Linear approximation

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.

5 Must Know Facts For Your Next Test

1. 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 $x = a$.
2. It is also known as the tangent line approximation.
3. Linear approximation works best when $x$ is close to $a$ because the function behaves almost linearly near this point.
4. The error in linear approximation can be analyzed using higher-order derivatives and Taylor series.
5. It provides a quick way to estimate values without computing complex functions.

Review Questions

• What is the formula for linear approximation?
• Why does linear approximation work best near the point of tangency?
• How can you determine if your linear approximation has significant error?

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.