Data$=$DataRead$[\text{'}$Data$\text{'}]$

In the $2$D array 'Data', rows correspond to data points, $1^{st}$ to $5^{th}$ columns correspond to time, strain E$11,$ strain L$22,$ stress S$1,$ and stress S$22,$ respectively. $11$ and $22$ stand for two directions in the biaxial test. For the following questions, use built$-$in functions whenever possible.

$(1)$ Using linear spline interpolation, find strain values E$11$ when stress S$11$ is at $100$ kPa and $2000$ kPa $.$ Repeat the same process and find strain values E$22$ when stress S$22$ is at $100$ kPa and $2000$ kPa $.$ Print the results with an informative statement. $(10$ pts$)$

$(2)$ Find the slope of the strain$-$stress curve $($E$11-$S$11)$ when stress S$11$ is at $100$ kPa and $2000$ kPa $.$ Repeat the same process and find the slope of the strain$-$stress curve $($E$22-$S$22)$ when stress S $22$ is at $100$ kPa and $2000$ kPa $.($Hint: We may not have stress data at exactly $100$ kPa and $2000$ kPa $.$ But we can find the closest point by calculating the distance between the array containing stress data and $100$ kPa or $2000$ kPa $.$ Consider finding the index of the minimum distance using $\text{'}$np$.$argmin'. $\text{'}$np$.$argmin' returns the index of the minimum value in an array$).$

Print the calculated slopes $(4$ slopes in total$)$ with an informative statement. $(20$ pts $)$

$(3)$ Plot the data points, linear spline interpolation curves, and the slope lines at $100$ kPa and $2000$ kPa $.$ Plot E$11-$S$11$ and E$22-$S$22$ in the same graph. Adjust the range of the slope line so the shape of the original stress$-$strain curves looks like the letter $"$ J $".$ The result should look like the granh helow $($von may use different colors$).(10$ pts $)$

$(4)$ Strain energy density can be obtained by integrating stress w$.$r$.$t$.$ strain. i$.$e$.,W(T)={\displaystyle \underset{0}{\overset{E11(T)}{}}}S11dE11+{\displaystyle \underset{0}{\overset{E22(T)}{}}}S22dE22,$ where E$11($T$),$ E$22(T)$ are strains at time T$.$ Using a for loop, compute the strain energy density at each time point $t$ with trapezoidal and Simpson's $\frac{1}{3}$ rule, then plot the strain energy density versus E$11.$ Calculate the relative percentage error between trapezoidal and Simpson's $\frac{1}{3}$ rule using the last data point and print the error on the graph. The result should look like the graph below $($you may use different colors$).(30$ pts $)$