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Calculus I · Unit 2B · exploration

Sensitivity, Elasticity, and Condition Numbers

How Input Error Becomes Output Error

The approximation

Δf≈f′(x)Δx\Delta f\approx f'(x)\Delta x

measures absolute sensitivity. Dividing by f(x)f(x) and rewriting gives

Δff(x)≈xf′(x)f(x)Δxx.\frac{\Delta f}{f(x)} \approx \frac{x f'(x)}{f(x)}\frac{\Delta x}{x}.

The factor

E(x)=xf′(x)f(x)E(x)=\frac{x f'(x)}{f(x)}

is called elasticity in economics and a relative condition number in numerical analysis, up to absolute value.

For f(x)=xnf(x)=x^n, E(x)=nE(x)=n. A 1%1\% relative input error produces approximately an n%n\% relative output error. This explains why volume r3r^3 is three times as sensitive, in relative terms, as radius.

Absolute sensitivity uses f'(x). Relative sensitivity uses the dimensionless factor E(x)=x f'(x)/f(x), which predicts the approximate percentage output change caused by a one-percent input change.
Read this graph as text

Elasticity measures how relative input error is amplified. Absolute sensitivity uses f'(x) . Relative sensitivity uses the dimensionless factor E(x)=x f'(x)/f(x) , which predicts the approximate percentage output change caused by a one-percent input change. The first box measures error as a fraction of the input rather than in raw units. Elasticity multiplies that fraction. The result is a fractional output error, so the comparison works across quantities with different units and scales. Large |E(x)| warns that small relative input errors may be strongly amplified.

Every relationship in elasticity measures how relative input error is amplified is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.

Why it matters: This visual distinguishes absolute derivative sensitivity from relative conditioning. It should help students understand why a derivative of 1000 is not automatically "large" and why percentage comparisons often provide the meaningful scale.

Visual study

Absolute sensitivity uses f'(x). Relative sensitivity uses the dimensionless factor E(x)=x f'(x)/f(x), which predicts the approximate percentage output change caused by a one-percent input change.

Modeling lab

A badly conditioned subtraction

When two nearly equal measured numbers are subtracted, the result may be tiny compared with either input. Small absolute measurement errors can then become enormous relative errors in the difference. Derivatives and condition numbers help identify such unstable calculations before they are trusted.

After the explanation

Use the section idea

Reading lens

Use local sensitivity, convexity, and convergence results to explain when familiar application methods become reliable global tools.

Mental model

Advanced results connect derivative evidence to error amplification, convergence speed, or global optimality under explicit hypotheses.

Decision

State the hypotheses before the conclusion and test the result on a concrete numerical or graphical example.

Common trap

Quoting elasticity, quadratic convergence, or convexity without checking units, root simplicity, or the relevant domain.

Check yourself

Can you describe both what the theorem guarantees and the failure mode its hypotheses exclude?

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Original instruction with traceable references.

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