Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 2 - Derivatives - 2.1 Derivatives and Rates of Change - 2.1 Exercises - Page 114: 27


If $f(x) = 3x^2 - x^3$, find $f'(1)$ and use it to find an equation of the tangent line to the curve $y = 3x^2 - x^3$ at the point $(1,2)$.

Work Step by Step

To find $f'(1)$, differentiate $f(x)$ using the power and difference rule. If $f(x) = 3x^2 - x^3$, $f'(x) = 6x-3x^2$ Then, plug in $x=1$ to the $f'(x)$ equation. This yields $f'(1) = 3$. We now have the slope of the tangent line and a point on the tangent line. With this information we can find the equation of the tangent line using point-slope form. $y-2 = 3(x-1)$ Rearranging the terms, we get $y = 3x - 1$ This is the equation of the tangent line to $y = 3x^2 - x^3$ at $(1,2)$.
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