Lecture 13: Derivative and product of Laplace transforms
In this lecture, we will . . .
introduce the derivative and product of Laplace transforms.
Advanced Engineering Mathematics
Week 5, Wednesday Lecture - 2
L13: 1 / 14
Product of Transforms
s
1
s
= 2
MATH283 - Autumn 2015
Workshop Week 8
Mid-session Exam Revision Sheet
This revision sheet does NOT cover all materials for the Mathematics topic.
Answers only will be provided for these revision questions.
1. if z = x cos(x + y) determine
z z 2 z 2 z 2 z
Lecture 15: Introduction to Fourier series
In this lecture, we will . . .
introduce Fourier series
Advanced Engineering Mathematics
Week 6, Wednesday Lecture - 1
L15: 1 / 22
Overview of Fourier Series
In mathematics, representation of a function by infini
MATH283/STAT291 - 2015
Outline Solutions to Tutorial Sheet - Week 8
1. (a) P (A) = 1 0.5 = 0.5
(b) P (A B) = P (B) = 0.2
(c) P (A B) = P (A) = 0.5
(d) P (A|B) =
P (AB)
P (B)
=
0.2
0.2
= 1 (if B occurs, we know that A occurs)
(e) P (B|A) =
P (AB)
P (A)
=
0
MATH283 - 2015
Tutorial Sheet - Week 5
Outline solutions available as of Friday afternoon at the MATH283 Moodle
1. Sketch the graph of the function f (x) with the formulae:
(a)
f (x) = xH(x)
(b)
f (x) = x [H(x) H(x 2)]
2. Sketch the graph of the function
MATH283 - 2015
Tutorial Sheet - Week 2
Outline solutions available as of Friday afternoon at the MATH283 Moodle site.
1. Find the domain and range for the following functions:
1
(a) z = y x2
(b) z =
xy
2. Find the domain and range for the following functi
MATH283/STAT291 - 2015
Tutorial Sheet - Week 8
Outline solutions available as of Friday afternoon at the MATH283 eLearning site.
1. For the Venn diagram shown at the right, evaluate
(a) P (A),
A
(b) P (A B),
B
(c) P (A B),
0.2
(d) P (A|B),
0.5
(e) P (B|A)
MATH283/STAT291 - 2015
Tutorial Sheet - Week 9
Outline solutions available as of Friday afternoon at the MATH283 eLearning site.
1. The following table contains interdivision times x in hours of cells exposed to beryllium, and a
natural logarithm transfor
MATH283 - 2015
Workshop - Week 3
Partial Differentiation
Aims of this workshop:
Learn to use the Symbolic Toolbox in MATLAB to plot functions and to find partial
derivatives
Required knowledge:
Knowledge of functions of more than one variable, and parti
Lecture 6: Reverse order of integration, Area and Volume
as double integrals
In this lecture, we will learn to . . .
set up an equivalent integral in reverse order for a given
integral
compute area and volume by evaluating double integrals
Advanced Engine
MATH283 - 2015
Solutions to Workshop Exercises - Week 12
GROUPWORK
1. (a) As the formula used to generate x3 involves x1 and differences which are independent
of both x1 and x2, but no terms involving x2. Therefore x2 and x3 are independent
samples. This
MATH283 - 2015
Answers to Revision Sheet - Week 8
1.
2.
z
z
2z
2z
= cos(x + y) x sin(x + y),
= x sin(x + y),
= 2 sin(x + y) x cos(x + y),
= x cos(x + y),
2
x
y
x
y 2
2z
2z
=
= sin(x + y) x cos(x + y).
xy
yx
df
1
= e 2e ln(12) 13.96cm/s. z is decreasing wi
MATH283/STAT291 - 2014
Outline Solutions to Tutorial Sheet - Week 9
1. (a) The stem-and-leaf plots with truncated data and split rows are as follows.
(Other variations are also possible.)
(leaf unit = 1 hours)
1 3
1 5677899
2 67
2 0111123
(leaf unit = 0.1
Lecture 18: Fourier series in general interval
In this lecture, we will . . .
Introduce Fourier series in general interval
Advanced Engineering Mathematics
Week 7, Wednesday Lecture
L18: 1 / 11
Fourier series in general interval
The Fourier series for f (
Lecture 13: Derivative and product of Laplace transforms
In this lecture, we will . . .
introduce the derivative and product of Laplace transforms.
Advanced Engineering Mathematics
Week 5, Wednesday Lecture - 2
L13: 1 / 14
Product of Transforms
s
1
s
= 2
Lecture 7: Double integral in Polar form
In this lecture, we will . . .
define double integral in Polar coordinates
introduce conversion between Cartesian and Polar integrals.
Advanced Engineering Mathematics
Week 3, Wednesday Lecture - 2
L7: 1 / 17
Revie
Lecture 8: Special functions - Gamma, Beta & error
functions
In this lecture, we will . . .
define Gamma, beta and error functions
learn their properties
Advanced Engineering Mathematics
Week 3, Thursday Lecture
L8: 1 / 18
Definition
The Gamma function (r
Lecture 10: Laplace transform - Linearity
In this lecture, we will . . .
define Laplace transform
introduce the linearity of Laplace transforms
find the Laplace transforms and the inverse Laplace
transforms of simple functions
Advanced Engineering Mathema
Estimation
Confidence Intervals
I
P. Davy
I
Week 10 Thursday (Statistics)
c MATH283/STAT291 2015
P. Davy
Week 10 Thursday (Statistics)
Slide 10A.1,
Point and Interval Estimates
c MATH283/STAT291 2015
P. Davy
I
But X is a single number (a point
estimate
MATH283 - 2015
Revision Sheet - Week 1
Outline solutions available as of Friday afternoon at the MATH283 Moodle site.
2
x 2,
x 2,
Find
1. A function f is dened by the rule f (x) =
1, 2 < x < 1,
x 3,
x 1.
(a)
f (2)
(b)
f (1)
(c)
(d)
f (3) + 3f (5)
(e)
va
Lecture 10: Laplace transform - Linearity
In this lecture, we will . . .
define Laplace transform
introduce the linearity of Laplace transforms
find the Laplace transforms and the inverse Laplace
transforms of simple functions
Advanced Engineering Mathema
MATH283 - 2015
Tutorial Sheet - Week 4
Outline solutions available as of Friday afternoon at the MATH283 Moodle
1. Reverse the order of integration and then evaluate the resulting integral.
1
2
4 cos(x2 )dxdy
(a)
0
2
4
2
xey dydx.
(b)
2y
0
x2
2. (a) Sketc
MATH283 - 2015
Answers to Workshop Sheet - Week 3
Partial Dierentiation
EXERCISES
1. (a) The logarithmic function is only dened for positive arguments. So f (x, y) is dened in the interior of the
ellipse x2 + 2y 2 = 1.
(b) g(x, y) is dened throughout the
MATH283 - 2015
Tutorial Sheet - Week 3
Outline solutions available as of Friday afternoon at the MATH283 Moodle site.
1. Find all the second order partial derivatives of z = sin(xy).
2. (a) Given f (x, y) = ex cos y + 2xy 2 , determine the partial derivat
MATH283/STAT291 - 2015
Tutorial Sheet - Week 12
Outline solutions available as of Friday afternoon at the MATH283 eLearning site.
1. The actual resistance in (ohm) was measured for a random sample of 10 resistors which
are labelled as 100 ohm. It can be a
MATH283 - 2015
Outline Solutions to Tutorial Sheet - Week 3
1. First of all the rst partial derivatives are found:
z
= y cos(xy),
x
z
= x cos(xy).
y
Then each of these is dierentiated with respect to x:
( )
( )
2z
z
2z
z
2
=
= y sin(xy),
=
= xy sin(xy) +
MATH283 - 2015
Workshop - Week 4
Evaluating Double Integrals
Aims of this workshop:
Learn to create polar plots in MATLAB and use the Symbolic Toolbox to evaluate double integrals
in Cartesian and Polar coordinates.
Tasks of this workshop:
By the end of
TABLE OF LAPLACE TRANSFORMS
x(t)
X(s)
1
s
1
tn , n = 0, 1, 2, . . .
n!
sn+1
tr , r > 1
(r + 1)
sr+1
ebt
1
s+b
sin nt
s2
n
+ n2
cos nt
s
s2 + n2
sinh nt
n
s2 n2
cosh nt
t sin nt
t cos nt
sin nt nt cos nt
s2
s
n2
(s2
2sn
+ n2 )2
s2 n2
(s2 + n2 )2
(s2
2n3
+
MATH283 - 2014
Tutorial Sheet - Week 2
Outline solutions available as of Friday afternoon at the MATH283 Moodle site.
1. Find the domain and range for the following functions:
1
(a) z = y x2
(b) z =
xy
2. Find the domain and range for the following functi