Explanation Given: The function y = f ( x ) has x -intercepts at ( 0 , 0 ) , ( π , 0 ) , and ( 2 π , 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 decreasing on the intervals ( 0 , π ) as the slope of the tangent lines over the intervals ( 0 , π ) is negative. Similarly, the graph of y = f ′ ( x ) is decreasing on the intervals ( π , 2 π ) as the slope of the tangent lines over the intervals ( π , 2 π ) is positive. Form the given Figure; make a Table with the values of f ′ ( x ) and their corresponding x values. x f ′ ( x ) 0 1 π − 1 2 π 1 Hence, from the above Table the function f ′ ( x ) = cos x is the required function that corresponds to the points ( 0 , 1 ) , ( π , − 1 ) , and ( 2 π , 1 ) . Therefore, the required trigonometric function is f ′ ( x ) = cos x . Determine the trigonometric function y = f ( x ) as follows. Given that graph of y = f ( x ) is defined on the interval 0 ≤ x ≤ 2 π with the condition that f ( x ) has x -intercepts at ( 0 , 0 ) , ( π , 0 ) , and ( 2 π , 0 ) . That is, the graph of f ( x ) cut x -axis at x = 0 , x = π , and x = 2 π . Note that the given graph is the graph of the trigonometric function f ′ ( x ) = cos x which is the differentiation of the trigonometric function f ( x ) = sin x . Substitute x = 0 in f ( x ) as, f ( 0 ) = sin ( 0 ) = 0 Substitute x = π in f ( x ) as, f ( π ) = sin ( π ) = 0 Substitute x = 2 π in f ( x ) as, f ( π ) = sin ( 2 π ) = 0 Hence, f ( x ) = sin x is the required function which has the x -intercepts at x = 0 , x = π and x = 2 π . Substitute x = π 2 in f ( x ) as, f ( 0 ) = sin ( π 2 ) = 1 This gives f ( x ) has maximum value at x = π 2 , since − 1 ≤ sin x ≤ 1 where x ∈ [ 0 , 2 π ]

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

15th Edition

ISBN: 9781323047620

Author: Barnett, Ziegler, Byleen

Publisher: Pearson Education

See similar textbooks

Videos

Question

Chapter 8.3, Problem 1ED

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 its local maxima and minima, and also sketch the graph of y=f(x) where 0≤x≤2π.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 8.3, Problem 4MP

Chapter 8.3, Problem 1E

Students have asked these similar questions

Suppose we have a linear program in standard equation form maximize cx subject to Ax = b, x > 0. and suppose u, v, and w are all optimal solutions to this linear program. (a) Prove that z = u+v+w is an optimal solution. (b) If you try to adapt your proof from part (a) to prove that that u+v+w is an optimal solution, say exactly which part(s) of the proof go wrong. (c) If you try to adapt your proof from part (a) to prove that u+v-w is an optimal solution, say exactly which part(s) of the proof go wrong.

Can the expert solve an Integral In detall? Hoxto³ W. 1 w = dw 大 90x103 80*10³ ⑥M = 1 1012 221 JW 70x10 80x103 © P= ± Sin (lw/+1) dw 70*10*A

Schoology → C Cportsk12.com bookmarks Sis Grades and Attendance Al Detector - the Original Al Che X GPTZero + portsmouth.schoology.com/common-assessment-delivery/start/7747152192?action=onresume&submissionId=1600790102 New Tab Home | Schoology Quadrilateral Quiz English If WXYZ is a square, and WY = 32, find XY. Round your answer to the nearest tenth. Z XY = R X Y POSSIBLE POINTS: 5 2 of 20 48 21 1 2 345678910 Next ▸ Δ ㄖㄨ All Bookmarks Schoology Help Center | PRIVACY POLICY | Terms of Use PowerSchool ©2025

Chapter 8 Solutions

Calculus for Business, Economics, Life Sciences, and Social Sciences - Boston U.

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 its local maxima and minima , and also sketch the graph of y = f ( x ) where 0 ≤ x ≤ 2 π .

Solution Summary: The author identifies the trigonometric functions f(x), and explains its concavity and the locations of its local maxima and minima.

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 its local maxima and minima, and also sketch the graph of y=f(x) where 0≤x≤2π.

Want to see the full answer?

Chapter 8 Solutions