Written by the Fiveable Content Team • Last updated August 2025
Written by the Fiveable Content Team • Last updated August 2025
Definition
Linear approximation is a method of estimating the value of a function near a given point using the tangent line at that point. It leverages the fact that the tangent line closely resembles the function in the vicinity of the point.
The formula for linear approximation is $L(x) = f(a) + f'(a)(x - a)$, where $f(a)$ is the function value and $f'(a)$ is its derivative at $x = a$.
It is also known as the tangent line approximation.
Linear approximation works best when $x$ is close to $a$ because the function behaves almost linearly near this point.
The error in linear approximation can be analyzed using higher-order derivatives and Taylor series.
It provides a quick way to estimate values without computing complex functions.
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Related terms
Tangent Line: A straight line that touches a curve at a single point without crossing it, representing the instantaneous rate of change of the curve at that point.
Derivative: A measure of how a function changes as its input changes, represented by $f'(x)$ for a function $f(x)$.
Taylor Series: An infinite series of mathematical terms that when summed together approximate a mathematical function, based on derivatives at a single point.