Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.2 The Derivative as a Function - Exercises - Page 116: 86



Work Step by Step

We find the derivative: $$ f^{\prime}(x)=nx^{n-1} $$ Then at $x=c, m=f^{\prime}(c)=nc^{n-1}$. Hence, the tangent line is: $$ \begin{aligned} \frac{y-y_{1}}{x-x_{1}} &=m \\ \frac{y-c^n}{x-c} &=nc^{n-1} \\ y &=nc^{n-1} (x-c)+c^n \end{aligned} $$ Since the tangent line intersects with the $x-$axis at $x=0,$ then $Q$ has the coordinates $(c-\frac{c}{n},0), R$ has coordinates $(c,0)$, and the subtangent is $$ c-\left(c-\frac{c}{n}\right)=\frac{c}{n} $$
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