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 - Page 180: 2

Answer

(a) See the graph below. The graph crosses the $y$-axis at $x=0$, $f(0)=1$ (b) $f(x)=e^x$ is an exponential function (Blue graph in the image) $g(x)=x^e$ is a power function (Red graph in the image) $\frac{d}{dx}(f)=e^x$ $\frac{d}{dx}(g)=ex^{e-1}$ (c) $f(x)=e^x$ grows more rapidly than $g(x)=x^e$ for large values of $x$.
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Work Step by Step

(a) To sketch the graph by hand, we have to calculate several points of the function $f(x)=e^x$ and then connect these points (note: it's an exponential function, so it is continuous). $f(-1) = e^{-1}=\frac{1}{e}\approx\frac{1}{2.718}\approx0.368$ $f(0)=e^0=1$ $f(1)=e^1\approx2.718$ $f(2)=e^2\approx7.387$ The graph crosses the $y$-axis when $x=0$, so at $f(0)$, $y=e^0=1$ We can also make use of the fact that $(e^x)'=e^x$. Thus, the slope of the function at each point is equal to the $y$ value (so the slope and $y$ value at $x=0$ is $1$). (b) $f(x)=e^x$ is an exponential function (Blue graph in the image) $g(x)=x^e$ is a power function (Red graph in the image) $\frac{d}{dx}(f)=e^x$ $\frac{d}{dx}(g)=ex^{e-1}$ (c) The exponential function ($f(x)$) grows more rapidly than the power function ($g(x)$) for large values of $x$.
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