Sensitivity Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium

A weather model uses

T(P)=0.1P^2

where

P

is pressure. If

P=10

and measurement error is

\Delta P=\pm0.5

, estimate the resulting error in

T

Solution

1
Sensitivity at $P=10$ : $T'(P)=0.2P$ , so $T'(10)=2$ .
2
Estimated error: $\Delta T \approx T'(10)\cdot\Delta P = 2\times0.5=1.0$ . So $T$ has error of approximately $\pm1.0$ .

Answer

Error in

T

\approx \pm 1.0

The derivative provides a linear approximation for how errors propagate. Multiplying the input error by the derivative (sensitivity) gives the approximate output error. This is the basis of error propagation analysis.

About Sensitivity

In the context of functions, sensitivity measures how much the output changes in response to a small change in the input — high sensitivity means small input changes cause large output changes.

Learn more about Sensitivity →

More Sensitivity Examples

Example 1 easy

Compute the sensitivity [formula] for [formula] at [formula] with perturbation [formula].

Example 2 hard

A function [formula] is highly sensitive near large [formula]. Compare the sensitivity at [formula]

Example 3 easy

For [formula], compute the sensitivity at [formula] with [formula] and compare to [formula].

← All Sensitivity Examples Sensitivity Concept

Related Concepts

Rate of Change Derivative