To determine Find the equation for the line tangent for expression x = 2 cos t , y = 2 sin t , t = π / 4 and the obtain the value of d 2 y / d x 2 . Explanation Given data: x = 2 cos t (1) y = 2 sin t (2) t = π / 4 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 π / 4 for t in equation (1) as follows. x = 2 cos ( π / 4 ) = 2 ( 0.707 ) = 1.414 = 2 Substitute π / 4 for t in equation (2) as follows. y = 2 sin ( π / 4 ) = 2 ( 0.707 ) = 1.414 = 2 Find d y d t . d y d t = d d t ( 2 sin t ) = 2 cos t Find d x d t . d x d t = d d t ( 2 cos t ) = − 2 sin t Find slope of the curve. Substitute 2 cos t for d y d t and − 2 sin t for d x d t in equation (3) as follows. d y d x = 2 cos t − 2 sin t = − cos t sin t Find value of d y d x at t = π / 4 . d y d x | t = π / 4 = − cos ( π / 4 ) sin ( π / 4 ) = − 0.707 0.707 = − 1 Find tangent line using point ( 2 , 2 ) and slope − 1 . y − 2 = − 1 ( x − 2 ) y − 2 = − x + 2 y = − x + 2 + 2 y = − x + 2 2 Thus, the equation for the line tangent for expression x = 2 cos t , y = 2 sin t , t = π / 4 is y = − x + 2 2 _

14th Edition

ISBN: 9780134438986

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 11.2, Problem 1E

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Find the equation for the line tangent for expression x=2cost, y=2sint,t=π/4 and the obtain the value of d2y/dx2.

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Chapter 11.1, Problem 58E

Chapter 11.2, Problem 2E

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