Task $2(17$ points$)$

A student collected data from a tensile testing experiment $($load in Newtons as a function of time in

seconds$),$ but there was electronic noise in the load cell used to record the data. The recorded data is

available in noisy $2.x$ lsx$.$

The student needs to smooth the data using a least$-$squares approach. In this case, there is a polynomial

relationship between $x$ and $y$ values, but it is clear that the relationship is not linear.

Week $15$ Assignment

Introduction to Structured Programming $($MAE $1090)$

Due Sunday, May $5,11$:$59$ pm $($for all sections$)$

The built$-$in MATLAB function, polyfit, will determine values of vector $p$ that will fit a polynomial of a

specified order using a least$-$squares approach to data given as $x$ and $y$ vectors, where the general form

of a polynomial of order $n$ approximating the data is:

$p_1{x}^n+p_2{x}^{n-1}+$cdots$+p_{n}x+p_{n+1}$

Once $p$ has been determined, a vector $y^{**}$ can be determined, which gives the values of $y$ approximated

using an $n^{th}$ order polynomial, using the built$-$in MATLAB function, polyval.

For instance, a $5^{th}$ order polynomial is given as: $y^{**}=p_1{x}^5+p_2{x}^4+p_3{x}^3+p_4{x}^2+p_{5}x+p_6.$

$($See the MATLAB help for polyfit and polyval.$)$

Write MATLAB code to approximate the data for the student, as described below.

Include comments explaining your code.

The main function should call a subfunction to read the data $($using readtable$)$ from the provided

spreadsheet file.

The main function should call a second subfunction to approximate the data using polynomials of

orders $3-6($using polyfit and polyval$).$

Output a plot containing the original data plotted with black circles with a marker size of $15$ and a

linewidth of $2.$ Superimpose plots of the approximated load values for polynomials of order $3-6$

using lines with a linewidth of $4$ and different colors for each polynomial order. Set the $x$ and $y$ axes

limits so that data points are not located on the edge of the plot. Create a legend with labels for

the original data and the approximated values, specifying the polynomial order for each color line.

Label the $x$ and $y$ axes appropriately $($include units$).($Make sure the legend is positioned where it

doesn't cover any data.$)$

For extra credit, create a second plot showing the errors associated with each polynomial. $($Plot

the values of $(y^{**}-y)$ vs$.x.)$ Use dots of different colors for each polynomial $($matching the colors

for polynomial order in the first plot$).$ The plot symbols should have a marker size of $20.$ Provide

appropriate labels for the $x$ and $y$ axes, create a legend with labels for polynomial order, and include

an appropriate plot title with a fontsize of $14.).($Make sure the legend is positioned where it doesn't

cover any data.$)$