Calculus (3rd Edition)

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

Chapter 3 - Differentiation - 3.1 Definition of the Derivative - Exercises - Page 103: 33


$ f'(0)=1$ Equation of the tangent line: $ y=x $

Work Step by Step

Recall that $ f'(a)=\lim\limits_{h \to 0}\frac{f(a+h)-f(a)}{h}$ Given that $ a=0$. Therefore, $ f'(0)=\lim\limits_{h \to 0}\frac{f(0+h)-f(0)}{h}=\lim\limits_{h \to 0}\frac{(h^{3}+h)-(0^{3}+0)}{h}$ $=\lim\limits_{h \to 0}\frac{h^{3}+h}{h}=\lim\limits_{h \to 0}(h^{2}+1)=1$ Equation of the tangent line is of the form $ y-f(a)=f'(a)(x-a)$ Knowing that $ f(a)=f(0)=0, f'(a)=f'(0)=1$ and $ a=0$, we obtain the equation of the tangent line through $ a $ as below. $ y-0=1(x-0)$ Or $ y=x $
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