Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 12 - Parametric Equations, Polar Coordinates, and Conic Sections - Chapter Review Exercises - Page 638: 16


$$ \frac{d y}{d x}=6t^2-1 $$ and at $(0,2)$, we have $$ \frac{d y}{d x}=5. $$

Work Step by Step

Since $x=\ln t$ and $y=3t^2- t$, then we have $$ \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{y^{\prime}(t)}{x^{\prime}(t)}=\frac{6t-1}{1/t} =6t^2-1 $$ and at $(0,2)$, we have $\ln t =0$, that is $t=1$ $$ \frac{d y}{d x}= 6-1=5. $$
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