Category: Differentials

Linear Approximations

After studying the differentials, we know that if $$$\Delta{y}={f{{\left({a}+\Delta{x}\right)}}}-{f{{\left({a}\right)}}}$$$ and $$${d}{y}={f{'}}{\left({x}\right)}\Delta{x}$$$. This means that $$$\Delta{x}$$$ becomes very small, i.e. if we let $$$\Delta{x}\to{0}$$$, we can write that $$${d}{y}\approx\Delta{y}$$$.


Suppose that we are given a function $$$y={f{{\left({x}\right)}}}$$$. Consider the interval $$${\left[{a},{a}+\Delta{x}\right]}$$$. The corresponding change in $$${y}$$$ is $$$\Delta{y}={f{{\left({a}+\Delta{x}\right)}}}-{f{{\left({a}\right)}}}$$$.

Using Differentials to Estimate Errors

Suppose that we measured some quantity $$$x$$$ and know error $$$\Delta{y}$$$ in measurements. If we have function $$$y={f{{\left({x}\right)}}}$$$, how can we estimate error $$$\Delta{y}$$$ in measurement of $$${y}$$$?