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

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Chapter 8.3, Problem 1ED

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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.

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Chapter 8.3, Problem 4MP

Chapter 8.3, Problem 1E

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Determine whether each function is an injection and determine whether each is a surjection.The notation Z_(n) refers to the set {0,1,2,...,n-1}. For example, Z_(4)={0,1,2,3}. f: Z_(6) -> Z_(6) defined by f(x)=x^(2)+4(mod6). g: Z_(5) -> Z_(5) defined by g(x)=x^(2)-11(mod5). h: Z*Z -> Z defined by h(x,y)=x+2y. j: R-{3} -> R defined by j(x)=(4x)/(x-3).

Determine whether each function is an injection and determine whether each is a surjection.

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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...

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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.

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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.

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Chapter 8 Solutions