Part $1$ Exercises $($MATLAB$)$

$1$A$)$ Open a new script $($top left button in Home menu$)$ and type your name and student ID number as comments

in the top two lines of the script, as designated below.

$\%$ First$\_$Name Last$\_$Name

$\%$ Student ID Number

For the remainder of this project, $a=$ last digit of your student ID$,b=$ second to last digit of your student ID$,c=$

third to last digit of your student ID$,$ and $d=$ the sum of the last three digits of your student ID$.$ Complete the

following operations in your script, making a new line for each operation. For these exercises, use the functions

given below.

$y_1=f(x)=(a+1)x{e}^{-x}sin(\frac{b+1}{x}),y_2=g(x)=(c+1)xco{s}^{-1}(\frac{x}{d+1})$

$1$B$)$ Define $y_1=f(x)$ in MATLAB, then use MATLAB to determine its limit from the left, from the right, and its

two$-$sided limit at $x=0.$ Declare these as P$1$Bleft, P$1$Bright, and P$1$Btwo. Next, compute its first and second

derivatives using MATLAB and declare these results as P$1$Bdydx and P$1$Bdyydxx$.$

$1$C$)$ Plot the original function $y_1=f(x),$ its first derivative, and its second derivative together on the same set of

axes in MATLAB, but limited by $\frac{c+1}{10}x\frac{d+25}{5}$ and $-20y20($figure $1).$ Give the plot a title and

correctly label both axes.

$1$D$)$ Define $y_2=g(x)$ in MATLAB, then plot it over its entire domain, namely $-(d+1)x(d+1)($figure

$.$ Next, determine all critical value locations over its entire domain. Call this result P$1$D$.$

$1$E$)$ The Mean Value Theorem $($MVT$)$ states that for a continuous function defined over a closed interval I that

is differentiable over the interval I $($except for maybe its endpoints$),$ then there is at least one point $x_0$ such that the

tangent line at $x_0$ is parallel to the secant line $($formal equation given below$).$ Use subs and solve together to find

an $x_0$ that satisfies the MVT for $y_2=g(x)$ over the domain $-(d+1)x(d+1).$ Call this result P$1$E$.$

$g^{\text{'}}(x_0)=\frac{g(d+1)-g(-(d+1))}{(d+1)-(-(d+1))}$

Once you have these operations written in proper MATLAB syntax, go to your Editor menu and click Run. If

there are values in your Command Window for all operations without error messages, then your code has

successfully run! If you have error messages, read them carefully and try to resolve them, then click Run again.