## Curve fitting

Curve fitting is a process of trying to find function of a certain form that is the best fit to some data. The form of the function is defined by parameters. For instance, a straight line $ax+b$ is described by two parameters, $a$ and $b$. Fitting data to a straight line involves finding the 'best' values of $a$ and $b$. For a least squares fit of $N$ data points $(x_i,y_i)$ to a three parameter function $f(x,a,b,c)$, the following function is defined,

$$\large S=\sum\limits_{i=1}^N \left(y_i-f(x_i,a,b,c)\right)^2.$$

The paramters $a$, $b$, and $c$ that minimize $S$ are then calculated numerically. Below is an app that will calculate find the best fit of a three parameter function $f(x,a,b,c)$ to a set of data. The program requires a reasonable starting guess for the parameters. If you don't want to use a parameter, delete the contents of the starting guess field.

$f\left(x, a, b, c\right)$ =
$a_{\text{start}}$ =
$b_{\text{start}}$ =
$c_{\text{start}}$ =

$a_{\text{fit}}$ =
$b_{\text{fit}}$ =
$c_{\text{fit}}$ =

### Examples

 some random function of the form $ax \exp(-bx)$. An object falls down to the ground and has a certain initial height and velocity. Use the fit to calculate those quantities! (Note: x is the time in this example. The function value is the height of the object)