To determine Find the equation for the line tangent for expression x = − t + 1 , y = 3 t , t = 3 and the obtain the value of d 2 y / d x 2 . Explanation Given data: x = − t + 1 (1) y = 3 t (2) t = 3 Formula used: Consider the Parametric Formulas. d y d x = d y / d t d x / d t (3) d 2 y d x 2 = d y ′ / d t d x / d t (4) Here, y ′ = d y d x (5) Calculation: Find points ( x , y ) . Substitute 3 for t in equation (1) as follows. x = − 3 + 1 = − 4 = − 2 Substitute 3 for t in equation (2) as follows. y = 3 ( 3 ) = 9 = 3 Find d y d t . d y d t = d d t ( 3 t ) = d d t ( 3 t ) 1 2 = 1 2 ( 3 t ) ( 1 2 − 1 ) ( 3 ) = 3 2 ( 3 t ) ( − 1 2 ) d y d t = 3 2 3 t Find d x d t . d x d t = d d t ( − t + 1 ) = − d d t ( t + 1 ) 1 2 = − 1 2 ( t + 1 ) 1 2 − 1 = − 1 2 ( t + 1 ) − 1 2 d x d t = − 1 2 t + 1 Find slope of the curve. Substitute 3 2 3 t for d y d t and − 1 2 t + 1 for d x d t in equation (3) as follows. d y d x = 3 2 3 t − 1 2 t + 1 = − 6 t + 1 2 3 t = − 3 t + 1 3 t Find value of d y d x at t = 3 . d y d x | t = 3 = − 3 ( 3 ) + 1 3 ( 3 ) = − 3 4 9 = − 3 ( 2 ) 3 = − 2 Find tangent line using point ( − 2 , 3 ) and slope − 2 . y − 3 = − 2 [ x − ( − 2 ) ] y − 3 = − 2 [ x + 2 ] y − 3 = − 2 x − 4 y = − 2 x − 4 + 3 y = − 2 x − 1 Thus, the equation for the line tangent for expression x = − t + 1 , y = 3 t , t = 3 is y = − 2 x − 1 _ . Find y ′ . Substitute d y / d t d x / d t for d y d x in equation (5) as follows

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ISBN: 9780134654874

Author: WEIR

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Chapter 11.2, Problem 8E

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Find the equation for the line tangent for expression x=−t+1, y=3t,t=3 and the obtain the value of d2y/dx2.

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Chapter 11.2, Problem 7E

Chapter 11.2, Problem 9E

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