M243: Calculus II
Section 11.11
Applications of Taylor Series¶
We begin this final section of Calculus II notes with a cautionary tale.
Consider the function
$$
f(x) = \begin{cases}
e^{-1/x^2} & x \neq 0, \\
0 & x = 0.
\end{cases}
$$
It's not hard to show that this function is continuous at $x_0 = 0$. (Do it!)
We'd like to consider the Maclaurin series for this function. As we saw in the previous section of the notes, a necessary condition for $f$ to have a power series representation at $x_0$ is that $f$ must be infinitely-differentiable at $x_0$. We won't check that this $f$ has all infinitely many derivatives at $x_0 = 0$, but we should at least check that the first derivative exists.
Video solution, part I
If we believe that this function is inifinitely differentiable at $x_0 = 0$ (it is), then we might guess that it has a Maclaurin series representation on some interval containing the origin.
Let's see what Taylor's Remainder Theorem has to say about this.
Video solution, part II
A short video goes here.
We've just shown that this function does not equal its Maclaurin series, without even writing down its Maclaurin series. We may interpret this as proof that it is not worth our time to even attempt to do so.
Given all of the information about this function that we've just discovered, take a look at its graph in an interval containing the origin. Can you explain in terms of the graph why $f$ is not equal to its Maclaurin series?
See the graph
For the remainder of these notes we will deal only with functions that equal their Taylor series. However it is very important not to take this fact for granted, as the previous discussion illustrates.
In practice, one of the most frequent uses of Taylor series is to approximate a given function by one of its Taylor polynomials (partial sums) near a point $x_0$. When doing this, it is very important to be able to approximate the error introduced in truncating the Taylor series.
We look at some examples below.
1. Example¶
Find the $5^\text{th}$ degree Taylor polynomial for $f(x) = \sinh(2x)$ centered at $x_0 = 0$, and estimate the maximum error on the interval $[-1,1]$.
Video solution
2. Example¶
How many terms of the Taylor-Maclaurin series for $f(x) = \ln(1 + x)$ are needed to estimate $\ln(1.4)$ correct to within $0.001$?
Video solution
3. Example¶
An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are $q$ and $-q$ and are located at a distance $d$ from each other, then the electric field $E$ at the point $P$ is
$$
E = \frac{q}{D^2} - \frac{q}{(D+d)^2},
$$
where $D$ is the distance from a point $P$ to the dipole.
By expanding this expression for $E$ as a series in powers of $d/D$, show that $E$ is approximately proportional to $1/D^3$ when $P$ is far away from the dipole.
Video solution
4. Newton's Method¶
Recall from Calculus I that Newton's method for appoximating a root $r$ of the equation $f(x) = 0$ with initial approximation $r \approx x_0$ is to iteratively plug in $x_k$ to the formula
$$
x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)},
$$
for $k = 0,1,2,...$.
Use Taylor's Inequality with $n=1$, $x_0 = x_k$, and $x = r$ to show that if $f''(x)$ exists on an interval $I$ containing $r$, $x_k$, and $x_{k+1}$, and $\vert f''(x)\vert \leq M$, $\vert f'(x) \vert \geq K > 0$ on $I$, then
$$
\vert x_{k+1} - r \vert \leq \frac{M}{2K} \;\vert x_k - r \vert^{\,2}.
$$
Interpret this result.
Video solution
We conclude this course with a celebration of unequivocal beauty.
5. Euler's Formula¶
Use the Maclaurin series for $e^z$, $\cos z$, and $\sin z$ to show that
$$
e^{i\theta} = \cos\theta + i\,\sin\theta
$$
for all real numbers, $\theta$.
Once you believe this formula, plug in $\theta = \pi$.
Video solution
Discussion¶
Questions? You can ask them here.