Error and Relative Error of Approximations Examples 1

# Error and Relative Error of Approximations Examples 1

Recall from the Error and Relative Error of Approximations page that if $x_T$ represents a true value and $x_A$ represents an approximation of $x_T$, then the error of $x_A$ from $x_T$ is given by the formula:

(1)
\begin{align} \quad \mathrm{Error} (x_A) = x_T - x_A \end{align}

Furthermore, the relative error of $x_A$ from $x_T$ is given by the formula:

(2)
\begin{align} \quad \mathrm{Rel} (x_A) = \frac{\mathrm{Error} (x_A)}{x_T} = \frac{x_T - x_A}{x_T} \end{align}

We will now look at some more examples of computing errors and relative errors.

## Example 1

An sample yields a true value of $x_T = 25044.33$. A mathematical model estimates the experiment for which this sample was taken and approximates $x_T$ as $x_A = 24993.53$. Find the error and relative error of $x_A$ from $x_T$.

Applying the error formula from above and we have that:

(3)
\begin{align} \quad \mathrm{Error} (24993.53) = 25044.33 - 24993.53 = 50.8 \end{align}

And applying the relative error formula from above and we have that:

(4)
\begin{align} \quad \mathrm{Rel} (24993.53) = \frac{50.8}{25044.33} \approx 0.002028... \end{align}

## Example 2

A function $p$ is defined on the interval $[0, 1]$ such that $0 ≤ p(x) ≤ 2$ for all $x \in [0, 1]$. Suppose that $p$ approximates a function $f(x) = x^2$ on the interval $[0, 1]$. Find the largest possible error of $p$ from $f$.

We are given that $0 ≤ p(x) ≤ 2$ for $x \in [0, 1]$. Furthermore, we note that the function we are approximating, $f$ is also bounded on $[0, 1]$. Furthermore, $f$ is increasing on $[0, 1]$. We can see this from the fact that $f(x) = x^2$ is a parabola that opens up whose vertex is at $(0, 0)$, or alternatively, we can see this since $f'(x) = 2x > 0$ for $x \in [0, 1]$. The minimum value of $f$ on $[0, 1]$ is $0$ and occurs at $x = 0$ since $f(0) = 0$, and furthermore, the maximum value of $f$ on $[0, 1]$ is $1$ and occurs at $x = 1$ since $f(1) = 1$. Therefore $0 ≤ f(x) ≤ 1$ for $x \in [0, 1]$. The largest possible error is thus $-2$ which occurs if $p(0) = 2$ and $f(0) = 0$.

From Example 2 above, we have bounded our error, that is for $x \in [0, 1]$ we have that:

(5)
\begin{align} \quad \mid \mathrm{Error} (p(x)) \mid ≤ 2 \end{align}

We will see that this will be important later when the true values for which we want to compute error for are unknown.