Calculus: Early Transcendentals (2nd Edition)

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

Chapter 3 - Derivatives - 3.1 Introducing the Derivative - 3.1 Execises - Page 133: 17


$a.$ The slope is $m_{tan} = -7$. $b.$ The equation is $y=-7x$.

Work Step by Step

$a.$ Using the formula from definition (2) with $a=-1$ and $f(a)=7$ (coordinates of the point $P(-1,7)$) we have $$m_{tan}=\lim_{h\to0}\frac{f(-1+h)-f(-1)}{h}=\lim_{h\to0}\frac{-7(-1+h)-7}{h}=\lim_{h\to0}\frac{7-7h-7}{h}=\lim_{h\to0}\frac{-7h}{h}=\lim_{h\to0}-7h =-7.$$ $b.$ Using the formula $y-f(a)=m_{tan}(x-a)$ with the same values for $a$ and $f(a)$ as in part $a$ and the calculated value $m_{tan} = -7$ we get $$y-7=-7(x-(-1))\Rightarrow y-7=-7x-7$$ which gives $$y=-7x.$$
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