HW 8 ungraded exam #3 practice problems
Math 527,
These are practice problems to help you prepare for exam #3. The actual exam will have fewer problems
and different kinds of problems, but their level of difficulty will be about the same.
INSTRUCTIONS (worth some points)
1. Write your name legibly in pen on each page and name and section number on this page.
2. Show your work and put a box or circle around your answers.
3. Always write equations. Partial credit will be given only for work written clearly in equations.
Problem 1. Compute the Laplace transform or inverse Laplace transform.
(a) L −1
e
−as 1
s
4
=
(b) L −1
1
s
2 − 2s
=
HW 8 ungraded Math 527, University of New Hampshire
Problem 1, cont’d. Compute the Laplace transform or inverse Laplace transform.
(c) L
U (t − 2) e
−3t
(t + 4)
=
(d) L −1
2s + 5
s
2 + 6s + 34
=
HW 8 ungraded Math 527, University of New Hampshire
Problem 2. Express f(t) in terms of Heaviside functions and then compute L {f(t)}.
f(t) =
sin t, 0 ≤ t < π
0, π ≤ t
HW 8 ungraded Math 527, University of New Hampshire
Problem 3. Find the solution of the initial value problem using Laplace transforms. Derivatives y
0
, y00
are with respect to t.
y
00 + 4y
0 + 8y = e
−t
, y(0) = 0, y0
(0) = 1
HW 8 ungraded Math 527, University of New Hampshire
Problem 4.
(a) Compute the matrix-vector product
3 2 1
4 1 0
−2 5 −1
3
1
5
=
(b) Express the system of equations as an Ax = b problem, where A is a matrix, b is a known vector,
and x is an unknown vector.
x + y − 2z = 14
2x − y + z = 0
6x + 3y + 4z = 1
(c) Solve the Ax = b equation from (b) using Gauusian elimination or Gauss-Jordan elimination. (Note:
this problem is a natural follow-on to (b), but we haven’t hit it in lecture yet, so don’t expect it to appear
on the exam).
