MATH135 Tutorial 3: Comments/solutions
1. For each of the functions below, sketch the inverse. For your convenience, the line y = x is repeatedly shown.
Comment:
Any intersections on the y = x axis stay fixed.
Asymptotes get reflected. Why? For sufficie
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Math 135 Tutorial 4, Third Semester 2002
A=
1 2
3 1
B=
0 1
1 0
C=
1 2
0 1
D=
a11 a12 a13 a14
G = a21 a22 a23 a24
a31 a32 a33 a34
1
0
L=
0
0
0
1
0
0
0
0
1
0
2
0
0
1
1 0
0 3
1
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Math 135 Homework Assignment 7
Due Monday 27 January 2003
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Calculus
Elementary Linear Algebra
James Stewart
Howard Anton
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Math 135 Homework Assignment 1
Due Monday 11 November 2002
Page A32
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Calculus
First Year Mathematics Notes
James Stewart
William W.
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Math 135 Homework Assignment 3
Due Monday 2 December 2002
Page 113
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Calculus
Elementary Linear Algebra
James Stewart
Howard Anton
(43, 45a ), (49b , 50c )
(1
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Math 135 Homework Assignment 2
Due Friday 22 November 2002
Page 81
Page 92
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Calculus
First Year Mathematics Notes
James Stewart
William W. L. Chen
(13), 36a
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Math 135 Complex Numbers Tutorial , Third Semester 2002
This tutorial contains several complex number questions. Answers to all of the questions are contained in this tutoria
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Math 135 Tutorial 2, Third Semester 2002
1. Simplify the following complex expressions. Example solutions to can be found on page 6 number 12.
1i
1 ix
13
5 (3 4i)
1 (x + iy
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Math 135 Homework Assignment 5
Due Monday 6 January 2003
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Calculus
Elementary Linear Algebra
James Stewart
Howard Anton
(1135)odda
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Math 135 Tutorial 1, Third Semester 2002
This tutorial contains some basic graphing, vector calculations, and some complex number questions.
1. Very carefully sketch the grap
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Math 135 Homework Assignment 6
Due Friday 17 January 2003
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Calculus
Elementary Linear Algebra
James Stewart
Howard Anton
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Math 135 Tutorial 3, Third Semester 2002
1. Give at least three formulas for continuous real valued functions f (x) which satisfy lim f (x) = 0.
x
2. Give at least three form
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Math 135 Homework Assignment 8
Due Friday 7 February 2003
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a Make
b plot
Calculus
Elementary Linear Algebra
James Stewart
Howard A
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Math 135 Homework Assignment 4
Due Friday 13 December 2002
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Calculus
Elementary Linear Algebra
James Stewart
Howard Anton
35
(1939)oddb , (49, 51),
Chris Gordon
Congruence Notes
1 Divisibility Tests
A common task in computing is to test to see
p if a number N is prime or not; one way of doing this
is to test N for divisibility for primes up to N. Then quick ways need to be developed to test for
divis
718
Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS
13.1
DISPLACEMENT VECTORS
Suppose you are a pilot planning a ight from Dallas to Pittsburgh. There are two things you must
know: the distance to be traveled (so you have enough fuel to make it) and in what
MATH135 Tutorial 4: Comments/solutions
1. Let y = f (t) be the (blue) function shown (twice) below, and let p(t) = f (t 1) and q(t) = f (t + 1). (That
is, the formula for p is obtained by replacing t with t 1 in the formula for f .) Is the orangle functio
MATH135 Tutorial 2: Comments/solutions
1. Write down explicit expressions (for example cfw_1, 2, 3) for the following sets:
(a) cfw_x N : x2 7x + 10 < 0;
Comment: Factor the quadratic and make a quick sketch of the graph; the answer only has two
elements.
MATH135 Tutorial 6: Comments/solutions
1. Let y = f (x) = 50(1 2x ). Draw a neat graph of f . (It is suggested that you show how f can be drawn
from 2x by a sequence of transformations.) You will now prove that f (x) can be made as close as you please
to
MATH135 Tutorial 7: Comments/solutions
1. Use the definition of the derivative to calculate:
(a) f 0 (3) where f (x) =
2x
x2
Solution: We have
2x
6
f (x) f (3)
= x2
x3
x3
2x 6x + 12
=
(x 2)(x 3)
4(x 3)
=
(x 2)(x 3)
4
=
x2
4
as x 3.
(b) f 0 (a) where f (x
MATH135 Tutorial 5: Comments/solutions
1. A metallic spherical ball is needed for a scientific experiment. The ideal volume required is 400 cm3 . The
machine that manufactures the ball sets the radius, r. Since the volume is a function of the radius, we h
MATH135 Tutorial 8: Comments/solutions
1. In week 7, we showed that if a function has a positive derivative on an interval (a, b), and is continuous on
[a, b], then f is increasing on that interval. If the converse of this statement true?
Comment: The con
MATH135 S216
Mathematics IA
Assignment 3
NAME: Ami Mushtaqali
Student Id: 44921977
Tutorial Group: D, Fri 13:00, W5C 309
Department of
Mathematics
Tutor: Joel Couchman
Due 21:00, 8/11 2016
Please sign the declaration below, and staple this sheet to the fr
DEPARTMENT OF
MATHEMATICS
MATH135 S216
Mathematics IA
Tutorial Week 5
NAME: Ami Mushtaqali
Student Id: 44921977
Tutorial Group: D, Fri 13:00, W5C 309
Tutor: Joel Couchman
Use this page as the coverage for your post-tutorial submission next week.
1
MATH135
Chapter 36 Diffraction
(continued)
PHYS 143 Week 6
!
36-1 Single-slit diffraction
36-2 Single-slit diffraction: intensity
36-3 Diffraction by a circular aperture
36-4 Diffraction by a double-slit
!
[36-5 not covered]
1
find the dark fringes, we shall use
Chapter 17 Waves II
PHYS 143 Week 2
Andrei Zvyagin
Department of Physics & Astronomy
Macquarie University
17-1 Speed of sound
17-2 Travelling sound waves
17-3 Interference [17-4, 17-5 not covered]
17-6 Beats
17-7 The Doppler effect [17-8 not covered]
Soun
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Math 135 Mathematics 1A
Unit Outline
Semester 3, 2002
28 October 15 February
Lecturers: Dr Edwin Franks and Christopher Gordon
October 15, 2002
Page 1 of 25
2002_3
Math 135 U
MATH135 Mathematics IA
Week 11 (Linear Systems)
Chris Gordon
chris.gordon@mq.edu.au
web.science.mq.edu.au
Systems of linear equations
Consider the linear equations
a11x1 + + a1nxn = b1,
.
.
am1x1 + + amnxn = bm.
This system has m equations and n unknowns.
MATH135 Assignment 2: Solutions
1. As part of a high performance telescope, NASA wishes to construct a reflective cube, so that regardless of its orientation, a light source can be moved to shine a laser beam directly onto a face.
Ideally, the cube must b