Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 3 - Derivatives - 3.2 Working with Derivatives - 3.2 Exercises - Page 141: 6

Answer

\[f'\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {2,{\text{ }}x \leqslant 1} \\ { - 1,{\text{ }}x > 1} \end{array}} \right.\]

Work Step by Step

$$\eqalign{ & {\text{From the graph we have for }}x \leqslant 1{\text{ the points }}\left( {1,5} \right){\text{ and }}\left( {0,3} \right) \cr & {\text{The equation of the line is given by}} \cr & y - {y_1} = m\left( {x - {x_1}} \right) \cr & {\text{Using the points }}\left( {1,5} \right){\text{ and }}\left( {0,3} \right){\text{ we obtain:}} \cr & m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} \cr & m = \frac{{3 - 5}}{{0 - 1}} \cr & m = 2 \cr & {\text{The equation of the line is}} \cr & y - 3 = 2\left( {x - 0} \right) \cr & y - 3 = 2x \cr & y = 2x + 3 \cr & \cr & {\text{From the graph we have for }}x > 1{\text{ the points }}\left( {1,5} \right){\text{ and }}\left( {2,4} \right) \cr & {\text{The equation of the line is given by}} \cr & y - {y_1} = m\left( {x - {x_1}} \right) \cr & {\text{Using the points }}\left( {1,5} \right){\text{ and }}\left( {2,4} \right){\text{ we obtain:}} \cr & m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} \cr & m = \frac{{4 - 5}}{{2 - 1}} \cr & m = - 1 \cr & {\text{The equation of the line is}} \cr & y - 4 = - \left( {x - 2} \right) \cr & y - 4 = - x + 2 \cr & y = - x + 6 \cr & \cr & {\text{The function }}y = f\left( x \right){\text{ can be written as}} \cr} $$ \[\begin{gathered} f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {2x + 3,{\text{ }}x \leqslant 1} \\ { - x + 6,{\text{ }}x > 1} \end{array}} \right. \hfill \\ {\text{Differentiating we obtain}} \hfill \\ f'\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {2,{\text{ }}x \leqslant 1} \\ { - 1,{\text{ }}x > 1} \end{array}} \right. \hfill \\ \end{gathered} \] \[{\text{The graph of }}f'\left( x \right){\text{ is shown below}}\]
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.