Means and Variances for some commonly used
distributions
Distribution
Mean
Variance
Bernoulli-
(1 )
Binomial B (n, )
n
n(1 )
Poisson Po()
Uniform U (a, b)
a +b
2
Normal N (, 2 )
(ba)2
12
2
Standard Normal N (0, 1)
0
1
Exponential Exp()
1
1
2
G Glonek (Uni
MATHS 2201/7201, Engineering Mathematics 1
Tutorial 6, Week 7
1. The functions y1 (x) = 1, y2 (x) = sin x and y3 (x) = cos x are all solutions of the third-order
ODE
y (3) + y 0 = 0
(1)
for < x < . Calculate the Wronskian of y1 , y2 and y3 . Are these fun
MATHS 2201 Engineering Mathematics I Tutorial Exercise 3
Week of 19 March 2012
1. The paper Ultimate Load Capacities of Expansion Anchor Bolts (J. Engr. Energy, 1993, pp.
139-158) reports the following summary data on shear strength for a sample of 3/8 in
MATHS 2201/7201, Engineering Mathematics IIA
Tutorial 10, Week 11
1. Consider the function f (x) = | sin x| for < x < and f (x + 2) = f (x) for all x.
Sketch the function and find its Fourier series.
2. Consider the function
f (x) =
x
`
0<x`
L x
L`
`<x<L
MATHS 2201/7201, Engineering Mathematics IIA
Tutorial 6, Week 7
1. The functions y1 (x) = 1, y2 (x) = sin x and y3 (x) = cos x are all solutions of the third-order
ODE
y (3) + y 0 = 0
(1)
for < x < . Calculate the Wronskian of y1 , y2 and y3 . Are these f
MATHS 2201 Engineering Mathematics IIA Tutorial 2
1. Suppose the diameters of pistons produced by a certain manufacturer are normally distributed
with mean 120mm and standard deviation 0.2mm. The diameters of the corresponding cylinders
are normally distr
MATHEMATICS IB - SEMESTER 1, 2012
Tutorial 3 Algebra Solutions
ALGEBRA
Problem: A vector u is said to be orthogonal to a subspace V Rn if u is orthogonal to every v V . Prove that u
is orthogonal to V if it is orthogonal to each vector in any basis for V
MATHEMATICS IB - SEMESTER 2, 2012
Tutorial 4 Solutions
ALGEBRA
Question:Let V be a subspace of Rn and u a vector in Rn . We showed in lectures that we can
write u = w1 + w2 , where w1 V and w2 is orthogonal to V. Show that this decomposition of u is
uniqu
MATHS 2201 Engineering Mathematics IIA Tutorial Exercise 1
1. In cars with ABS brakes, faults occasionally occur with the wheel sensors and a warning light is
illuminated. It can also happen that the warning light comes on even when no fault is present.
I
MATHS 2201 Engineering Mathematics I Tutorial Exercise 4
Week of 26 March 2012
1. For each of ten streets with bicycle lanes, investigators measured the distance between the centre
line and a cyclist in the bicycle lane. They used photography to determine
MATHS 2201 Engineering Mathematics I Tutorial Exercise 1
Week of 5 March 2012
1. A pair of fair dice are rolled and the total of the two faces is calculated.
(a) Write down the sample space for this experiment.
Solution:
Note: In these solutions, the dots
MATHS 2201 Engineering Mathematics I Tutorial Exercise 1
Week of 5 March 2012
1. A pair of fair dice are rolled and the total of the two faces is calculated.
(a) Write down the sample space for this experiment.
(b) Find the probability that the total is a
MATHS 2201/7201, Engineering Mathematics 1
Tutorial 5, Week 6
1. Flow out of Torrens Lake is controlled by a sluice gate, which can be raised to create
an opening at the bottom through which water can flow. Suppose that at time t = 0, an
electrical malfun
MATHS 2201/7201, Engineering Mathematics 1
Tutorial 5, Week 6
1. Flow out of Torrens Lake is controlled by a sluice gate, which can be raised to create
an opening at the bottom through which water can flow. Suppose that at time t = 0, an
electrical malfun
MATHS 2201 Engineering Mathematics I Tutorial Exercise 3
Week of 19 March 2012
1. The paper Ultimate Load Capacities of Expansion Anchor Bolts (J. Engr. Energy, 1993, pp.
139-158) reports the following summary data on shear strength for a sample of 3/8 in
MATHS 2201/7201, Engineering Mathematics 1
Tutorial 9, Week 10
1. Consider the initial value problem
y 00 + xy 0 + (2x2 + 1)y = 0,
y(0) = 1,
y 0 (0) = 1.
(a) Find the first six terms of the power series solution of (1)
(b) Calculate successive approximati
MATHS 2201 Engineering Mathematics I Tutorial Exercise 2
Week of 12 March 2012
1. Suppose the diameters of pistons produced by a certain manufacturer are normally distributed
with mean 120mm and standard deviation 0.2mm. The diameters of the corresponding
MATHS 2201/7201, Engineering Mathematics 1
Tutorial 8, Week 9
1. A simple model for torsional oscillation of a suspension bridge (such as the Tacoma Narrows
bridge) consists of a rigid rod of mass m and length 2l supported at either end by elastic
cables
MATHS 2201 Engineering Mathematics I Tutorial Exercise 2
Week of 12 March 2012
1. Suppose the diameters of pistons produced by a certain manufacturer are normally distributed
with mean 120mm and standard deviation 0.2mm. The diameters of the corresponding
MATHS 2201 Engineering Mathematics I Tutorial Exercise 4
Week of 26 March 2012
1. For each of ten streets with bicycle lanes, investigators measured the distance between the centre
line and a cyclist in the bicycle lane. They used photography to determine
MATHS 2201/7201, Engineering Mathematics IIA
Tutorial 8, Week 9
1. A simple model for torsional oscillation of a suspension bridge (such as the Tacoma Narrows
bridge) consists of a rigid rod of mass m and length 2l supported at either end by elastic
cable
THE UNIVERSITY OF ADELAIDE
SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING
GEOTECHNICAL ENGINEERING II (C&ENVENG 2069)
ASSIGNMENT No. 2, 2013
COMPACTION and VERTICAL STRESS IN SOILS
Deadline: 5 pm, Thursday, 5 September in Schools Geotechnical Engineering I
THE UNIVERSITY OF ADELAIDE
SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING
GEOTECHNICAL ENGINEERING II (C&ENVENG 2069)
SOLUTIONS TO ASSIGNMENT No. 5, 2013
STRENGTH of SOILS
1.
(i)
confining stress 3
deviator stress q
So, 1=c+q
At failure, 3'=3uf
1'= 1uf
=10
THE UNIVERSITY OF ADELAIDE
SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING
GEOTECHNICAL ENGINEERING II (C&ENVENG 2069)
SOLUTIONS TO ASSIGNMENT No. 4, 2013
COMPRESSIBILITY and CONSOLIDATION of SOILS
1.
(i)
0.950
0.900
Void Ratio, e
0.850
0.800
0.750
0.700
0.
THE UNIVERSITY OF ADELAIDE
SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING
GEOTECHNICAL ENGINEERING II (C&ENVENG 2069)
ASSIGNMENT No. 5, 2013
SHEAR STRENGTH OF SOILS
Deadline: 5 pm, Monday, 4 November in Geotechnical Engineering II submission box. Late subm
MATHS 2201 Engineering Mathematics IIA Tutorial 3
1. The paper Ultimate Load Capacities of Expansion Anchor Bolts (J. Engr. Energy, 1993, pp.
139-158) reports the following summary data on shear strength for a sample of 3/8 inch anchor
bolts: n = 78, x
=
THE UNIVERSITY OF ADELAIDE
SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING
GEOTECHNICAL ENGINEERING II (C&ENVENG 2069)
SOLUTIONS TO ASSIGNMENT No. 2, 2013
COMPACTION and VERTICAL STRESS IN SOILS
1.
Convert bulk unit weights to dry unit weights using:
d =
b
MATHS 2201/7201, Engineering Mathematics IIA
Tutorial 10, Week 11
1. Consider the function f(a:) = [sin 2:] for 7f < a: < 7r and f(a: + 271) : f(:z:) for all :3.
Sketch the function and nd its Fourier series. 3. 8- 5! 693 1.61%.
. TI"
.,
453 S Soon warm d
MATHS 2201/7201, Engineering Mathematics IIA
Tutorial 8, Week 9
1. A simple model for torsional oscillation of a suspension bridge (such as the Tacoma Narrows
bridge) consists of a rigid rod of mass m and length 2l supported at either end by elastic
cable
THE UNIVERSITY OF ADELAIDE
SCHOOL OF CIVIL AND ENVIRONMENTAL ENGINEERING
GEOTECHNICAL ENGINEERING II (C&ENVENG 2069)
SOLUTIONS TO ASSIGNMENT No. 3, 2013
FLOW OF WATER THROUGH SOILS
1.
Given the datum at the surface of the downstream, we have
(i)
Elevation