Linear Approximation Error Estimator

Math Calculus • Applications of Derivatives

Written by STEM Calculators Team Published December 23, 2025 Updated February 24, 2026

9. Linear Approximation Error Estimator

Estimate \(f(x)\approx f(a)+f'(a)(x-a)\) and bound the error using \(\lvert R(x)\rvert\le \dfrac{M}{2}(x-a)^2\), where \(M\ge \max\limits_{t\in I}\lvert f''(t)\rvert\).

Inputs

Function \(f(x)\)

Supported: + − * / ^, variable x, constants pi, e, sin cos tan, ln log, sqrt abs exp. Implicit multiplication: 2x, (x+1)(x-1), 3sin(x).

Output format

Expansion point \(a\)

You can use pi or e.

Target point \(x\)

Slider for \(x\) near \(a\)

Drag slider to update \(x\) quickly.

Slider half-range \(\Delta\)

Slider spans \(x\in[a-\Delta, a+\Delta]\).

Error-bound \(M\) mode Estimate \(M\) numerically (sampling) Use manual \(M\) (guaranteed upper bound)

If you need a guaranteed bound, provide a safe \(M\) yourself.

Manual \(M\)

Used only if “manual \(M\)” is selected.

Interval \(I\) for \(\max |f''|\) Use \(I\) between \(a\) and \(x\)

If Auto is on, the boxes update to \([\,\min(a,x),\,\max(a,x)\,]\).

Sampling points

Higher = slower, but less chance to miss peaks.

Graph options Show grid Show error band \(L(x)\pm \dfrac{M}{2}(x-a)^2\) (on \(I\)) Mark \(a\), \(x\), and approximation

Ready

Graph

Drag to pan • wheel/pinch to zoom • shows \(f(x)\), linearization \(L(x)\), and (optional) error band on \(I\).

Graph center \(c\)

Auto-fit recommended.

Graph window half-width \(w\)

Graph uses \(x\in[c-w,c+w]\).

x: 0, y: 0, zoom(px/unit): 80 Tip: For huge scales, zoom out or use Auto fit.

Result

Enter \(f(x)\), choose \(a\) and \(x\), then click Estimate.

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9. Linear Approximation Error Estimator — Theory

The linear approximation (also called the linearization) of a differentiable function \(f\) at a point \(x=a\) is the tangent line approximation:

\[ L(x) = f(a) + f'(a)(x-a). \]

This approximation is especially accurate when \(x\) is close to \(a\).

Why the error is about quadratic

If \(f\) is twice differentiable near \(a\), then the error behaves like a second-order term. The classical remainder estimate (from Taylor’s theorem) gives:

\[ f(x) = f(a) + f'(a)(x-a) + R(x), \qquad \lvert R(x)\rvert \le \frac{M}{2}(x-a)^2, \] \[ \text{where } M \ge \max_{t\in I} \lvert f''(t)\rvert \text{ and } I \text{ is an interval containing } a \text{ and } x. \]

How to use the bound

Choose your expansion point \(a\) and the target point \(x\).
Compute \(f(a)\) and \(f'(a)\), then form \(L(x)=f(a)+f'(a)(x-a)\).
Find a number \(M\) that is at least as large as \(\max_{t\in I}\lvert f''(t)\rvert\) for an interval \(I\) between \(a\) and \(x\).
Compute the error bound \(\dfrac{M}{2}(x-a)^2\).

Worked example

Approximate \(\sin(0.1)\) using \(a=0\).

\[ f(x)=\sin(x),\quad f(0)=0,\quad f'(x)=\cos(x)\Rightarrow f'(0)=1. \] \[ L(x)=0+1\cdot(x-0)=x. \quad\Rightarrow\quad \sin(0.1)\approx L(0.1)=0.1. \]

For the remainder bound we use \(f''(x)=-\sin(x)\), so \(\lvert f''(x)\rvert\le 1\) for all \(x\). Taking \(M=1\) gives a simple (conservative) bound:

\[ \lvert R(0.1)\rvert \le \frac{1}{2}(0.1)^2 = 0.005. \]

The actual error is much smaller: \(\sin(0.1)\approx 0.0998334\), so \(\lvert 0.0998334 - 0.1\rvert \approx 0.0001666\).

Important notes

The bound requires a correct \(M\). Any number \(M\) larger than \(\max_{t\in I}\lvert f''(t)\rvert\) works, but larger \(M\) makes the bound looser.
If you estimate \(M\) numerically by sampling, it may underestimate the true maximum. For a guaranteed bound, choose a safe manual \(M\).
The approximation is best when \(\lvert x-a\rvert\) is small, because the bound grows like \((x-a)^2\).

Frequently Asked Questions

What is the linear approximation (linearization) of a function at x=a?

The linear approximation is the tangent-line model L(x)=f(a)+f'(a)(x-a). It is most accurate when x is close to a.

How is the linear approximation error bound computed?

If M is an upper bound for max |f''(t)| on an interval I containing a and x, then the remainder satisfies |R(x)| <= (M/2)(x-a)^2. The calculator uses your chosen M mode and interval to compute this bound.

What does M represent and why does it matter?

M bounds the maximum magnitude of the second derivative on the interval, so it controls how large the error bound can be. A larger M produces a looser (more conservative) bound.

Why might the calculator offer both estimated M and manual M?

Sampling can miss sharp peaks of |f''(t)| and may underestimate the true maximum, so it is not guaranteed. Manual M lets you provide a safe upper bound when you need a guaranteed error bound.

How do I choose the interval I for the error bound?

A common choice is the interval between a and x, because the Taylor remainder bound uses a region containing both points. Using a wider interval can increase max |f''| and make the bound larger.

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Frequently Asked Questions

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