Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 13 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - Chapter Review - Review Exercises - Page 926: 35



Work Step by Step

The general formula for the average rate of change from $x$ to $y$ can be written as: $\dfrac{f(y)-f(x)}{y-x}$. As $y\rightarrow x$, the rate of change becomes instantaneous (derivative). Here, we have: $f(x)=2x^2+3x+2$ $\lim\limits_{x\to 1}\dfrac{f(x)-f(1)}{x-1}=\lim\limits_{x\to 1}\dfrac{(2x^2+3x+2)-[(2)(1)^2+(3)(1)+2]]}{x-1} \\=\lim\limits_{x\to 1}\dfrac{2x^2+3x-5}{x-1} \\=\lim\limits_{x\to 1} \dfrac{(x-1)(2x+5)}{x-1} \\=\lim\limits_{x\to 1} (2x+5) \\=(2)(1)+5 \\=7$
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