Summary of Linear Approximations and Differentials

A differentiable function [latex]y=f(x)[/latex] can be approximated at [latex]a[/latex] by the linear function
[latex]L(x)=f(a)+f^{\prime}(a)(x-a)[/latex]
For a function [latex]y=f(x)[/latex], if [latex]x[/latex] changes from [latex]a[/latex] to [latex]a+dx[/latex], then
[latex]dy=f^{\prime}(x) \, dx[/latex]

is an approximation for the change in [latex]y[/latex]. The actual change in [latex]y[/latex] is

[latex]\Delta y=f(a+dx)-f(a)[/latex]
A measurement error [latex]dx[/latex] can lead to an error in a calculated quantity [latex]f(x)[/latex]. The error in the calculated quantity is known as the propagated error. The propagated error can be estimated by
[latex]dy\approx f^{\prime}(x) \, dx[/latex]
To estimate the relative error of a particular quantity [latex]q[/latex], we estimate [latex]\dfrac{\Delta q}{q}[/latex]

Key Equations

differential: the differential [latex]dx[/latex] is an independent variable that can be assigned any nonzero real number; the differential [latex]dy[/latex] is defined to be [latex]dy=f^{\prime}(x) \, dx[/latex]

differential form: given a differentiable function [latex]y=f^{\prime}(x)[/latex], the equation [latex]dy=f^{\prime}(x) \, dx[/latex] is the differential form of the derivative of [latex]y[/latex] with respect to [latex]x[/latex]

linear approximation: the linear function [latex]L(x)=f(a)+f^{\prime}(a)(x-a)[/latex] is the linear approximation of [latex]f[/latex] at [latex]x=a[/latex]

propagated error: the error that results in a calculated quantity [latex]f(x)[/latex] resulting from a measurement error [latex]dx[/latex]

relative error: given an absolute error [latex]\Delta q[/latex] for a particular quantity, [latex]\dfrac{\Delta q}{q}[/latex] is the relative error.

tangent line approximation (linearization): since the linear approximation of [latex]f[/latex] at [latex]x=a[/latex] is defined using the equation of the tangent line, the linear approximation of [latex]f[/latex] at [latex]x=a[/latex] is also known as the tangent line approximation to [latex]f[/latex] at [latex]x=a[/latex]