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 $x$.
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 $x$.
