Differentials and Approximations

Differentials and Approximations

In engineering, exact measurements are often impossible. Differentials provide a powerful tool to approximate changes in functions and analyze error propagation.

Linear Approximation

We can approximate the value of a function $f(x)$ near a known point $a$ using the tangent line at that point. This is called the linearization $L(x)$ of $f$ at $a$:

$$L(x) = f(a) + f'(a)(x - a)$$

For $x$ close to $a$, $f(x) \approx L(x)$.

Error Propagation

If a quantity $x$ is measured with a possible error $dx$ (or $\Delta x$), the propagated error in a calculated quantity $y = f(x)$ is approximately $dy$.

• Absolute Error: $dy \approx f'(x) dx$
• Relative Error: $\frac{dy}{y} \approx \frac{f'(x) dx}{f(x)}$
• Percentage Error: Relative Error $\times 100\%$

This concept is crucial in civil engineering surveying and materials testing, where measurement tolerances must be accounted for in design calculations.