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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  ? A hint comes from the following sketch of \(\sin(x)\) for \(x\) around \(0\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

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 
  • is a good approximation to \(f(x)\) for \(x\) values close to \(a\text{.}\)

Futher, 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.