Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 3 - Section 3.1 - Derivatives of Polynomials and Exponential Functions - 3.1 Exercises: 52

Answer

$$S'(100)=\frac{0.743}{100^{0.158}}\approx0.359 (tree/m^2)$$ $S'(100)$ denotes the instantaneous rate of increase of the number of trees every $m^2$ at an area of $100m^2$. At an area of $100m^2$, the instantaneuous rate of increase of the number of trees every $m^2$ is $0.359tree/m^2$.

Work Step by Step

$$S(A)=0.882A^{0.842}$$ 1) Find the derivative of $S(A)$ $$S'(A)=\frac{d}{dA}(0.882A^{0.842})$$ $$S'(A)=0.882\times(0.842A^{-0.158})$$ $$S'(A)\approx\frac{0.743}{A^{0.158}}$$ ($S'(A)$ is in $tree/m^2$) Therefore, $$S'(100)=\frac{0.743}{100^{0.158}}\approx0.359 (tree/m^2)$$ 2) The derivative of a function can be seen as the rate of change of that function. We know that $S(A)$ is the function of the number of tree species S in a given area A. Therefore, $S'(A)$ would be the rate of change of the number of trees species S every $m^2$ at an area A. That means $S'(100)$ denotes the instantaneous rate of increase of the number of trees every $m^2$ at an area of $100m^2$. At an area of $100m^2$, the instantaneuous rate of increase of the number of trees every $m^2$ is $0.359tree/m^2$.
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