Category: Differentials

Linear Approximations

After studying differentials we know that if `Delta y=f(a+Delta x)-f(a)` and `dy=f'(x)Delta x` then making `Delta x` very small, i.e. if we let `Delta x->0` we can write that `dy~~Delta y`.

This can be rewritten as `f(a+Delta x)-f(a)~~f'(x)Delta x`.


Suppose that we are given function `y=f(x)`. Consider interval `[a,a+Delta x]`. Corresponding change in `y` is `Delta y=f(a+Delta x)-f(a)`.

We are interested in the following question: is there exist constant `C` such that `Delta y~~C Delta x` when `Delta x->0`?

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(x)`, how can we estimate error `Delta y` in measurement of `y`?

Since error is very small we can write that `Delta y ~~dy`, so error in measurement is differential of the function. Since `dx=Delta x`, then error in measurement of `y` can be caluclated using formula `dy=f'(x)dx`.