# List of Notes - 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.

## Differentials

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.