Exercises

(1)
Determine \(F(1)\) where \(F(s)\) is the Laplace transform of \(f(t)=\left \{\begin{array}{lr} 4t & \mbox { if }0\leq t<2\\ \\ t^2 & \mbox { if }t\geq 2 \end{array}\right .\). Round to the nearest hundredth.
(2)
Determine \(F(2)\) where \(F(s)\) is the Laplace transform of \(\displaystyle f(t)=\left \{\begin{array}{lr} t^2 & \mbox { if }0\leq t<1\\ \\ 1 & \mbox { if }1\leq t<2\\ \\ 2-\dfrac {t}{2} & \mbox { if }2\leq t<4 \\ 0 & \mbox { if }t\geq 4\\ \end{array}\right .\) Round to the nearest thousandth.
The graph of a piecewise continuous function. [Picture]
Figure 1: Graph of a piecewise continuous function
(3)
Which is the Laplace transform of \(\displaystyle f(t)=\left \{\begin{array}{lr} t^3 & \mbox { if }0\leq t<1\\ \\ 2-t & \mbox { if }t\geq 1 \end{array}\right .\)?
(a)
\(\displaystyle \frac {6+e^{-s}(2s^3-s^2)}{s^4}\)
(b)
\(\displaystyle \frac {6-2e^{-s}(2s^2+3s+3))}{s^4}\)
(c)
\(\displaystyle \frac {6+2e^{-s}(2s^2+3s+3))}{s^4}\)
(d)
\(\displaystyle \frac {6-2e^{-s}}{s^4}\)
(e)
\(\displaystyle \frac {6e^{-s}}{s^4}\)
(f)
None of these
(4)
Suppose a \(1\)-kg mass, attached to a spring whose stiffness is \(100\) N/m, is undamped and begins at rest at equilibrium when an external upward force of \(10\) N is applied for just \(\frac {\pi }{10}\) seconds, followed by no external force (free motion). Find velocity of the mass in m/s at \(t=1\) second. Round to the nearest tenth.
(5)
Let \(y(t)\) be the solution to the IVP \(y\,^{\prime \prime }+2y\,^{\prime }+y=\left \{\begin{array}{lr} 4t & \mbox { if }0\leq t<2\\ \\ t^2 & \mbox { if }t\geq 2 \end{array}\right .\), \(y(0)=1\), \(y\,^{\prime }(0)=0\). Find \(y(1)\). Round to the nearest tenth.