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: 11

Answer

$a.$ The slope is $m_{tan}=-5$ $b.$ The equation is $y=-5x+1$ $c.$ The graph is on the figure below.
1510687001

Work Step by Step

$a.$ For the slope of the tangent line at point $P(a,f(a))$ we have by definition (1) with $a=1$ and $f(a) = -4$: $$m_{tan} = \lim_{x\to1}\frac{f(x)-f(1)}{x-1}= \lim_{x\to1}\frac{-5x+1-(-4)}{x-1} = \lim_{x\to1}\frac{-5x+5}{x-1} = \lim_{x\to1}\frac{-5(x-1)}{x-1}= \lim_{x\to1}(-5)=-5.$$ $b.$ From definition (1) we have $y-f(a) = m_{tan} (x-a)$. Using this formula with $a=1$, $f(a)=-4$ and $m_{tan}=-5$: $$y-(-4)=-5(x-1)\Rightarrow y+4=-5x+5$$ which gives $$y=-5x+1.$$ $c.$ The graph is on the figure below. The function is solid and the tangent is dashed. (do not be surprised that the two lines are one over another since the tangent of the straight line is that same straight line)
Small 1510687001
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