Linear Approximation Using Derivatives
Simple Explanation
Near a point a where f(a) and f'(a) are both known exactly, the tangent line L(x) = f(a) + f'(a)(x−a) closely hugs the curve — so L(x) gives a quick, accurate estimate of f(x) for x near a, without needing to compute f(x) exactly.
Why Do We Need It?
Linear approximation gives fast, useful estimates when an exact value is hard or impossible to compute by hand — a technique used constantly in engineering, physics, and numerical methods.
See It
A square-root curve with a straight tangent line touching it at the point (4,2), the two nearly overlapping close to that point
Formula
Linear Approximation
L(x) = f(a) + f'(a)(x−a)
Uses the tangent line at a known point a to estimate the value of f at a nearby point x — since the tangent line closely hugs the curve near the point of tangency.
- a
- — a known point where f(a) and f'(a) can be computed exactly
- x
- — a nearby point where f(x) is being estimated
- L(x)
- — the linear approximation (estimate) of f(x)
When to use it: Whenever a quick estimate of a function's value is needed near a point where the exact value is already known.
Worked Example
Approximate √4.1 using a tangent line
Use linear approximation at a=4 to estimate √4.1, where f(x) = √x.
Why Does This Work?
The tangent line at a shares both the value f(a) and the slope f'(a) with the curve at that exact point — the best straight-line match to the curve's local behavior — so for x close to a, where the curve has not had room to bend away from that straight line yet, L(x) stays very close to f(x).
Real-Life Example
Estimating a measurement without a calculator
An engineer needs a quick estimate of √4.1 in the field, without access to a calculator.
Linear approximation using the easy, exact nearby value √4=2 gives a fast, sufficiently accurate estimate — 2.025 — without needing to compute the square root directly.
Practice
Using f(x)=√x and a=9, estimate √9.2 with linear approximation. (f'(9) = 1/6.)
MediumCommon mistake
Using linear approximation for an x-value far away from a — the tangent line only stays close to the curve near the point of tangency, and the estimate gets progressively worse the farther x is from a.
Quick Review
- L(x) = f(a) + f'(a)(x−a): the tangent line at a, used as an estimate for f(x) near a.
- Choose a to be a nearby point where f(a) and f'(a) are both easy to compute exactly.
- Accuracy degrades as x moves farther from a.