Explanation Given: 1) Generating circle of cycloid has radius 1. 2) Need to prove tangent line at a point P on cycloid passes through top point of rolling circle. Formula used: 1) Point-slope formula states that at point ( x 1 , y 1 ) having slope m , equation of the line can be written as ( y − y 1 ) = m ( x − x 1 ) 2) Slope of the tangent line = d y d x 3) Chain rule states that d y d x = d y d t . d t d x 4) d d x ( k ) = 0 where k is any constant value 5) d d x ( cos x ) = − sin x 6) d d x ( sin x ) = cos x 7) d d x ( x ) = 1 8) ( a − b ) 2 = a 2 − 2 a b + b 2 9) sin 2 θ + cos 2 θ = 1 Calculation: Parametric equation of cycloid is represented by following equation x = r ( t − sin t ) and y = r ( 1 − cos t ) Where r = radius of circle generating cycloid t = angular displacement of the generating circle Let’s first evaluate co-ordinates of point P at t x P = r ( t − sin t ) and y P = r ( 1 − cos t ) Slope of tangent at P will be d y d x = d y d t . d t d x d x d t = d d t ( t − sin t ) d x d t = d d t ( t ) − d d t ( sin t ) d x d t = 1 − cos t [using formula d d x ( x ) = 1 and d d x ( sin x ) = cos x ] d y d t = d d t ( 1 − cos t ) d y d t = d d t ( 1 ) − d d t ( cos t ) d y d t = 0 − ( − sin t ) [using formula d d x ( k ) = 0 and d d x ( cos x ) = − sin x ] d y d t = sin t Slope of tangent at P is d y d x = sin t 1 − cos t Calculating co-ordinates of point O Assuming that circle started rolling at origin where centre of circle is ( 0 , ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Chapter 12.1, Problem 82E

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Need to prove that the tangent line at given point on the cycloid always passes through the top point on the unit circle generating the cycloid, as indicated in given fig.

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Chapter 12.1, Problem 81E

Chapter 12.1, Problem 83E

Chapter 12 Solutions

CALCULUS (CLOTH)

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

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Need to prove that the tangent line at given point on the cycloid always passes through the top point on the unit circle generating the cycloid, as indicated in given fig.

Solution Summary: The author explains that the tangent line at a point P on cycloid always passes through top point of rolling circle.

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Need to prove that the tangent line at given point on the cycloid always passes through the top point on the unit circle generating the cycloid, as indicated in given fig.

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Chapter 12 Solutions