Introduction to Laplace Transforms for Engineers
C.T.J. Dodson, School of Mathematics, Manchester University
1
What are Laplace Transforms, and Why?
This is much easier to state than to motivate! We state the denition in two ways,
rst in words to explain
University of Windsor, Mathematics 62-126-01/02
Quiz 1 (b), September 12, 2008
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1. (4 marks) Write a vector equation for the line which passes through points P (3, 4, 2, 7)
and Q(1, 9, 2, 3).
Solution. A direction vecto
University of Windsor, Mathematics 62-126-01/02
Quiz 2 (a) Solutions, October 3, 2008
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1. (3 marks) Use a projection to nd the minimum distance from the point P (2, 1, 3)
to the plane x1 x2 + x3 = 5.
The point Q(5, 0, 0
University of Windsor, Mathematics 62-126-01/02
Quiz 2 (b), October 3, 2008
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1. (3 marks) Use a projection to nd the minimum distance from the point P (5, 2, 1)
to the plane x1 + 2x2 + x3 = 1.
The point Q(1, 0, 0) lies
University of Windsor, Mathematics 62-126-01/02
Quiz 4 (b) Solutions, November 7, 2008
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1. (3 marks) Determine a spanningset for the nullspace of the linear mapping L with
1 0 0 1
3
1 5 .
matrix A = 0 1 0
0 0 1 2
8
Solu
University of Windsor, Mathematics 62-126-01/02
Quiz 3 (a), October 31, 2008
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1. (3 marks) Find the standard matrix of the reection in the line x1 2x2 = 0 in R2 .
3
Solution. re(1,2) (1, 0) = (1, 0) 2proj(1,2) (1, 0) =
University of Windsor, Mathematics 62-126-01/02
Quiz 4 (a) Solutions, November 7, 2008
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1. (3 marks) Determine a spanningset for the nullspace of the linear mapping L with
100
1 4
5 .
matrix A = 0 1 0 2
001
3 6
Solution
University of Windsor, Mathematics 62-126-01/02
Quiz 3 (b), October 31, 2008
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1. (3 marks) Find the standard matrix of the reection in the line 3x1 + x2 = 0 in R2 .
4
Solution. re(3,1) (1, 0) = (1, 0) 2proj(3,1) (1, 0)
University of Windsor, Mathematics 62-126-01/02
Quiz 5 (a), November 28, 2008
1. (4 marks) Let S = cfw_1+x+x2 , 3+2x+2x2 +x3 , 5+4x+4x2 +x3 , 4+2x+3x2 +2x3 P3 ,
and let U be the vector space of polynomials spanned by S . Find a basis for U .
Solution. Le
University of Windsor, Mathematics 62-126-01/02
Quiz 1 (a), September 12, 2008
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1. (4 marks) Write a vector equation for the line which passes through points P (2, 5, 3, 7)
and Q(0, 2, 1, 4).
Solution. A direction vecto
University of Windsor, Mathematics 62-126
Test 2, November 21, 2008
1. (a) (2 marks) Let R be the linear mapping of reection in R2 in the line x + y = 0.
Determine the matrix of R (relative to the standard basis of R2 ).
Solution: A normal vector to the g
University of Windsor, Mathematics 62-141-05
Quiz 2 (a), Jan. 23, 2009
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1. (5 points) Find g (x) if
sin x
g (x) =
(t4 + 2)8 dt.
x3 +2
Solution. Choose any constant a, then
sin x
g (x) =
4
x3 +2
8
(t + 2) dt
a
(t4 + 2)8
University of Windsor, Mathematics 62-141-05
Quiz 2 (b), Jan. 23, 2009
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1. (5 points) Find g (x) if
2+ln x
g (x) =
(t6 + 4)7 dt.
x5
Solution. Choose any constant a, then
2+ln x
g (x) =
6
x5
7
(t + 4) dt
a
(t6 + 4)7 dt.
University of Windsor, Mathematics 62-141-05
Quiz 4 (b), Mar. 6, 2009
1. (6 marks) A force of 20 N is required to maintain a spring stretched from its
natural length of 20 cm to a length of 30 cm. How much work is done in stretching
the spring from 22 cm
University of Windsor, Mathematics 62-141-05
Quiz 4 (a), Mar. 6, 2009
1. (6 marks) A force of 30 N is required to maintain a spring stretched from its
natural length of 12 cm to a length of 15 cm. How much work is done in stretching
the spring from 12 cm
University of Windsor, Mathematics 62-141-05
Quiz 3 (b), Feb. 27, 2009
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1. (7 marks) Evaluate
x2
1
dx.
x2 9
Solution. Let x = 3 sec t, 0 < t < /2 or < t < 3/2.
Then dx = 3 sec t tan t dt, and x2 9 = 9 sec2 t 9 = 3 tan t
University of Windsor, Mathematics 62-141-05
Quiz 3 (a), Feb. 27, 2009
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1. (7 marks) Evaluate
x2
1
dx.
x2 16
Solution. Let x = 4 sec t, 0 < t < /2 or < t < 3/2.
Then dx = 4 sec t tan t dt, and x2 16 = 16 sec2 t 16 = 4 t
University of Windsor, Mathematics 62-141-05
Quiz 5 (a), Mar. 27, 2009
1. (7 marks) Find the area of the surface obtained by rotating the curve
y = 1 + 2x2 , 0 x 1, about y -axis.
Solution.
dy
dx
= 4x. The surface area is
1
S=
2x
1+
0
dy
dx
2
1
dx =
2x 1
University of Windsor, Mathematics 62-141-05
Quiz 5 (b), Mar. 27, 2009
1. (7 marks) Find the area of the surface obtained by rotating the curve
y = 1 + 5 x2 , 0 x 1, about y -axis.
2
Solution.
dy
dx
= 5x. The surface area is
1
S=
2x
0
dy
1+
dx
2
1
dx =
2x
University of Windsor, Mathematics 62-126-01/02
Quiz 5 (b), November 28, 2008
1. (4 marks) Let S = cfw_1+x+x3 , 2+3x+x2 +2x3 , 1+3x+2x2 +x3 , 2+4x+2x2 +3x3 P3 ,
and let U be the vector space of polynomials spanned by S . Find a basis for U .
Solution. Le
University of Windsor, Mathematics 62-126-01/02
Quiz 6(a), December 5, 2008
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3
0
with
1 2
respect to the standard basis S of R2 . Determine the matrix of L with respect to the
basis B = cfw_(1, 1), (1, 0) of R2 .
1. (4
The University of Windsor
Department of Mechanical, Automotive and Material Engineering
92-321 Final Examination
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Policy:
1. This is a closed-book examination.
2. Students can bring in one double-sided sheet of formulas with no explan
88-324/92-321Mid-Term Test Summer 2005
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1.(30) The transfer function model of a system is given as follows:
G(s) =
5(s + 80)
Y (s)
=
,
2 + 20s + 36)
R(s)
s(s
where Y (s) and R(s) are the Laplace transforms of the output signal y (t) a