Linearization

Definition

Linearization is the process of approximating a function near a given point using the tangent line at that point. It provides a simpler linear function that closely matches the original function in the vicinity of the point.

5 Must Know Facts For Your Next Test

1. The linear approximation of a function $f(x)$ at $x = a$ is given by $L(x) = f(a) + f'(a)(x - a)$.
2. Linearization is most accurate close to the point of tangency and less accurate as you move further away.
3. The derivative $f'(a)$ represents the slope of the tangent line used for linearization.
4. Linear approximations can be used to estimate values of functions that are difficult to compute exactly.
5. Differentials can be used alongside linearization to approximate changes in function values for small changes in input.

Review Questions

Related terms

Tangent Line: A straight line that touches a curve at a single point without crossing it, representing the instantaneous rate of change (slope) at that point.

Derivative: A measure of how a function's output value changes as its input value changes, formally defined as $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.

Differential: An infinitesimal change in a function's value resulting from an infinitesimal change in its input, often denoted as $dy = f'(x)dx$.