Explanation Given information : The given curve is y = x 3 . Formula used : Let y = f ( x ) be given curve. Then the slope at any point ( a , b ) is given by m = ( d y d x ) ( a , b ) . Proof: The given curve is y = x 3 . Let P ( h , k ) be a point on the curve then k = h 3 . Now, slope of the tangent m = ( d y d x ) = ( d ( x 3 ) d x ) = 3 x 2 &ems... (1) Then slope of tangent at P is m P = ( d y d x ) ( h , k ) = 3 h 2 &ems... (2) The equation of the tangent at point P is y - k = m ( x - h ) ⇒ y - k = 3 h 2 ( x - h ) y - k = 3 h 2 ( x - h ) ⇒ y - k = 3 h 2 x − 3 h 3 y = 3 h 2 x - 2 h 3 &ems... (3) Now, we find point Q , the point of intersection of tangent and the curve from equation y = x 3 and equation (3) as y = 3 h 2 x - 2 k ⇒ x 3 = 3 h 2 x - 2 h 3 ⇒ x 3 - 3 h 2 x + 2 h 3 =0 ⇒ x 3 - h 2 x - 2 h 2 x + 2 h 3 =0 ⇒ x ( x 2 − h 2 )

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3, Problem 98SP

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Toprove:If the tangent at point P to the curve meets the curve again at point Q , then slope of tangent at point Q is four time that the slope at P .

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Chapter 3, Problem 97SP

Chapter 3, Problem 99SP

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Chapter 3 Solutions

Calculus

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Toprove: If the tangent at point P to the curve meets the curve again at point Q , then slope of tangent at point Q is four time that the slope at P .

Solution Summary: The author proves that the slope of the tangent at point P to the curve is four times higher than that of point Q.

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Toprove:If the tangent at point P to the curve meets the curve again at point Q , then slope of tangent at point Q is four time that the slope at P .

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