Summary of Linear Approximations and Differentials

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

is an approximation for the change in $y$ . The actual change in $y$ is

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

Key Equations

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

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

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

propagated error: the error that results in a calculated quantity $f(x)$ resulting from a measurement error $dx$

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

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