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 117: 91


$$\frac{a}{n} $$

Work Step by Step

We know that: $$ f^{\prime}(x)=nx^{n-1} $$ Then at $x=a, m=f^{\prime}(a)=na^{n-1},$ hence the tangent line is $$ \begin{aligned} \frac{y-y_{1}}{x-x_{1}} &=m \\ \frac{y-a^n}{x-a} &=na^{n-1} \\ y &=na^{n-1} (x-a)+a^n \end{aligned} $$ since the tangent line intersect with $x-$ axis at $x=0,$ then $Q$ has coordinates $(a-\frac{a}{n},0), R$ has coordinates $(c,0)$ and the subtangent is $$ a-\left(a-\frac{a}{n}\right)=\frac{a}{n} $$
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