### Subsection3.4.1Zeroth 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 $x=a\text{.}$

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..