Department of Mathematics
MATHS 108
Tutorial 4 Solutions
) 1.70
1. (a) = cos1 ( 144
5
(b) 2u + 12 v = (1, 2, 3, 4, 5)
2. ku + vk2 = kuk2 + kvk2
(u + v) (u + v) = u u + v v
(u + v) u + (u + v) v = u u + v v
uu+vu+uv+vv = uu+vv
u u + 2u v + v v = u u +
Department of Mathematics
MATHS 108
Tutorial 7 Solutions
1 3
1. (a) 2D C T = 5 4
1 1
(b) The matrix
undefined.
0
(c) DC = 0
0
A(B C) is undefined as B and C have different dimensions so their difference is
1 2
2 4
1 2
(d) The matrix CB is undefined as C
Department of Mathematics
MATHS 108
1. lim
x2
g(x)
f (x)
Tutorial 2 Solutions
= lim
x2
x2
1
x2
= .
+ 3x 10
7
2. (a) Yes this is a function.
(b) Not a function. For some x R there exists more than one y value.
(c) Not a function. For all x R there exist in
Department of Mathematics
MATHS 108
Tutorial 3 Solutions
1. Change 3x2 1 to 4x2 1.
2. (a) w R such that w 6= 17
(b) The time taken by an experienced worker to assemble the item in the factory.
3. f is discontinuous, g is continuous and h is discontinuous.
Department of Mathematics
MATHS 108
Tutorial 5 Solutions
1. t = 2, so (x, y, z) = (1, 8, 7)
2. (a) No. There is no value of t which satisfies both equations.
(b) The line is parallel to the plane.
3. (a) Intersection point is (5, 5).
(b) Intersection poin
Department of Mathematics
MATHS 108
Tutorial 2
1. Let f (x) = x2 + 3x 10 and g(x) = x 2.
g(x)
g(x)
Determine whether or not lim
exists. If it does exist, find lim
. If it does not exist,
x2 f (x)
x2 f (x)
explain why not.
2. Determine if the following rel
MATHS 108
Department of Mathematics
Tutorial 1
Work together in a group of 2 to 5 students to answer these questions.
1. Solve the inequality 2x + 1 > 7.
2. Solve the inequality |x + 1| > 2.
3. Solve the inequality |x + 5| > |x 5| using the following meth
Department of Mathematics
MATHS 108
Tutorial 1 Solutions
1. x > 3
2. x (, 3) (1, )
3. x > 0
4. Methods: quadratic formula, find a root, factorise, or sketch the graph: to find x = 1 and x = 3/4
5. x cfw_1, 0, 1
6. FALSE all assessments count for a student
Page 1 of 2
Department of Mathematics
MATHS 108 Assignment 1 Due: 4pm, Wednesday March 30st
Hand your completed assignment in to the correct box in the Student Resource Centre (G38 Building 301) before the due date. Please
use a Mathematics Department cov
MATHS 363
THE UNIVERSITY OF AUCKLAND
FIRST SEMESTER, 2015
Campus: City
MATHEMATICS
Advanced Modeling and Computation
(Time allowed: 2.0 hours)
NOTE: Do all questions. Calculators are not permitted. Show your working. Total marks 60.
Page 1 of 4
MATHS 363
DEPARTMENT OF MATHEMATICS
MATHS 190
Tutorial 9
Tutorials in Maths 190 are collaborative tutorials. You should work in groups of 2 or
3 students, discussing the situations and puzzles listed below, or issues arising from
lectures. Part of your final mark d
DEPARTMENT OF MATHEMATICS
MATHS 190
Tutorial 6
Tutorials in Maths 190 are collaborative tutorials. You should work in groups of 2 or 3 students,
discussing the situations and puzzles listed below, or issues arising from lectures. Part of your final
mark d
DEPARTMENT OF MATHEMATICS
MATHS 190
Lecture 1 Summary
In this lecture we illustrate two important observations:
Mathematics involves logical and creative thinking.
Thinking can be fun.
In Lecture 1 there are two main examples
Dodge Ball. Instructions fo
DEPARTMENT OF MATHEMATICS
MATHS 190/190G
Tutorial 4
Tutorials in Maths 190 are collaborative tutorials. You should work in groups
of 3 or 4 students, discussing the situations and questions listed below, or issues arising
from lectures. Part of your final
Department of Mathematics
MATHS 190/190G
Assignment 1
Due: 4pm, March 30th
Hand your completed assignment in to the correct box in the Student Resource Centre (G402
Building 301) before the due date. Please use a Mathematics Department cover sheet availab
Department of Mathematics
MATHS 190/190G
Assignment 2
Due: 4pm, Tuesday 3 May
Hand your completed assignment in to the correct box in the Student Resource Centre (G38
Building 301) before the due date. Please use a Mathematics Department cover sheet avail
DEPARTMENT OF MATHEMATICS
MATHS 190
Tutorial 8
Tutorials in Maths 190 are collaborative tutorials. You should work in groups of 2 or
3 students, discussing the situations and puzzles listed below, or issues arising from
lectures. Part of your final mark d
DEPARTMENT OF MATHEMATICS
MATHS 190/190G
Tutorial 3
Tutorials in Maths 190 are collaborative tutorials. You should work in groups of 2 or
3 students, discussing the situations and puzzles listed below, or issues arising from lectures.
Part of your final m
DEPARTMENT OF MATHEMATICS
MATHS 190/190G
Tutorial 2
Tutorials in Maths 190 are collaborative tutorials. You should work in groups of 2 or
3 students, discussing the situations and puzzles listed below, or issues arising from lectures.
Part of your final m
DEPARTMENT OF MATHEMATICS
MATHS 190/190G
Tutorial 1
Tutorials in Maths 190 are collaborative tutorials. You should work in groups of 3 or
4 students, discussing the situations and puzzles listed below, or issues arising from lectures.
Part of your final m
Solutions to the exercises of Chapter 3
Exercises of Section 3.1
1.
Since the space is infinitely-dimensional we cannot find eigenvalues using the
characteristic polynomial. But it is easy to find eigenvectors. Let f (x) = a0 + a1 x + . . . +
an xn with a
Decomposed Graph
This shows 3 features:
1) The Time Series Plot (you have already
discussed the Raw Data AND the Trend!)
2) Seasonal effects
3) Residuals
millions of litres
Now, get your decomposed graph for Visitors to NZ
Seasonal effects seasonal variat
13. EQUATIONS AND IDENTITIES 2, Factorj'ze the following:
EXERCISE 13A Meaning ofldentities
(a) 2x2 w 8
1. State whether the foifowing retations are an equation or . 2
. . (b) 3.3 27'
an Identity. 2
(a) x“—i=(x—I)(x+1){x3+1) (0)83 ~3
(b) X2+2X+3=4x
(c) 2(
9. Make p the subject of the formula xjp + 1) = M
8. USE OF FORMULAE
1‘ Subject of Formulae
ke h the subject of the formula.
EXERCISE 8A Change 0
1. H 721: 6(1) + C), express 317a in terms of b and c. 10' Ma E2 1_ k
x - r - - T = me2 1 + h 1
2 In the form
12. SIMULTANEOUS LINEAR EQUATIONS IN TWO UNKNOWNS
EXERCISE 12A Revision of Linear Equations in one unknown
1- Solve the foﬁowing equations: 3. If 2;: — 1 is a factor of 4x2+ kx — 1, than k =
(1) 1—73- 1»% (2) 1343213 4. Ifx+3isafactorofx2+6x+k,thenk=
2 9