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: 47


$a.$ True; $b.$ Not true; $c.$ True.

Work Step by Step

$a.$ The graph of the linear function is a straight line. Any secant or a tangent of a straight line is that very same straight line, so this statement is true. $b.$ This doesn't have to be true. One example is the linear function where the slopes are equal. Another conterexample is in the next part. $c.$ The slope of the secant line is given by $$m_{sec}=\frac{f(x+h)-f(x)}{h}=\frac{(x+h)^2-x^2}{h}=\frac{x^2+h^2+2xh-x^2}{h}=\frac{h^2+2xh}{h}=h+2x.$$ The slope of the tangent at point $P$ is given by $$m_{tan}=\lim_{h\to0}m_{sec}=\lim_{h\to0}(h+2x)=0+2x=2x.$$ Since $h>0$ we see that $2x+h>2x$ which means that $m_{sec}>m_{tan}$ so the statement is correct.
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