$10$ pts$.$ Write a python function to smooth your data with a moving average. The function

should return an array that is the moving average of an input array of data that contains

noise or errors in it$.$

$10$ pts$.$ Write a python function to return an array that is the first derivative of an array

with respect to another array. The first input array is the independent variable. The

second input array is the dependent variable. The function should compute the derivative

using first order forward finite difference.

$10$ pts$.$ Write a python function to return an array that is the first derivative of an array

with respect to another array. The first input array is the independent variable. The

second input array is the dependent variable. The function should compute the derivative

using second order central finite difference. Note: this second derivative function may be

the same as the previous one but using an optional input variable to select first order,

second order, or even $3^{rd},4^{th},$dots, order $($BONUS$)$ accurate differencing.

$10$ pts$.$ Each. Write a python script to use your functions to analyze data.

a$.$ Implement python forms of the $1968$ Mustang GT equations given on the class

slides for $x,v,$ and $a.$

b$.$ Get $x_{\text{N}\text{o}\text{i}\text{s}\text{y}}=x+,$ where $$ is random noise $($e$.$g$.,$ numpy.random.rand$(50)/50).$

You may add noise to $v$ and $a$ to compare $({}_{v,\text{n}\text{o}\text{i}\text{s}\text{e}}$ and ${}_{a,\text{n}\text{o}\text{i}\text{s}\text{e}}).$

c$.$ Use your smoothing function on $x_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ to get $x_{smth}.$

d$.$ Use your differentiation functions on $x_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $x_{smth}$ to compute $v_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and

$v_{smth}.$ Note that you will have $2$ first derivatives for each order and a $2^{nd}$

order$).$

e$.$ Use your differentiation functions on $v_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $v_{smth}$ to compute $a_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and

$a_{smth}.$ Note that you will again have $2$ first derivatives for each order and a

$2^{nd}$ order$).$

f$.$ Generate a figure with sub$-$plots of $x,v,$ and $a$ in three rows, respectively, with

one $t$ horizontal axis. Note that there will be $x,x_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $x_{smth}$ in the first sub$-$

plot. Note also that the second sub$-$plot will show $v,1^{st}$ order $v_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $v_{smth}$;

$2^{nd}$ order $v_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $v_{smth}.$ Note also that the third sub$-$plot will show $a,1^{st}$ order

$a_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $a_{smth}$;$2^{nd}$ order $a_{\text{N}\text{o}\text{i}\text{s}\text{y}}$ and $a_{smth}.$ You can plot more if you wish, as

long as they are relevant.

g$.$ Print $($with nice statements, units, etc.$)$ the mean square error $($MSE$)$ between

properties plotted, where

MSE$(x)=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}(x_i-x_{\text{N}\text{o}\text{i}\text{s}\text{y}\text{,}i}{)}^2$

That MSE is indeed like the variance between dependent variables and a good way to see how

close the curves, or kinematic properties, are. A statistics course is highly recommended to learn

more about MSE... A sample implementation of MSE is numpy.mean $((x-x_{\text{N}\text{o}\text{i}\text{s}\text{y}}{)}^2).$

Try different numbers of points $($not just $50$ but $100,1000,$dots $).$ Try different equations. Have fun

w$/$this$!$

Note that the numerical differentiations will lead to arrays $1$ element less than the array being

differentiated. You may choose to extrapolate to get the arrays to be the same length $($using

material from a previous semester$).$ Alternatively, you may choose to make sure to plot against a

$t$ array with $1$ less element.