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7.4E: The Unit Step Function (Exercises)

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

     

    This page titled 7.4E: The Unit Step Function (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.