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 ${d}{y}\approx\Delta{y}$.

Differentials

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 ${y}$?