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: 47

Answer

$a.$ 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.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.