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Subsection 3.4.1 Zeroth Approximation — the Constant Approximation

The simplest functions are those that are constants. And our zeroth  3 It barely counts as an approximation at all, but it will help build intuition. Because of this, and the fact that a constant is a polynomial of degree 0, we'll start counting our approximations from zero rather than 1. approximation will be by a constant function. That is, the approximating function will have the form \(F(x)=A\text{,}\) for some constant \(A\text{.}\) Notice that this function is a polynomial of degree zero.

To ensure that \(F(x)\) is a good approximation for \(x\) close to \(a\text{,}\) we choose \(A\) so that \(f(x)\) and \(F(x)\) take exactly the same value when \(x=a\text{.}\)

\begin{gather*} F(x)=A\qquad\text{so}\qquad F(a)=A=f(a)\implies A=f(a) \end{gather*}

Our first, and crudest, approximation rule is

An important point to note is that we need to know \(f(a)\) — if we cannot compute that easily then we are not going to be able to proceed. We will often have to choose \(a\) (the point around which we are approximating \(f(x)\)) with some care to ensure that we can compute \(f(a)\text{.}\)

Here is a figure showing the graphs of a typical \(f(x)\) and approximating function \(F(x)\text{.}\)

At \(x=a\text{,}\) \(f(x)\) and \(F(x)\) take the same value. For \(x\) very near \(a\text{,}\) the values of \(f(x)\) and \(F(x)\) remain close together. But the quality of the approximation deteriorates fairly quickly as \(x\) moves away from \(a\text{.}\) Clearly we could do better with a straight line that follows the slope of the curve. That is our next approximation.

But before then, an example:

Use the constant approximation to estimate \(e^{0.1}\text{.}\)

Solution First set \(f(x) = e^x\text{.}\)

  • Now we first need to pick a point \(x=a\) to approximate the function. This point needs to be close to \(0.1\) and we need to be able to evaluate \(f(a)\) easily. The obvious choice is \(a=0\text{.}\)
  • Then our constant approximation is just
    \begin{align*} F(x) &= f(0) = e^0 = 1\\ F(0.1) &= 1 \end{align*}

Note that \(e^{0.1} = 1.105170918\dots\text{,}\) so even this approximation isn't too bad..