7.4E: The Unit Step Function (Exercises)

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Jan 11, 2021
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- 7.4: The Unit Step Function
- 7.5: Constant Coefficient Equations with Piecewise Continuous Forcing Functions

Zoya Kravets
Mission College

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Q7.4.1

In Exercises 7.4.1-7.4.6 find the Laplace transform. Then express the given function $f$ in terms of unit step functions and find ${\cal L}(f)$ . Graph $f$ for Exercises 7.4.3 and 7.4.4.

1. $f(t)=\left\{\begin{array}{cl} {1,}&{0 \le t<4,}\\ {t,} & {t\ge4.} \end{array}\right.$

2. $f(t)=\left\{\begin{array}{cl} t,&0 \le t<1,\\[4pt] 1,& t\ge1.\end{array}\right.$

3. $f(t)=\left\{\begin{array}{cl} 2t-1,& 0\le t<2,\\[4pt] t,&t\ge2.\end{array}\right.$

4. $f(t)=\left\{\begin{array}{cl}1, &0\le t<1,\\[4pt] t+2,&t\ge1.\end{array}\right.$

5. $f(t)=\left\{\begin{array}{cl} t-1,& 0\le t<2,\\[4pt] 4,&t\ge2.\end{array}\right.$

6. $f(t)=\left\{\begin{array}{cl} t^2,& 0\le t<1,\\[4pt] 0,&t\ge1.\end{array}\right.$

Q7.4.2

In Exercises 7.4.7-7.4.18 express the given function $f$ in terms of unit step functions and find $\cal{L} (f)$ . Graph $f$ for Exercises 7.4.15-7.4.18.

7. $f(t)=\left\{\begin{array}{cl} 0, &0\le t<2,\\[4pt] t^2+3t,&t\ge2.\end{array}\right.$

8. $f(t)=\left\{\begin{array}{cl} t^2+2, &0\le t<1,\\[4pt] t,&t\ge1.\end{array}\right.$

9. $f(t)=\left\{\begin{array}{cl} te^t,& 0\le t <1,\\[4pt] e^t,&t\ge1.\end{array}\right.$

10. $f(t)=\left\{\begin{array}{cl} e^{\phantom{2}-t}, &0\le t<1,\\[4pt] e^{-2t},&t\ge1.\end{array}\right.$

11. $f(t)=\left\{\begin{array}{cl} -t,&0 \le t<2,\\[4pt] t-4,&2\le t<3,\\[4pt] 1,&t\ge3. \end{array}\right.$

12. $f(t)=\left\{\begin{array}{cl} 0,&0 \le t<1,\\[4pt] t,&1\le t<2,\\[4pt] 0,&t\ge2.\end{array}\right.$

13. $f(t)=\left\{\begin{array}{cl} t,&0 \le t<1,\\[4pt] t^2,&1\le t<2,\\[4pt] 0,&t\ge2. \end{array}\right.$

14. $f(t)=\left\{\begin{array}{cl} t,&0\le t<1,\\[4pt] 2-t,&1\le t<2,\\[4pt] 6,&t > 2. \end{array}\right.$

15. $f(t)=\left\{\begin{array}{cl} {\sin t,}&{0\leq t<\frac{\pi }{2}}\\{2\sin t,}&{\frac{\pi }{2}\leq t<\pi }\\{\cos t,}&{t\geq \pi } \end{array} \right.$

16. $f(t)=\left\{\begin{array}{cl}\phantom{-} 2,&0\le t<1,\\[4pt]-2t+2,&1\le t<3,\\[4pt]\phantom{-}3t,&t\ge 3.\end{array}\right.$

17. $f(t)=\left\{\begin{array}{cl}3,&0\le t<2,\\[4pt]3t+2,&2\le t<4,\\[4pt]4t,&t\ge 4.\end{array}\right.$

18. $f(t)=\left\{\begin{array}{ll}(t+1)^2,&0\le t<1, \\[4pt](t+2)^2,&t\ge1.\end{array}\right.$

Q7.4.3

In Exercises 7.4.19-7.4.28 express the inverse transforms in terms of step functions, and then find distinct formulas the for inverse transforms on the appropriate intervals, as in Example 7.4.7. Graph the inverse transform for Exercises 7.4.21, 7.4.22, and 7.4.25.

19. $H(s)={e^{-2s}\over s-2}$

20. $H(s)={e^{-s}\over s(s+1)}$

21. $H(s)={e^{-s}\over s^3}+ {e^{-2s}\over s^2}$

22. $H(s)=\left({2\over s}+{1\over s^2}\right) +e^{-s}\left({3\over s}-{1\over s^2}\right)+e^{-3s}\left({1\over s}+{1\over s^2}\right)$

23. $H(s)=\left({5\over s}-{1\over s^2}\right) +e^{-3s}\left({6\over s}+{7\over s^2}\right)+{3e^{-6s}\over s^3}$

24. $H(s)={e^{-\pi s} (1-2s)\over s^2+4s+5}$

25. $H(s)=\left({1\over s}-{s\over s^2+1}\right)+e^{-{\pi\over 2}s}\left({3s-1\over s^2+1}\right)$

26. $H(s)= e^{-2s}\left[{3(s-3)\over(s+1)(s-2)}-{s+1\over(s-1)(s-2)}\right]$

27. $H(s)={1\over s}+{1\over s^2}+e^{-s}\left({3\over s}+{2\over s^2}\right) +e^{-3s}\left({4\over s}+{3\over s^2}\right)$

28. $H(s)={1\over s}-{2\over s^3}+e^{-2s}\left({3\over s}-{1\over s^3}\right) +{e^{-4s}\over s^2}$

Q7.4.4

29. Find ${\cal L}\left(u(t-\tau)\right)$ .

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