Calculus I · Unit 2B · lesson

Error Estimation with Differentials

Concept

Learning objectives

Use differentials to estimate propagated measurement error.

Absolute and Relative Error

Explanation

Before the formulas

Newton's method in Error Estimation with Differentials repeatedly uses local linearity to solve a nonlinear equation. Each tangent line supplies an easier zero, which becomes the next estimate. The method can be spectacularly fast near a suitable root and unreliable from a poor starting point.

Record the iteration formula and a table of estimates. Check the function value as well as the digits of $x_n$ . A sequence that appears stable on a calculator display may still have converged to the wrong root, entered a cycle, or encountered a nearly horizontal tangent.

Explanation

Error propagation is derivative interpretation with uncertainty as the input change

If a measured quantity has uncertainty $|dx|$ , then $|dy|\approx|f'(x)|\,|dx|$ estimates the resulting output uncertainty. A large derivative magnifies measurement error; a small derivative dampens it.

Relative error compares uncertainty with the size of the quantity itself. State clearly whether the problem asks for absolute error, relative error, or percentage error, because they answer different practical questions.

Measurements are never infinitely precise. If a radius, voltage, mass, or time is known only within a tolerance, the derivative estimates how that uncertainty affects a computed output. Large derivative magnitude means the output is locally sensitive to input error.

Absolute error and relative error answer different questions. An error of one cubic centimeter may be negligible for a tank and disastrous for a microfluidic device; relative error compares the error with the size of the quantity being estimated.

If an input measurement has possible error $|dx|$ , the propagated absolute output error is approximately

|dy|=|f'(x)||dx|.

Relative error is

\frac{|dy|}{|y|},

and percentage error is the relative error multiplied by $100\%$ .

Guided walkthrough

Error in the volume of a sphere

A sphere's radius is measured as $8$ cm with possible error $0.02$ cm. Estimate the maximum error in the computed volume and the relative error.

Answer reveal

Worked solution

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Exercise

Estimate the error in $A=\pi r^2$ when $r=5\pm0.01$ .

Exercise

A cube side is measured with $0.2\%$ relative error. Estimate the relative error in volume.

Exercise

Explain why relative error is often more informative than absolute error when comparing quantities of different sizes.

Modeling lab

Manufacturing tolerance in a cylindrical part

A cylinder has fixed height $h=20$ mm and measured radius $r=5.00\pm0.02$ mm. With

V=\pi r^2h, \qquad dV=2\pi rh\,dr,

the estimated volume error is

|dV|\approx2\pi(5)(20)(0.02)=4\pi\text{ mm}^3.

The relative error is approximately

\frac{|dV|}{V}=\frac{2|dr|}{r}=0.008,

or $0.8\%$ . Squaring the radius doubles the relative sensitivity.

Optional advanced note

Sensitivity and condition numbers

The derivative measures absolute sensitivity. A dimensionless relative sensitivity is often written

\kappa(x)=\left|\frac{x f'(x)}{f(x)}\right|.

It estimates the relative output error produced by a relative input error. Numerical analysts call this a condition number. A large $\kappa$ warns that even accurate input data may produce an unstable output.

After the explanation

Use the section idea

Reading lens

A differentiable curve behaves like its tangent line over a small enough neighborhood, with concavity explaining the direction of error.

Mental model

Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.

Decision

Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.

Common trap

Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.

Check yourself

Can you give the estimate, error direction, units, and a reason the local line is trustworthy here?

Interactive checkapp-tolerance-01

A cylinder radius is $5.00\pm0.02$ mm with fixed height. Estimate relative volume error as a percent.

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Show hint

For $V\propto r^2$ , relative error is about twice the radius relative error.

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