7. Approximating using Taylor's Polynomials

Problem

The anti-derivative of some functions, eg. $\cos (x^2)$, $\sin (x^2)$, and $(e^x{)}^2$, cannot be expressed in terms of elementary functions. Other antiderivatives are hard to figure out, eg.

$\int \frac{{\tan }^3(\ln (x))}{x}dx$

Solution: Approximate by Taylor’s polynomials.

Taylor Polynomials

Theorem

Suppose $T_n(x)$ is the nth Taylor polynomial of a function $f(x)$. Then

$\int T_n(x)dx$

is the $(n+1)$th order Taylor polynomial of an anti-derivative $F(x)$ of $f(x)$.

Taylor Polynomials - Accuracy

Corollary

Suppose $T_n(x)$ is the nth order Taylor polynomial of the function $f(x)$ about the point $x=a$.

Taylor’s inequality implies that the approximation error

$R_n(x)=|T_n(x)-f(x)|$

is small when $|x-a|$ is smaller.

When we use Taylor polynomials about a point $x=a$ to approximate the integral of $f(x)$ over the interval $[b,c]$ then the entire interval should be close to the point $x=a$.

Absolute Value of an Integral

Theorem

Let $f(x)$ be a function that is integrable over $[a,b]$ then

$|{\int }_a^{b}f(x)dx|\le {\int }_a^b|f(x)|dx$

Taylor’s Inequality - For Integrals

Corollary

Let $f$ be a function and $T_n$ its Taylor polynomial at the point $a$.

Let $[b,c]$ be an interval that contains the point $a$ and $M$ an upper bound for the function $|f^{(n+1)}(x)|$ on $D$, in other words:

$|f^{(n+1)}(x)|\le M$ holds for all $x$ in $D$.

Then

$|{\int }_b^{c}f(x)dx-{\int }_b^c{T}_n(x)dx|\le {\int }_b^c\frac{M}{(n+1)!}|x-a{|}^{n+1}dx$

Summary

Approach to approximate ${\int }_b^{c}f(x)dx$:

1. Compute Taylor polynomial $T_n(x)$ about $x=a$ for $f(x)$ where $a$ should be close to $[b,c]$.
2. Compute ${\int }_b^c{T}_n(x)dx$, this is your approximation
3. Taylor’s inequality for integrals provides an upper bound on approximation error.