2. Approximating Function Values

We can discuss the properties of functions. For example, $f(x)=\sqrt{x}$. The derivative would be $f^{\prime }(x)=\frac{1}{2\sqrt{x}}$. The maximal domain would be $D=[0,\infty )$. Then lastly, we have the functional values. The values can be like $f(36)=6$, $f(100)=10$, $f(0.25)=0.5$, but how can we express $f(101)$?

When we utilize decimal expansion, we can get some obstacles.

Attention

1. Some decimal expansions have no repeating patterns.
2. Computation can cost a lot of time or other resources.

So in this case, we use approximations.

Definition

Given a function f(x) an approximation of a function value f(a) is a real number $\widehat{f(a)}$ that is “close” to $f(a)$.

We denote this as $f(a)\approx \widehat{f(a)}$.

Example

$\sqrt{101}\approx \sqrt{100}=10$

Approximation Error

Definition

Suppose $f(a)\approx \widehat{f(a)}$ then the approximate error $E$ is defined as

$E=|f(a)-\widehat{f(a)}|$

Similarly, the relative approximation error $\eta$ is defined as

$\eta =|\frac{f(a)-\widehat{f(a)}}{f(a)}|$

$\eta$ is the Greek alphabet Eta.

Example

If we consider that $\sqrt{101}\approx \sqrt{100}=10$ then

$E=|\sqrt{101}-10|\le 0.05$

If we consider that $\sqrt{101}\approx 10.05$ then

$E=|\sqrt{101}-10.05|\le 0.000125$

Approximation methods often come with restrictions so the corresponding approximation error depends on values to which it is applied.