Sensitivity Math Example 1

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

Example 1

easy
Compute the sensitivity ฮ”F/ฮ”x\Delta F / \Delta x for F(x)=3x2F(x) = 3x^2 at x=5x = 5 with perturbation ฮ”x=0.1\Delta x = 0.1.

Solution

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

Answer

Sensitivity โ‰ˆ30.3\approx 30.3; derivative Fโ€ฒ(5)=30F'(5)=30
Sensitivity ฮ”F/ฮ”x\Delta F/\Delta x measures how much the output changes per unit change in input. As ฮ”xโ†’0\Delta x\to0, this ratio converges to the derivative. High sensitivity means small input changes produce large output changes.

About Sensitivity

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

Learn more about Sensitivity โ†’

More Sensitivity Examples

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].

Example 4 medium

A weather model uses [formula] where [formula] is pressure. If [formula] and measurement error is [f

โ† All Sensitivity Examples Sensitivity Concept

Related Concepts

Rate of ChangeDerivative