$3.$ Fill in LAB$05$ex$2.$m to reproduce Fig. $2$ and then answer the following questions.

$($a$)$ For what minimal time t$1$ will the mass$-$spring system satisfy $|$y$($t$)|<mtext\; id="EditorElement\_1912710"></mtext><mn\; id="EditorElement\_1912711">0</mn>$:$06$ for all t $>$ t$1?$ You can answer the question either by magnifying the MATLAB figure using the magnify button, or use the following MATLAB commands $($explain$)$:

for i$=1$: length $($y$)$

M$($i$)=$ max$($abs$($y$($i: end $)))$;

end

i $=$ find $($M $<mtext\; id="EditorElement\_1912940"></mtext><mn\; id="EditorElement\_1912941">0</mn><mtext\; id="EditorElement\_1912942"></mtext><mn\; id="EditorElement\_1912943">.</mn><mn\; id="EditorElement\_1912944">0</mn><mn\; id="EditorElement\_1912945">6</mn><mtext\; id="EditorElement\_1912946"></mtext><mi\; id="EditorElement\_1912947">)</mi>$; i $=$ i$(1)$

disp $([\text{'}|$y$|<mn\; id="EditorElement\_1912973">0</mn><mn\; id="EditorElement\_1912974">.</mn><mn\; id="EditorElement\_1912975">0</mn><mn\; id="EditorElement\_1912976">6</mn>$ for t $>$ t$1$ with $\text{'}$ num$2$str$($t$($i $-1))\text{'}$ t$1<mi\; id="EditorElement\_1913020">\text{'}</mi>$ num$2$str$($ t$($i$))])$

$($b$)$ What is the largest $($in magnitude$)$ velocity attained by the mass, and when is it attained? Answer by using the magnify button.

$($c$)$ How does the size of c affect the motion? To support your answer, run the file LAB$05$ex$2.$m for c $=28,$ c $=56,$ and c $=114.$ Include the corresponding graphs with a title indicating the value of c used.

$($d$)$ Determine analytically the smallest $($critical$)$ value of c such that no oscillation appears in the solution.