Explanation Use online graphing calculator and sketch the graph of r = sin 2 θ as shown below in Figure 1. From Figure 1, the exact value of y -coordinates of the highest point on curve r = sin 2 θ is y = 0.77 . The equation for the variable x is as, x = r cos θ (1) The equation for the variable y is as, y = r sin θ (2) Substitute ( sin 2 θ ) for r in equation (1) as, x = r cos θ = ( sin 2 θ ) cos θ (3) Substitute ( sin 2 θ ) for r in equation (2) as, y = r sin θ = ( sin 2 θ ) sin θ (4) Use calculus method to obtain the exact value of maximum height as follows. Differentiate the variable y with respect to θ . d y d θ = d d θ ( sin θ sin 2 θ ) = 2 sin θ cos 2 θ + cos θ sin 2 θ = 2 sin θ ( 2 cos 2 θ − 1 ) + cos θ ( 2 sin θ cos θ ) = 2 sin θ ( 2 cos 2 θ − 1 + cos 2 θ ) = 2 sin θ ( 3 cos 2 θ − 1 ) To obtain maximum height, the derivative has to be 0. That is, d y d θ = 0 . Observe that since the curve is symmetric about the origin, the case of first can be applied here

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Chapter 10.3, Problem 66E

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To find: The exact value of y-coordinates of the highest point on curve r=sin2θ using a graph; also use calculus to find the exact value.

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Chapter 10.3, Problem 65E

Chapter 10.3, Problem 67E

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Chapter 10 Solutions

Calculus, Early Transcendentals

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The exact value of y -coordinates of the highest point on curve r = sin 2 θ using a graph; also use calculus to find the exact value.

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