Calculus I · 2B · lesson
Error Estimation with Differentials
Learn error estimation with differentials with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Local linearity, differentials, and Newton's methodWhat this section is building
Learn error estimation with differentials with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Linearization, differentials, and Newton's method reuse one local line for estimation, uncertainty, or a better root guess.
Choose a nearby easy input, record the local slope, and state why the requested change is small enough for the model.
Presenting a tangent estimate as exact or running Newton iterations without checking residuals and failure modes.
Learning objectives
Use differentials to estimate propagated measurement error.
Absolute and Relative Error
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 . 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.
Error propagation is derivative interpretation with uncertainty as the input change
If a measured quantity has uncertainty , then 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 , the propagated absolute output error is approximately
Relative error is
and percentage error is the relative error multiplied by .
Error in the volume of a sphere
A sphere's radius is measured as cm with possible error cm. Estimate the maximum error in the computed volume and the relative error.
Worked solution
Write a real attempt before opening the supplied answer.
Estimate the error in when .
A cube side is measured with relative error. Estimate the relative error in volume.
Explain why relative error is often more informative than absolute error when comparing quantities of different sizes.
Manufacturing tolerance in a cylindrical part
A cylinder has fixed height mm and measured radius mm. With
the estimated volume error is
The relative error is approximately
or . Squaring the radius doubles the relative sensitivity.
Sensitivity and condition numbers
The derivative measures absolute sensitivity. A dimensionless relative sensitivity is often written
It estimates the relative output error produced by a relative input error. Numerical analysts call this a condition number. A large warns that even accurate input data may produce an unstable output.
app-tolerance-01A cylinder radius is mm with fixed height. Estimate relative volume error as a percent.
Your work stays on this device. No account or AI grader is used.
Show hint
For , relative error is about twice the radius relative error.
Attempt once to unlock the solution
Submit an answer first. The hint is available now.
Source & rights
Original instruction with traceable references.
BetterGrades-original composition declared by source handoff; owner provenance review required before public release
Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.