$(1)$ Write the loop$-$closure $($position$)$ equation in terms of ${}_2(5$ points$),$ and analytically solve $(5$

points$)$ and plot $(5$ points$)$ by Excel the position and velocity of the slider $C$ as a function of ${}_2$

$),$ where $R_{AO}=1in,R_{BA}=2\mathrm{.}5in,$ and $R_{CB}=7in$; $($assume the angular velocity of

the crank $2,{}_2,$ is $\frac{}{5}ra\frac{d}{s})$

$(2)$ Based on the position function, using Forward, Backward and Centered Finite$-$Divided$-$

Difference Approximations of the First Derivative, respectively, numerically solve and plot $(10$

points$)$ the velocity of the slider $C$ as a function of time second$),$ where the angular

velocity of the crank $2,{}_2,$ is $\frac{}{5}ra\frac{d}{s}.$ Using Excel $($or Excel Macro$)$ or MatLab $($or MatLab

programing$)$ to calculate and plot the data of analytical and numerical solutions $(10$ points$).$

$(3)$ Using different time step size to compare the true error of the numerical velocity of the slider $C$

up to $t=10$ second and plot the graph of error $E_t$ vs$.$ time $t(10$ points$).$