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: 38


$a.$ The value of the derivative is $f'(0)=0$. $b.$ The equation is $y=0$ (the $x$ axis). $c.$ The graph is on the figure below.

Work Step by Step

$a.$ Using the definition of the derivative with $a=0$ we have $$f'(0)=\lim_{h\to0}\frac{f(0+h)-f(0)}{h}=\lim_{h\to0}\frac{3(0+h)^2-3\cdot0}{h}=\lim_{h\to0}\frac{3h^2}{h}=\lim_{h\to0}3h=3\cdot0=0.$$ $b.$ The equation of the tangent line through the point $(a,f(a))$ is given by $y-f(a)=f'(a)(x-a)$. Using $a=0$, $f(a) =f(1)= 3\cdot0^2=0$ and $f'(a)=f'(1)=0$ we have $$y-0=0\cdot(x-0)\Rightarrow y=0.$$ $c.$ The graph is on the figure below. The function is graphed by the solid line and the tangent is dashed (the tangent is $x$ axis actually).
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