Calculus Concepts: An Informal Approach to the Mathematics of Change 5th Edition

Published by Brooks Cole
ISBN 10: 1-43904-957-2
ISBN 13: 978-1-43904-957-0

Chapter 2 - Describing Change: Rates - 2.5 Activities - Page 172: 10

Answer

The rate of change of the function $m(p)=4p+p^2$ is equal to $4+2p$ and $\displaystyle\left.\frac{dm}{dp}\right|_{p=-2}=0$

Work Step by Step

\[m(p)=4p+p^2\] The derivative of the function $m(p)=4p+p^2$ is equal to \[\frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{m(p+h)-m(p)}{h}\] \[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{[4(p+h)+(p+h)^2]-[4p+p^2]}{h}\] \[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{[(4p+4h)+(p^2+h^2+2ph)]-[4p+p^2]}{h}\] \[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{h^2+4h+2ph}{h}\] \[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}\frac{(h+4+2p)h}{h}\] \[\Rightarrow \frac{dm}{dp}=\lim_{h\rightarrow 0}(h+4+2p)\] \[\Rightarrow \frac{dm}{dp}=0+4+2p=4+2p\] Substitute $p=-2$ \[\left.\frac{dm}{dp}\right|_{p=-2}=4+2(-2)=0\] Hence , The rate of change of the function $m(p)=4p+p^2$ is equal to $4+2p$ and $\displaystyle\left.\frac{dm}{dp}\right|_{p=-2}=0$
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