Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.3 Rules of Differentiation - 3.3 Exercises - Page 151: 36


The equation of the tangent line at the given point is $y=-2x-1$

Work Step by Step

$y=x^{3}-4x^{2}+2x-1$ $;$ $a=2$ First, evaluate the derivative of the given expression: $y'=3x^{2}-4(2)x+2$ $y'=3x^{2}-8x+2$ Substitute $x$ by $a=2$ in the derivative found to obtain the slope of the tangent line at the given point: $m_{tan}=3(2)^{2}-8(2)+2=3(4)-16+2=...$ $...=12-16+2=-2$ Substitute $x$ by $a=2$ in the original expression to obtain the $y$-coordinate of the point given: $y=2^{3}-4(2)^{2}+2(2)-1=8-4(4)+4-1=...$ $...=8-16+4-1=-5$ The point is $(2,-5)$ The slope of the tangent line and a point through which it passes are now known. Use the point-slope form of the equation of a line, which is $y-y_{1}=m(x-x_{1})$ to obtain the equation of the tangent line at the given point: $y-(-5)=-2(x-2)$ $y+5=-2x+4$ $y=-2x+4-5$ $y=-2x-1$ The graph of both the function and the tangent line are shown in the answer section.
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