Linear approximation

A function’s first-order Taylor polynomial near a point, using its value and derivative to estimate nearby values.

Tangent line: A line that touches a curve at a point and, when defined, shares its slope there. The approximation is the tangent line expressed as a formula for nearby function values.
Function value: The output assigned by a function to a specified input. The value at the chosen point anchors the approximation on the graph.
Newton's method: An iterative root-finding method that uses tangent lines to improve successive estimates. Each update solves the linear approximation to the function at the current estimate.
Finite difference: A quotient of function-value differences used to approximate a derivative numerically. It estimates the slope from sampled values rather than evaluating the derivative directly.
Derivative: The instantaneous rate of change of a function, defined by a limit of difference quotients. Its value at the expansion point supplies the approximation’s slope.
Secant line: A line passing through two points on a curve, representing average change between them. As the second point approaches the expansion point, its slope approaches the tangent slope.
Error propagation: The study of how uncertainties in inputs affect uncertainty in computed outputs. A derivative-based linear approximation estimates how input errors change an output.
Piecewise linear interpolation: An approximation that joins sampled function values with straight-line segments across an interval. It builds a global, data-based approximation rather than a tangent estimate at one point.
Differential: A linear map that gives the first-order change of a function under a small input change. It expresses the predicted output change as derivative times input change.
Limit of a function: A value that a function’s outputs approach as its input approaches a specified point. Limits define the derivative and the small-error condition behind first-order accuracy.

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