critical points are x=2 and x=6.

Now consider the table as;

Interval0<x<22<x<66<x<2πSign of y′(x)y′(x)>0y′(x)<0y′(x)>0yIncreasingDecreasingIncreasing

From above table, y′(x) is changing from positive to negative at x=2

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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
Sect-8.1 P-7SWUSect-8.1 P-8SWUSect-8.1 P-9SWUSect-8.1 P-10SWUSect-8.1 P-1ESect-8.1 P-2ESect-8.1 P-3ESect-8.1 P-4ESect-8.1 P-5ESect-8.1 P-6ESect-8.1 P-7ESect-8.1 P-8ESect-8.1 P-9ESect-8.1 P-10ESect-8.1 P-11ESect-8.1 P-12ESect-8.1 P-13ESect-8.1 P-14ESect-8.1 P-15ESect-8.1 P-16ESect-8.1 P-17ESect-8.1 P-18ESect-8.1 P-19ESect-8.1 P-20ESect-8.1 P-21ESect-8.1 P-22ESect-8.1 P-23ESect-8.1 P-24ESect-8.1 P-25ESect-8.1 P-26ESect-8.1 P-27ESect-8.1 P-28ESect-8.1 P-29ESect-8.1 P-30ESect-8.1 P-31ESect-8.1 P-32ESect-8.1 P-33ESect-8.1 P-34ESect-8.1 P-35ESect-8.1 P-36ESect-8.1 P-37ESect-8.1 P-38ESect-8.1 P-39ESect-8.1 P-40ESect-8.1 P-41ESect-8.1 P-42ESect-8.1 P-43ESect-8.1 P-44ESect-8.1 P-45ESect-8.1 P-46ESect-8.1 P-47ESect-8.1 P-48ESect-8.1 P-49ESect-8.1 P-50ESect-8.1 P-51ESect-8.1 P-52ESect-8.1 P-53ESect-8.1 P-54ESect-8.1 P-55ESect-8.1 P-56ESect-8.1 P-57ESect-8.1 P-58ESect-8.2 P-1CPSect-8.2 P-2CPSect-8.2 P-3CPSect-8.2 P-4CPSect-8.2 P-5CPSect-8.2 P-6CPSect-8.2 P-7CPSect-8.2 P-1SWUSect-8.2 P-2SWUSect-8.2 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P-7SWUSect-8.5 P-8SWUSect-8.5 P-9SWUSect-8.5 P-10SWUSect-8.5 P-11SWUSect-8.5 P-12SWUSect-8.5 P-13SWUSect-8.5 P-14SWUSect-8.5 P-15SWUSect-8.5 P-16SWUSect-8.5 P-17SWUSect-8.5 P-18SWUSect-8.5 P-19SWUSect-8.5 P-20SWUSect-8.5 P-1ESect-8.5 P-2ESect-8.5 P-3ESect-8.5 P-4ESect-8.5 P-5ESect-8.5 P-6ESect-8.5 P-7ESect-8.5 P-8ESect-8.5 P-9ESect-8.5 P-10ESect-8.5 P-11ESect-8.5 P-12ESect-8.5 P-13ESect-8.5 P-14ESect-8.5 P-15ESect-8.5 P-16ESect-8.5 P-17ESect-8.5 P-18ESect-8.5 P-19ESect-8.5 P-20ESect-8.5 P-21ESect-8.5 P-22ESect-8.5 P-23ESect-8.5 P-24ESect-8.5 P-25ESect-8.5 P-26ESect-8.5 P-27ESect-8.5 P-28ESect-8.5 P-29ESect-8.5 P-30ESect-8.5 P-31ESect-8.5 P-32ESect-8.5 P-33ESect-8.5 P-34ESect-8.5 P-35ESect-8.5 P-36ESect-8.5 P-37ESect-8.5 P-38ESect-8.5 P-39ESect-8.5 P-40ESect-8.5 P-41ESect-8.5 P-42ESect-8.5 P-43ESect-8.5 P-44ESect-8.5 P-45ESect-8.5 P-46ESect-8.5 P-47ESect-8.5 P-48ESect-8.5 P-49ESect-8.5 P-50ESect-8.5 P-51ESect-8.5 P-52ESect-8.5 P-53ESect-8.5 P-54ESect-8.5 P-55ESect-8.5 P-56ESect-8.5 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Solve the equations in Exercises 112 for x (mentally, if possible). x+5=0

Finite Mathematics

Use a calculator to check your answers for Problems 55-94. 55. 9128+56 56. 69+1187+14 57. 21+(17)11+15(10) 58. ...

Intermediate Algebra

Consider a sample x1, ... , xn with n even. Let xL and xU denote the average of the smallest n/2 and the larges...

Probability and Statistics for Engineering and the Sciences

In Problems 41 and 42, use a graphing calculator and graph each equation in the viewing window given. graph ea...

Mathematical Applications for the Management, Life, and Social Sciences

Evaluate the integral. 45. x5ex3dx

Single Variable Calculus: Early Transcendentals

The table factor for an annuity due is found by ____ one period to the number of periods of the annuity and the...

Contemporary Mathematics for Business & Consumers

Review Exercises Given: DE is a median of ADC BE-FD- EF-FD- Prove: ABCD is a

Elementary Geometry For College Students, 7e

Each of the following problems refers to triangle ABC. If a=3.7m,c=6.4cm,B=31, find b.

Trigonometry (MindTap Course List)

a. Find all Sylow 3-subgroups of the alternating group A4. b. Find all Sylow 2-subgroups of A4.

Elements Of Modern Algebra

Find the prime factorization of each number (use divisibility tests where helpful): 56

Elementary Technical Mathematics

Approximating the Sum of an Alternating Series In Exercises 31-34, approximate the sum of the series by using t...

Calculus of a Single Variable

HOW DO YOU SEE IT? Determine whether each labeled point is an absolute maximum or minimum, a relative maximum o...

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercise 1120, construct the dual problem associated with the primal problem. Solve the primal problem. Mini...

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercises 33 to 36, find the value of the indicated variable for each given formula. Y,ifm=YyXx,m=32,y=1,X=2...

Elementary Geometry for College Students

Limit and ContinuityIn Exercises 1318, find the limit (if it exists) and discuss the continuity of the function...

Multivariable Calculus

A normal vector to the plane 2x + 3y + 6z =–20 is:

Study Guide for Stewart's Multivariable Calculus, 8th

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

Calculus

What substitution should be made to evaluate sin2 x cos x dx? a) u = sin x b) u = cos x c) u = sin x cos x d) u...

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

Eagle Outfitters is a chain of stores specializing in outdoor apparel and camping gear.They are considering a p...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Self Check Graph: 2x2+2y2+4x-8y+2=0.

College Algebra (MindTap Course List)

An online survey by ShareBuilder, a retirement plan provider, and Harris Interactive reported that 60% of femal...

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

A researcher wants to describe how fine motor skills change as a group of infants age from 18 to 24 months. Des...

Research Methods for the Behavioral Sciences (MindTap Course List)

Sea bream are a type of fish that are often raised in large fish farming enterprises. These fish are usually fe...

Introduction To Statistics And Data Analysis

Describe the pattern that would appear in a scatter plot showing the data points for each of the following corr...

Research Methods for the Behavioral Sciences (MindTap Course List)

Express the following BCD (8421) numbers as decimal numbers. 11 0110 0101.0000 0111

Mathematics For Machine Technology

Checking Points on a Line In Exercises 5 and 6, determine whether each point lies on the line. x32=y78,z=+2 (a)...

Calculus (MindTap Course List)

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