TRIUMPHS Project 8.2.1.
Fourier’s Infinite Series Proof of the Irrationality of \(e\) by Kenneth M. Monks
2
https://digitalcommons.ursinus.edu/triumphs_calculus/14/Section 8.2 The Natural Exponential
In exactly the same way that we can define \(\text{sqr}(x)\) via any of the Initial Value Problems in Drill 8.1.2 we can declare, by fiat, the following:
Definition 8.2.2. The Natural Exponential Function.
The function which satisfies the IVP
\begin{align}
\dfdx{y}{x}=y,\amp{}\amp{} y(0)=1.\tag{8.2}
\end{align}
is called the natural exponential function, denoted \(\exp(x)\text{.}\)
But as we said earlier, simply giving the function a name does not tell us anything about it.
In Section 7.3 we investigated IVP (8.2) in some detail and we were able to draw an approximate graph of the natural exponential at least near the initial value, \(\exp(0)\text{.}\) We will explore some of the other properties of this function here.
Recall that the Taylor Polynomial of a function, as seen in Section 7.4, is given by
\begin{equation*}
T_n(x)=y(0) +\frac{y^{\prime}(0)}{1!}x
+\frac{y^{(2)}(0)}{2!}x^2+\frac{y^{(3)}(0)}{3!}x^3
+ \cdots +
\frac{y^{(n)}(0)}{n!}x^n,
\end{equation*}
can approximate the function \(y(x)\) well, near a given point. Since we know the value of the natural exponential at the initial value let’s try to generate the Taylor Polynomial approximation of \(\exp(x)\) near that point.
Notice that because \(\exp(x)\) satisfies IVP (8.2) it also satisfies each of the following (why?):
\begin{align}
\exp^{(2)}(x)=\exp^\prime(x)\amp =\exp(x)\notag\\
\exp^{(3)}(x)=\exp^{(2)}(x)=\exp^\prime(x)\amp =\exp(x)\notag\\
\amp \ \ \vdots\notag\\
\exp^{(n)}(x)=\cdots = \exp^{(2)}(x) = \exp^\prime(x)\amp =\exp(x)\tag{8.3}
\end{align}
Problem 8.2.3.
Let \(E_n(x)\) be the \(n\)th degree Taylor polynomial approximation of \(\exp(x)\text{.}\)
(a)
Use equation (8.3) and the initial value \(\exp(0)=1\) to show that
\begin{equation*}
E_n(x) =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}
+ \cdots + \frac{x^n}{n!}.
\end{equation*}
(b)
On the same set of axes graph \(E_n(x)\) for \(n=1, 2,
3, 4\text{,}\) and \(n=5\text{.}\)
Since the Taylor Polynomial of a given function at \(x=0\) approximates the function near zero it is reasonable to suppose that near zero the graphs in part (b) of Problem 8.2.3 look like the graph of \(\exp(x) \text{.}\)
