Propagation of Error in Evaluating Functions Examples 1

# Propagation of Error in Evaluating Functions Examples 1

Recall from the Propagation of Error in Evaluating Functions page that if $y = f(x)$ is a differentiable function for $x \in [a, b]$ and $x_A, x_T \in [a, b]$ then we can approximate the error of $f(x_A)$ to $f(x_T)$ with either of the formulas

(1)
\begin{align} \quad \mathrm{Error} (f(x_A)) \approx f'(x_T) (x_T - x_A) \quad \quad \mathrm{or} \quad \quad \mathrm{Error} (f(x_A)) \approx f'(x_A) (x_T - x_A) \end{align}

More generally, we can approximate the error of $f(x_A)$ to $f(x_T)$ by $\mathrm{Error} (f(x_A)) \approx f'(\xi) (x_T - x_A)$ where $\xi$ is between $x_T$ and $x_A$.

Furthermore, we can approximate the relative error of $f(x_A)$ to $f(x_T)$ with the following formula:

(2)
\begin{align} \quad \mathrm{Rel} (f(x_A)) \approx \frac{f'(x_T)}{f(x_T)} x^T \mathrm{Rel} (x_A) \end{align}

We will now look at some examples of applying the formulas above.

## Example 1

**Approximate the error