Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 2 - Section 2.1 - Functions - Exercises - Page 157: 50

Answer

$3a^{2}+3ah+h^{2}$

Work Step by Step

$f(x)=x^{3}$ Find $f(a)$ by substituting $x$ with $a$: $f(a)=a^{3}$ Find $f(a+h)$ by substituting $x$ with $a+h$: $f(a+h)=(a+h)^{3}=a^{3}+3a^{2}h+3ah^{2}+h^{3}$ We have $f(a)$ and $f(a+h)$, we can substitute them into the expression $\dfrac{f(a+h)-f(a)}{h}$: $\dfrac{f(a+h)-f(a)}{h}=\dfrac{a^{3}+3a^{2}h+3ah^{2}+h^{3}-a^{3}}{h}=...$ Simplify the numerator: $...=\dfrac{3a^{2}h+3ah^{2}+h^{3}}{h}=...$ Take out common factor $h$ from the numerator and simplify the fraction: $...=\dfrac{h(3a^{2}+3ah+h^{2})}{h}=3a^{2}+3ah+h^{2}$
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