Question 1
0 pts
Are the polynomials listed below linearly independent in P2 ?
1 3t, 1 + t 2 , 1 3t + t 2 .
1
Question 2
0 pts
Let B = cfw_1 + t, 2 t and C = cfw_1 t, t be two bases for P1 .
1. Expr
Q1
Q2
Q3
Q4
Q5
Total
The Australian National University
Midsemester Examination - March 2014
MATH1014 - Mathematics and Applications II.
Book A - Calculus
Questions 1 - 5
Student No.: . . . . . . . .
Q1
Q2
Q3
Q4
Q5
Total
The Australian National University
Midsemester Examination - March 2013
MATH1014 - Mathematics and Applications II.
Book B - Linear Algebra
Questions 1 - 5
Student No.: . . . . .
Q1
Q2
Q3
Q4
Total
The Australian National University
Midsemester Examination - March 2015
MATH1014 - Mathematics and Applications II.
Book B - Linear Algebra
Questions 1 - 4
Student No:
U
Important no
Q1
Q2
Q3
Total
The Australian National University
Midsemester Examination - March 2014
MATH1014 - Mathematics and Applications II.
Book B - Linear Algebra
Questions 1 - 3
Student No.: . . . . . . . .
U3232826
Cynthia Benjamin
MATH1014 - Assignment
Due May 8, 2017
Linear Algebra
Refer to assignment sheet for full problem.
W1
W3
W2
A)
Formulate a finite Markov Chain that describes this system. Be su
MATH 1014, Calculus Worksheet 01 with answers
1. Consider the integral
2
Z
x2 dx.
0
(a) Find the approximation M4 , where M4 is the midpoint Riemann
sum with 4 subintervals.
(b) Does M4 overestimate o
Question 3
Let R be the region bounded by the curves
y=
2
,
x2
y = 0,
14 points
x = 2,
x = 5.
(a) Find the area of R.
(6 points)
Solution:
y
4
3
2
1
y = x22
x
1
2
3
4
5
1
Z
5
2
dx
2
2 x
5
2
=
x 2
= 2
MATH1014 Assignment, Semester 2, 2016
Due on Thursday September 29 at 5pm.
Please submit your assignment in the assignment box labelled with
your tutors name, in the foyer of the John Dedman Building
Functions of Several Variables: Visualisation
So far we have considered function of just one variable: each value of a
single input x determines a value f (x).
But we often have to deal with functions
Sequences: introduction
The term sequence in mathematics is used to describe an infinite
succession of numbers. For example:
1, 2, 3, 4, 5, .
1, 3, 5, 7, 9, .
0, 2, 4, 6, 8, .
In the above cases, the
Q1
Q2
Q3
Q4
Total
The Australian National University
Midsemester Examination - September 2014
MATH 1014 - Mathematics and Applications II.
Book A - Calculus
Questions 1 - 4
Student number: . . . . . .
Department of Mathematics, ANU
MATH1014
Worksheet 7
1. Find all points on the curve
r2 = cos 2
where the tangent line is horizontal, vertical or it does not exist.
2. Using polar co-ordinates, find th
Department of Mathematics
MATH1014
Worksheet 3
1. Suppose that a mothball loses volume by evaporation at a rate proportional to its
instanteneous area. If the radius of the ball decreases from 2 cm to
Question 1
0 pts
Find the eigenvalues of the matrix
0 1 1
A = 1 2 1
1 1 0
and identify the dimension of each eigenspace.
1
Question 2
0 pts
If v1 and v2 are eigenvectors corresponding to differen
Department of Mathematics, ANU
MATH1014
Tutorial Worksheet 5
1. Prove that if a positive term series
2. Show that the series
P
an converges, then
P
1/an diverges.
X
(1)n
n=1
n5n
is convergent. How man
Question 1
0 pts
Define T : P3 M 22 by
p(3) p(1)
T (p) =
.
p(1) p(3)
1. Show that T is a linear transformation.
2 3
2. Find the matrix
of
T relative
to the
basis B = cfw_1, t, t , t for P3 and the
Question 1
0 pts
0.4 0.3
1
3
has eigenvectors u =
and v =
. Explain
0.4 1.2
2
2
0.5 0.75
k
why B approaches
as k .
1.0
1.5
The matrix B =
1
Question 2
0 pts
Consider the following 3 3 matrices:
(a) A
Question 1
0 pts
#o f rabbits (in 1000s)
When modelling a rabbit/fox ecosystem, with xk =
#o f f oxes
in the k-th period, two possible models are under consideration:
1.2 0.2
xk+1 =
x ,
(1)
0.4 0.6 k
Question 1
0 pts
If W is the plane ax + by + cz = 0 in R3 , describe W .
1
Question 2
0 pts
1. Recall the definition of an orthogonal matrix. If U is orthogonal, whats
special about U T U ? UU T ?
2.
Question 1 (Gram-Schmidt)
0 pts
The matrix A is given by
1
1
A =
0
0
0
1
1
0
0
0
.
1
1
1. Use Gram-Schmidt to find an orthogonal basis for col A.
2. Find a QR factorisation for A.
1
Question 2 (Least
Department of Mathematics
MATH1014
LF 2/09
Calculus Worksheet 9 : for tutorials in week 11
1. The dimensions of a box are 30cm, 40cm, and 50cm, with a possible error of
1/16cm in each measurement. Use
Department of Mathematics, ANU
MATH1014/Calculus
Worksheet 6
Suppose we wish to estimate sin 5 using a Taylor polynomial (note that 5 = /36
radians).
1. Find the Taylor series for sin about a = 0.
2.
Overview
Weve studied the geometric and algebraic behaviour of vectors in
Euclidean space. This week we turn to an abstract model that has many
of the same algebraic properties.
The importance of this
Overview
Given two bases B and C for the same vector space, we saw last week how
P and P . Such a matrix is
to find the change of coordinates matrices CB
BC
always square, since every basis for a vect
Overview
Last time we studied the evolution of a discrete linear dynamical system,
and today we begin the final topic of the course (loosely speaking).
Today well recall the definition and properties
Overview
Last time
we defined the dot product on Rn ;
we recalled that the word orthogonal" describes a relationship
between two vectors in Rn ;
we extended the definition of the word orthogonal" to d
Applications of Integration
Here, we shall see some applications of the definite integral. In particular,
we shall start with the application that originally motivated its definition.
That is, determi
Overview
Last time we discussed orthogonal projection. Well review this today
before discussing the question of how to find an orthonormal basis for a
given subspace.
From Lay, 6.4
A/Prof Steve Robert
Warm-up
Question
Do you understand the following sentence?
The set of 2 2 symmetric matrices is a subspace of the vector
space of 2 2 matrices.
A/Prof Steve Roberts (ANU)
MATH1014 Notes
First Semester