## Section3.4Approximating Functions Near a Specified Point — Taylor Polynomials

Suppose that you are interested in the values of some function $f(x)$ for $x$ near some fixed point $a\text{.}$ When the function is a polynomial or a rational function we can use some arithmetic (and maybe some hard work) to write down the answer. For example:

\begin{align*} f(x) &= \frac{x^2-3}{x^2-2x+4}\\ f(1/5) &= \frac{ \frac{1}{25}-3}{\frac{1}{25}-\frac{2}{5}+4 } = \frac{\frac{1-75}{25} }{\frac{1-10+100}{25}}\\ &= \frac{-74}{91} \end{align*}

Tedious, but we can do it. On the other hand if you are asked to compute $\sin(1/10)$ then what can we do? We know that a calculator can work it out

\begin{align*} \sin(1/10) &= 0.09983341\dots \end{align*}

but how does the calculator do this? How did people compute this before calculators  1 Originally the word “calculator” referred not to the software or electronic (or even mechanical) device we think of today, but rather to a person who performed calculations. ? A hint comes from the following sketch of $\sin(x)$ for $x$ around $0\text{.}$

The above figure shows that the curves $y=x$ and $y=\sin x$ are almost the same when $x$ is close to $0\text{.}$ Hence if we want the value of $\sin(1/10)$ we could just use this approximation $y=x$ to get

\begin{gather*} \sin(1/10) \approx 1/10. \end{gather*}

Of course, in this case we simply observed that one function was a good approximation of the other. We need to know how to find such approximations more systematically.

More precisely, say we are given a function $f(x)$ that we wish to approximate close to some point $x=a\text{,}$ and we need to find another function $F(x)$ that

• is simple and easy to compute  2 It is no good approximating a function with something that is even more difficult to work with.
• is a good approximation to $f(x)$ for $x$ values close to $a\text{.}$

Further, we would like to understand how good our approximation actually is. Namely we need to be able to estimate the error $|f(x)-F(x)|\text{.}$

There are many different ways to approximate a function and we will discuss one family of approximations: Taylor polynomials. This is an infinite family of ever improving approximations, and our starting point is the very simplest.