Sensitivity (Meta) Examples in Math

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

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

Concept Recap

The degree to which a result or output changes in response to small changes in its inputs, parameters, or assumptions.

Is this result stable, or does a tiny change blow everything up?

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: High sensitivity means results are unreliable or require high precision.

Common stuck point: Sensitivity is not the same as magnitude — a model can produce large outputs while being insensitive to inputs, or produce small outputs while being highly sensitive.

Sense of Study hint: Compute the answer with the original inputs, then recompute with a slightly changed input (say, add 0.01). Compare the two outputs; a large difference signals high sensitivity.

Worked Examples

Example 1

easy
Compute f(x) = x^3 at x = 2 and x = 2.1. Find the sensitivity: by what percentage does f change when x changes by 5%?

Solution

  1. 1
    f(2) = 8, f(2.1) = 9.261.
  2. 2
    Change in f: 9.261 - 8 = 1.261. Relative change in f: \frac{1.261}{8} \approx 15.8\%.
  3. 3
    Change in x: \frac{0.1}{2} = 5\%.
  4. 4
    Sensitivity: a 5% increase in x causes about a 15.8% increase in f. The function is sensitive — it amplifies errors by a factor of about 3.

Answer

\text{5\% change in }x \Rightarrow \approx 15.8\%\text{ change in }f
Sensitivity measures how much the output changes relative to the input. For f(x) = x^3, the exponent 3 amplifies percentage changes roughly threefold, which can be derived from the derivative: f'(x)/f(x) \approx 3 \Delta x/x.

Example 2

medium
In the compound interest formula A = P(1+r)^t, compute the sensitivity of A to a small change \Delta r in the interest rate, using the derivative \frac{dA}{dr}.

Practice Problems

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

Example 1

easy
For f(x)=\sqrt{x}, compute f(100) and f(104). By what percentage does f change when x increases by 4%?

Example 2

medium
A model predicts exam score S = 10\log_{10}(h) where h is hours of study. If h changes from 10 to 11 hours, compute the change in S and discuss sensitivity.
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Background Knowledge

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

local vs global behavior