Problem $1$:

Data was collected for the thermal conductivity $[$W$/$m$-$K$]$ of a silicon wafer at

different temperatures $[$K$].$ The data is shown in the table below.

$($K$)2468102040608010015025035050010001400$

$$

$($W$/$m$-$K$)46300820156023005000350021001350900400190120753020$

Write a script program that addresses the following features:

$($a$)$ Assign the data as row vectors to the variables T$\_$exp and k$\_$exp, respectively.

$($b$)$ Compute the natural log of both T$\_$exp and k$\_$exp. Then, create a plot of ln$()$ vs$.$ ln$()$ as

black circles for the data points. Include a title and axis labels.

$($i$)($Concept$)$ By looking at the plot, comment on the relationship between ln$()$ vs$.$ ln$().$

How does it differ from what you might expect the relationship to be$?$ Why?

$($c$)$ Using the natural log data from part $($b$),$ proceed to assume a power fit model. You can use the

least$-$squares method explicitly $($as shown and coded in class$)$ or use the polyfit command.

$($i$)($Concept$)$ Use the values of the found coefficients from the model to write an equation for

the thermal conductivity $$ as a function of the natural logarithm of temperature ln$().$

Show all your typed work to derive this equation from the model.

$($d$)$ Repeat part $($c$),$ but modify the power fit model to use a quadratic fit on the data from part $($b$)$

instead of the linear fit that a conventional power fit model uses.

$($e$)$ Use each of the assumed models in parts $($c$)$ and $($d$)$ to compute the predicted values for thermal

conductivity. Assign the results to the variables k$\_$power and k$\_$power$2,$ respectively.

$($f$)$ Create a second plot of the: data from part $($a$)($as black asterisk points$),$ predicted values in

k$\_$power from part $($e$)($as a red dashed line$),$ and predicted values in k$\_$power$2$ from part $($e$)$

$($as a blue dashed line$).$ Add a title, axis labels, a legend. Use axis$([052503750]).$

$($i$)($Concept$)$ Determine the sum of squares of the residual error $!="$

# for each of the

assumed models. Compare and comment on their accuracy. Which one is more accurate?

BONUS: Use numerical differentiation to compute $($most accurately$)$ the conductive gradient

relative to temperature $/$ of the original data from part $($a$).$ Save the temperature data T$\_$exp,

thermal conductivity data k$\_$exp, and $/$ in a mat file named dkdT$.$mat.

Submit the script code in your report, typed responses to concept questions, and screenshots

of each plot.