Section 10.5

1. $f (t) = u (t - 3) \cdot (t - 3)$ $f (t) = u (t - 3) \cdot (t - 3)$
2. $f (t) = (t - 1) u (t - 1) - (t - 3) u (t - 3)$ $f (t) = (t - 1) u (t - 1) - (t - 3) u (t - 3)$
3. $f (t) = u (t - 1) \cdot e^{- 2 (t - 1)}$ $f (t) = u (t - 1) \cdot e^{- 2 (t - 1)}$
4. $f (t) = e^{t - 1} u (t - 1) - e^{2} e^{t - 2} u (t - 2)$ $f (t) = e^{t - 1} u (t - 1) - e^{2} e^{t - 2} u (t - 2)$
5. $f (t) = u (t - π) \cdot \sin (t - π) = - u (t - π) \sin t$ $f (t) = u (t - π) \cdot \sin (t - π) = - u (t - π) \sin t$
6. $f (t) = u (t - 1) \cdot \cos π (t - 1) = - u (t - 1) \cos π t$ $f (t) = u (t - 1) \cdot \cos π (t - 1) = - u (t - 1) \cos π t$
7. $f (t) = \sin t - u (t - 2 π) \sin (t - 2 π) = [1 - u (t - 2 π)] \sin t$ $f (t) = \sin t - u (t - 2 π) \sin (t - 2 π) = [1 - u (t - 2 π)] \sin t$
8. $f (t) = \cos π t - u (t - 2) \cos π (t - 2) = [1 - u (t - 2)] \cos π t$ $f (t) = \cos π t - u (t - 2) \cos π (t - 2) = [1 - u (t - 2)] \cos π t$
9. $f (t) = \cos π t + u (t - 3) \cos π (t - 3) = [1 - u (t - 3)] \cos π t$ $f (t) = \cos π t + u (t - 3) \cos π (t - 3) = [1 - u (t - 3)] \cos π t$
10. $\begin{matrix} f (t) = 2 u (t - π) \cos 2 (t - π) - 2 u (t - 2 π) \cos 2 (t - 2 π) \\ = 2 [u (t - π) - u (t - 2 π)] \cos 2 t \end{matrix}$ $\begin{matrix} f (t) = 2 u (t - π) \cos 2 (t - π) - 2 u (t - 2 π) \cos 2 (t - 2 π) \\ = 2 [u (t - π) - u (t - 2 π)] \cos 2 t \end{matrix}$
11. $f (t) = 2 [1 - u_{3} (t)]$ $f (t) = 2 [1 - u_{3} (t)]$ ; $F (s) = 2 (1 - e^{- 3 s}) / s$ $F (s) = 2 (1 - e^{- 3 s}) / s$
12. $F (s) = (e^{- s} - e^{- 4 s}) / s$ $F (s) = (e^{- s} - e^{- 4 s}) / s$
13. $F (s) = (1 - e^{- 2 π s}) / (s^{2} + 1)$ $F (s) = (1 - e^{- 2 π s}) / (s^{2} + 1)$
14. $F (s) = s (1 - e^{- 2 s}) / (s^{2} + π^{2})$ $F (s) = s (1 - e^{- 2 s}) / (s^{2} + π^{2})$
15. $F (s) = (1 + e^{- 3 π s}) / (s^{2} + 1)$ $F (s) = (1 + e^{- 3 π s}) / (s^{2} + 1)$
16. $F (s) = 2 (e^{- π s} - e^{- 2 π s}) / (s^{2} + 4)$ $F (s) = 2 (e^{- π s} - e^{- 2 π s}) / (s^{2} + 4)$
17. $F (s) = π (e^{- 2 s} + e^{- 3 s}) / (s^{2} + π^{2})$ $F (s) = π (e^{- 2 s} + e^{- 3 s}) / (s^{2} + π^{2})$
18. $F (s) = 2 π (e^{- 3 s} + e^{- 5 s}) / (4 s^{2} + π^{2})$ $F (s) = 2 π (e^{- 3 s} + e^{- 5 s}) / (4 s^{2} + π^{2})$
19. $F (s) = e^{- s} (s^{- 1} + s^{- 2})$ $F (s) = e^{- s} (s^{- 1} + s^{- 2})$
20. $F (s) = (1 - e^{- s}) / s^{2}$ $F (s) = (1 - e^{- s}) / s^{2}$
21. $F (s) = (1 - 2 e^{- s} + e^{- 2 s}) / s^{2}$ $F (s) = (1 - 2 e^{- s} + e^{- 2 s}) / s^{2}$
28. $F (s) = (1 - e^{- a s} - a s e^{- a s}) / [s^{2} (1 - e^{- 2 a s})]$ $F (s) = (1 - e^{- a s} - a s e^{- a s}) / [s^{2} (1 - e^{- 2 a s})]$
31. $x (t) = \frac{1}{2} [1 - u (t - π)] \sin^{2} t$ $x (t) = \frac{1}{2} [1 - u (t - π)] \sin^{2} t$
32. $x (t) = g (t) - u (t - 2) g (t - 2),$ $x (t) = g (t) - u (t - 2) g (t - 2),$ where $g (t) = \frac{1}{12} (3 - 4 e^{- t} + e^{- 4 t})$ $g (t) = \frac{1}{12} (3 - 4 e^{- t} + e^{- 4 t})$
33. $x (t) = \frac{1}{8} [1 - u (t - 2 π)] (\sin t - \frac{1}{3} \sin 3 t)$ $x (t) = \frac{1}{8} [1 - u (t - 2 π)] (\sin t - \frac{1}{3} \sin 3 t)$
34. $x (t) = g (t) - u (t - 1) [g (t - 1) + h (t - 1)],$ $x (t) = g (t) - u (t - 1) [g (t - 1) + h (t - 1)],$ where $g (t) = t - \sin t$ $g (t) = t - \sin t$ and $h (t) = 1 - \cos t$ $h (t) = 1 - \cos t$
35. $x (t) = \frac{1}{4} {- 1 + t + (t + 1) e^{- 2 t} + u (t - 2) [1 - t + (3 t - 5) e^{- 2 (t - 2)}]}$ $x (t) = \frac{1}{4} {- 1 + t + (t + 1) e^{- 2 t} + u (t - 2) [1 - t + (3 t - 5) e^{- 2 (t - 2)}]}$
36. $x (t) = 2 | \sin t | \sin t$ $x (t) = 2 | \sin t | \sin t$
37. $x (t) = g (t) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} u (t - n π) g (t - n π),$ $x (t) = g (t) + 2 \sum_{n = 1}^{\infty} {(- 1)}^{n} u (t - n π) g (t - n π),$ where $g (t) = 1 - \frac{1}{3} e^{- t} (3 \cos 3 t + \sin 3 t)$ $g (t) = 1 - \frac{1}{3} e^{- t} (3 \cos 3 t + \sin 3 t)$