Explanation Given: The function is y = θ + sin 2 θ , [ 0 , π ] . Formula used: Test for concavity: Let f ″ ( x ) exist for all x ∈ ( a , b ) , then · y = f ″ ( x ) > 0 for all x ∈ ( a , b ) , here the function f is concave upward on ( a , b ) . · y = f ″ ( x ) < 0 for all x ∈ ( a , b ) , here the function f is concave downward on ( a , b ) . y = f ( x ) has a point of inflection if f ″ ( x ) = 0 . Calculation: The function is y = θ + sin 2 θ , [ 0 , π ] . Let f ( θ ) = θ + sin 2 θ . … (1) First, find the derivative. f ′ ( θ ) = 1 + 2 sin θ cos θ = 1 + sin 2 θ Now, find the second derivative. f ″ ( θ ) = 2 cos 2 θ Find the value of θ by substitute f ″ ( θ ) = 0 . 2 cos 2 θ = 0 cos 2 θ = cos π 2 cos 2 θ = cos 2 ( π 4 ) θ = π 4 Or 2 cos 2 θ = 0 cos 2 θ = cos 3 π 2 cos 2 θ = cos 2 ( 3 π 4 ) θ = 3 π 4 Thus, the zeroes of the f ″ ( θ ) are x = π 4 , 3 π 4 . This divides the x − axis into three intervals: ( 0 , π 4 ) , ( π 4 , 3 π 4 ) , and ( 3 π 4 , π )

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ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 4.4, Problem 8E

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Write the intervals on which the function is concave up or down. Also, find the points of inflection.

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Chapter 4.4, Problem 7E

Chapter 4.4, Problem 9E

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Sketch r = 3 sin(θ) on a standard x − y coordinate plane. Also, convert this equation to a recognizable form using x and y. Show all steps.

Chapter 4 Solutions

Calculus - Standalone book

Ch. 4.1 - Prob. 1PQ Ch. 4.1 - Prob. 2PQ Ch. 4.1 - Prob. 3PQ Ch. 4.1 - Prob. 4PQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E

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Write the intervals on which the function is concave up or down. Also, find the points of inflection.

Solution Summary: The author explains how the function f is concave up and down, and the points of inflection.

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