Precalculus: Mathematics for Calculus, 7th Edition

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

Chapter 2 - Section 2.5 - Linear Functions and Models - 2.5 Exercises - Page 197: 52

Answer

$(a)$ $\frac{f(x)-f(a)}{x-a}$ $(b)$ In general a linear function has a form $f(x)=mx+b$ In our case $m=c$ and $b=f(a)-ca)$ The slope is $c$ $y$-intercept is $f(a)-ca$

Work Step by Step

$(a)$ In general we have average rate of change formula: $c=\frac{f(x_2)-f(x_1)}{x_2-x_1}$ Then we simply input the $x$ and $a$ values: $\frac{f(x)-f(a)}{x-a}$ $(b)$ We can simplify the equation: $c=\frac{f(x)-f(a)}{x-a}$ $c(x-a)=f(x)-f(a)$ $f(x)=cx-ca+f(a)$ $f(x)=cx+(f(a)-ca)$ In general a linear function has a form $f(x)=mx+b$ In our case $m=c$ and $b=f(a)-ca)$ The slope is $c$ We have $y$-intercept when $x=0$ $f(0)=c\times 0(f(a)-ca)=f(a)-ca$
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