Explanation Formula Used: If the trajectory of a particle is given by c ( t ) = ( x ( t ) , y ( t ) ) , then its speed is given by v ( t ) = [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 Second derivative test: The point for which f ′ ( t ) = 0 is the critical point for the function and if the second derivative of the function is positive at this point, then f has a local minimum at that point. But, if for that critical point the second derivative is negative, then f has local maximum at that point. Calculations: Consider the cycloid c ( t ) = ( 4 t − 4 sin t , 4 − 4 cos t ) Then, x ( t ) = 4 t − 4 sin t y ( t ) = 4 − 4 cos t So, x ′ ( t ) = d x d t = 4 − 4 cos t y ′ ( t ) = d y d t = 4 sin t Now, the speed is evaluated as v ( t ) = [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 = ( 4 − 4 cos t ) 2 + ( 4 sin t ) 2 = 16 − 32 cos t + 16 cos 2 t + 16 sin 2 t = 16 ( 1 − 2 cos t + cos 2 t + sin 2 t ) = 16 ( 1 − 2 cos t + 1 ) = 16 ( 2 − 2 cos t ) = 4 2 ( 1 − cos t ) Now, the slope of the tangent line is given by d y d x = d y d t d x d t = 4 sin t 4 − 4 cos t , when d x d t ≠ 0 Tangent line is horizontal when d y d x = 0 ...

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

3rd Edition

ISBN: 9781319048518

Author: Colin Adams Jon Rogawski

Publisher: Macmillan Higher Education

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Chapter 11.2, Problem 23E

To determine

To find: speed of cycloid $c (t) = (4 t - 4 \sin t, 4 - 4 \cos t)$ at points where tangent line is horizontal.

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Chapter 11.2, Problem 22E

Chapter 11.2, Problem 24E

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Chapter 11 Solutions

Applied Calculus (with Infotrac) 3rd Edition By Waner, Stefan; Costenoble, Steven Published By Brooks Cole Hardcover

Ch. 11.1 - Prob. 1PQ Ch. 11.1 - Prob. 2PQ Ch. 11.1 - Prob. 3PQ Ch. 11.1 - Prob. 4PQ Ch. 11.1 - Prob. 5PQ Ch. 11.1 - Prob. 6PQ Ch. 11.1 - Prob. 7PQ Ch. 11.1 - Prob. 1E Ch. 11.1 - Prob. 2E Ch. 11.1 - Prob. 3E

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speed of cycloid c ( t ) = ( 4 t − 4 sin t , 4 − 4 cos t ) at points where tangent line is horizontal.

To find: speed of cycloid c(t)=(4t−4sint,4−4cost) at points where tangent line is horizontal.

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Chapter 11 Solutions

To find: speed of cycloid $c (t) = (4 t - 4 \sin t, 4 - 4 \cos t)$ at points where tangent line is horizontal.