Sensitivity Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sensitivity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

A sensitive scale notices tiny weight differences. An insensitive one doesn't.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Sensitivity = rate of change. Steep slope = high sensitivity.

Common stuck point: Sensitivity can vary—function might be sensitive in some regions, not others.

Sense of Study hint: Try changing the input by a small amount (like 0.1) at different points and compare how much the output changes each time.

Worked Examples

Example 1

easy
Compute the sensitivity \Delta F / \Delta x for F(x) = 3x^2 at x = 5 with perturbation \Delta x = 0.1.

Solution

  1. 1
    Compute F(5) = 3(25)=75 and F(5.1)=3(26.01)=78.03.
  2. 2
    \Delta F = 78.03-75=3.03. Sensitivity = \Delta F / \Delta x = 3.03/0.1 = 30.3.
  3. 3
    Compare to derivative: F'(x)=6x, so F'(5)=30. The sensitivity approximates the derivative, with a small discrepancy due to \Delta x being finite.

Answer

Sensitivity \approx 30.3; derivative F'(5)=30
Sensitivity \Delta F/\Delta x measures how much the output changes per unit change in input. As \Delta x\to0, this ratio converges to the derivative. High sensitivity means small input changes produce large output changes.

Example 2

hard
A function F(x)=e^x is highly sensitive near large x. Compare the sensitivity at x=0 and x=5 using a perturbation of \Delta x=0.01.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
For f(x)=10x, compute the sensitivity at x=3 with \Delta x=0.5 and compare to \Delta x=0.1.

Example 2

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.
← Back to Sensitivity Formula Explained Practice Mode

Related Concepts

Rate of ChangeDerivative

Background Knowledge

These ideas may be useful before you work through the harder examples.

rate of change