American College of Computer & Information Sciences MATH 2255 practice-midterm-2-solutions

Practice midterm I provides good practice problems for previous material.
** Bring a single double-sided 8:5 × 11 sheet of notes to use during the final.
1. Decide if the following statements are TRUE or FALSE. You do NOT need to justify your answers.
(a) (2 points) If Lff(t)g = F (s) and Lfg(t)g = G(s) then Lff(t)g(t)g = F (s)G(s)
Solution: F ( Lff(t) ∗ g(t)g = F (s)G(s).)
2. Give examples of the following. Be as explicit as possible. You do NOT need to justify your answers.
(a) (2 points) Give an example of a piecewise continuous function on the interval [0; 5] which is not
continuous on the interval [0; 5].
Solution: The Heavyside function u2(t).
3. (10 points) Using only the definition of the Laplace transform compute the Laplace transform Lfu2(t)tg.
Solution:
Lfu2(t)tg = Z01 e-stu2(t)t dt
= lim
A!1 Z0A e-stu2(t)t dt
= lim
A!1 Z2A te-st dt
u = t; dv = e-st dt; du = dt; v = e-st
-s
= lim
A!1
te-st
-s
A t
=2
- Z2A e--sst dt!
= lim
A!1
Ae-sA
-s
-
2e-2s
-s
-
e-st
s2
A t
=2!
= lim
A!1
Ae-sA
-s
-
2e-2s
-s
-
e-sA
s2 +
e-2s
s2 !
= lim
A!1
Ae-sA
-s ! + 2e-s 2s - Alim !1 e-s2sA ! + e-s22s
= lim
A!1
A
-sesA ! + 2e-s 2s - 0 + e-s22s ; s > 0
L’H^opital’s Rule
= lim
A!1
1
-s2esA ! + 2e-s 2s - 0 + e-s22s ; s > 0
= 0 +
2e-2s
s
- 0 +
e-2s
s2 ; s > 0
= 2s + s12  e-2s; s >