Explanation Given: The function f ( x ) has x -intercepts at ( − π 2 , 0 ) and ( π 2 , 0 ) . Formula used: Derivative involving Sine and Cosine: Basic Form: 1. d d x sin x = cos x 2. d d x cos x = − sin x Calculation: From the given Figure, it is observed that the graph of y = f ′ ( x ) is increasing on the intervals ( − π , − π 2 ) and ( π 2 , π ) as the slope of the tangent lines on the curve y = f ′ ( x ) over the intervals ( − π , − π 2 ) and ( π 2 , π ) is positive. Similarly, the graph of y = f ′ ( x ) is decreasing on the intervals ( − π 2 , 0 ) and ( 0 , π 2 ) as the slope of the tangent lines on the curve y = f ′ ( x ) over the intervals ( − π 2 , 0 ) and ( 0 , π 2 ) is negative. Form the given Figure; make a Table with the values of f ′ ( x ) and their corresponding x values. x f ′ ( x ) − π 0 − π 2 1 0 0 π 2 − 1 π 0 Hence, from the above Table the function f ′ ( x ) = − sin x is the required function that corresponds to the points ( − π , 0 ) , ( − π 2 , 1 ) , ( 0 , 0 ) , ( π 2 , − 1 ) , and ( π , 0 ) . Therefore, the required trigonometric function is f ′ ( x ) = − sin x . Determine the trigonometric function y = f ( x ) as follows. Given that graph of y = f ( x ) is defined on the interval − π ≤ x ≤ π , with the condition that f ( x ) has x -intercepts at ( − π 2 , 0 ) and ( π 2 , 0 ) . That is, the graph of f ( x ) cut x -axis at x = − π 2 and x = π 2 . Note that the given graph is the graph of the trigonometric function f ′ ( x ) = − sin x which is the differentiation of the trigonometric function f ( x ) = cos x . Substitute x = − π 2 in f ( x ) as, f ( − π 2 ) = cos ( − π 2 ) = 0 Substitute x = π 2 in f ( x ) as, f ( π 2 ) = cos ( π 2 ) = 0 Hence, f ( x ) = cos x is the required function which has x -intercepts at x = − π 2 and x = π 2

Pearson eText for Calculus for Business, Economics, Life Sciences, and Social Sciences, Brief Version -- Instant Access (Pearson+)

14th Edition

ISBN: 9780137400126

Author: Raymond Barnett, Michael Ziegler

Publisher: PEARSON+

See similar textbooks

Videos

Question

Chapter 8.3, Problem 29E

To determine

To identify: The functions f(x) and f′(x) as particular trigonometric functions and to explain where f(x) is increasing and decreasing, its concavity, and the locations of local maxima and minima, and also sketch the graph of y=f(x) where −π≤x≤π.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 28E

Chapter 8.3, Problem 30E

Students have asked these similar questions

In each of Problems 1 through 4, draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t → ∞. If this behavior depends on the initial value of y at t = 0, describe the dependency.1. y′ = 3 − 2y

A = 5.8271 ± 0.1497 = B 1.77872 ± 0.01133 C=0.57729 ± 0.00908 1. Find the relative uncertainty of A, B, and C 2. Find A-3 3. Find 7B 4. Find A + B 5. Find A B-B - 6. Find A * B 7. Find C/B 8. Find 3/A 9. Find A 0.3B - 10. Find C/T 11. Find 1/√A 12. Find AB²

Please help on question 16b attached. thx

Chapter 8 Solutions

Pearson eText for Calculus for Business, Economics, Life Sciences, and Social Sciences, Brief Version -- Instant Access (Pearson+)

Ch. 8.1 - Find the degree measure of 1 rad.Ch. 8.1 - Without using a calculator, find: (A)cot 45 (B)cos...Ch. 8.1 - Solve the right triangle in Figure 14. Round side...Ch. 8.1 - Solve the right triangle in Figure 16. Round...Ch. 8.1 - Repeat Example 5 assuming that the man is standing...Ch. 8.1 - Prob. 1ED Ch. 8.1 - Prob. 1E Ch. 8.1 - In Problems 18, mentally convert each degree...Ch. 8.1 - In Problems 18, mentally convert each degree...Ch. 8.1 - In Problems 18, mentally convert each degree...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, subject and related others by exploring similar questions and additional content below.

The functions f ( x ) and f ′ ( x ) as particular trigonometric functions and to explain where f ( x ) is increasing and decreasing, its concavity, and the locations of local maxima and minima , and also sketch the graph of y = f ( x ) where − π ≤ x ≤ π .

Solution Summary: The author identifies the functions f(x) & f'prime 'left' as trigonometric functions, and sketched the graph of the function.

Videos

To identify: The functions f(x) and f′(x) as particular trigonometric functions and to explain where f(x) is increasing and decreasing, its concavity, and the locations of local maxima and minima, and also sketch the graph of y=f(x) where −π≤x≤π.

Want to see the full answer?

Chapter 8 Solutions