MATH283/STAT291 - 2015
Workshop - Week 11
Central Limit Theorem, Confidence Intervals
This workshop shows computations related to the continuous Normal model. By the end of
the class, groups should have completed all work up to and including the starred (
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/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
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 - 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 - 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 - 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
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
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 2: Chain Rule & Second Order Partial Derivatives
In this lecture, we will . . .
introduce partial derivative of functions of n variables
introduce chain rule for functions of several variables
define higher order partial derivatives - second order
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 14: Laplace transform of periodic functions
In this lecture, we will . . .
introduce the Laplace transform of periodic functions.
Advanced Engineering Mathematics
Week 5, Thursday Lecture
L14: 1 / 16
Review of periodic functions
A function f (t) i
Lecture 4: Total differential and Small Errors
In this lecture, we will . . .
Introduce total differential
Applications of total differential
1
2
3
Estimate change
Sensitivity to Change
Estimating percentage error
Advanced Engineering Mathematics
Week 2,
Lecture 3: Higher Order Derivative & Implicit
Differentiation
In this lecture, we will . . .
introduce mixed derivative theorem
define higher order partial derivatives
introduce implicit differentiation
Advanced Engineering Mathematics
Week 2, Wednesday L
MATH283 - 2015
Answers to Workshop Sheet - Week 4
Evaluating Double Integrals
EXERCISES
1. (a) True
(b) False
sin x
2.
5
4
(x + y)dydx =
0
0
1
y
3. (a)
f (x, y)dxdy, R is the triangle region with vertices at (0, 0), (1, 1) and (0, 1).
0
0
/2
cos x
(b)
dyd
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 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 9: Heaviside step function, Dirac delta function
and Laplace transform
In this lecture, we will learn . . .
more about step function
Dirac delta function
Laplace transform
Advanced Engineering Mathematics
Week 4, Monday Lecture - 1
L9: 1 / 16
Defi
Lecture 10: Laplace transform method for solving ordinary
differential equations (ODEs)
In this lecture, we will . . .
define inverse Laplace transform
introduce Laplace transform of derivatives
learn to use Laplace transform to solve ordinary differentia
Lecture 4: Total differential and Small Errors
In this lecture, we will . . .
Introduce total differential
Applications of total differential
1
2
3
Estimate change
Sensitivity to Change
Estimating percentage error
Advanced Engineering Mathematics
Week 2,
Lecture 14: Introduction to Fourier series
In this lecture, we will . . .
introduce Fourier series
Advanced Engineering Mathematics
Week 5, Thursday Lecture
L14: 1 / 22
Overview of Fourier Series
In mathematics, representation of a function by infinite se
Lecture 2: Chain Rule & Second Order Partial Derivatives
In this lecture, we will . . .
introduce partial derivative of functions of n variables
introduce chain rule for functions of several variables
define higher order partial derivatives - second order