PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
8th Edition
ISBN: 9781337904285
Author: Larson
Publisher: CENGAGE L
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Chapter 5.3, Problem 89E

a.

To determine

To graph: The given function in the given interval and approximate maximum and minimum points (to four decimal places) of the graph.

a.

Expert Solution
Check Mark

Answer to Problem 89E

The approximate maximum and minimum points are:

Maxima: (1.0472,1.2500),(5.236,1.2500)

Minima: (0,1),(3.1416,−1),(6.2832,1)

Explanation of Solution

Given information:

The following function:

  f(x)=sin2x+cosx

The following interval:

  [0,2π)

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press Y= button.

Step 3: Enter the expression sin2x+cosx which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS), Chapter 5.3, Problem 89E

Interpretation:

Using TRACE feature on the calculator, the maximum and minimum points are obtained as:

Maxima: (1.0472,1.2500),(5.236,1.2500)

Minima: (0,1),(3.1416,−1),(6.2832,1)

b.

To determine

To solve: The given trigonometric equation using calculus and verify that the x-coordinates of the maximum and minimum points of f are among its solutions.

b.

Expert Solution
Check Mark

Answer to Problem 89E

The solutions of the trigonometric equation are:

  x=nπx=π3+2nπx=5π3+2nπ

Explanation of Solution

Given information:

The following equation:

  2sinxcosx−sinx=0

Calculation:

The given equation is solved as:

  2sinxcosx−sinx=0           [Given]       sinx(2cosx−1)=0               [Factor] sinx=0                                 [Set each factor to 0]  x=nπ                                   [Using inverse trigonometric function]2cosx=1                              cosx=12                              [Divide both sides by 2]x=π3+2nπ,5π3+2nπ          [Using inverse trigonometric function]

On substituting the x-coordinates of the maximum and minimum points of f , the following solutions are obtained:

  2sinxcosx−sinx=2sin0cos0−sin0=02sinxcosx−sinx=2sin(1.0472)cos(1.0472)−sin(1.0472)=02sinxcosx−sinx=2sin(5.236)cos(5.236)−sin(5.236)=02sinxcosx−sinx=2sin(3.1416)cos(3.1416)−sin(3.1416)=02sinxcosx−sinx=2sin(6.2832)cos(6.2832)−sin(6.2832)=0

Since every answer corresponds to 0 , the x-coordinates of the maximum and minimum points of fare among the solutions.

Previous
Chapter 5.3, Problem 88E
Next
Chapter 5.3, Problem 90E

Chapter 5 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

Ch. 5.1 - Prob. 1ECh. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 10E
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SEE MORE TEXTBOOKS
01 - What Is A Differential Equation in Calculus? Learn to Solve Ordinary Differential Equations.; Author: Math and Science;https://www.youtube.com/watch?v=K80YEHQpx9g;License: Standard YouTube License, CC-BY
Higher Order Differential Equation with constant coefficient (GATE) (Part 1) l GATE 2018; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=ODxP7BbqAjA;License: Standard YouTube License, CC-BY
Solution of Differential Equations and Initial Value Problems; Author: Jefril Amboy;https://www.youtube.com/watch?v=Q68sk7XS-dc;License: Standard YouTube License, CC-BY