MAM 1010F, 2017
1
Tutorial 14
Evaluate the following indefinite integrals.
sin3t cos t dt
1.1
1.3
1.2
dx
(2x 1) sin(3x2 3x + 1) dx
1.5
1.7
3x
dx
)
dx
1.4
(
1.6
4 cot x dx
dx
2
Consider the indefinite integral
cos sin d.
(a)
Evaluate this integral by makin
MAM1010F Bootcamp
Term 2 : Skills Test 1 - Page 1 of 2
19 May 2017
1. (9 points) Find dy/dx if (i) ey cos x = 1 + sin (xy) and (ii) y = (tan x)2x showing all your
working.
2. (7 points) Find the derivative of the function f (x) = 3arctan x + sec (xe2x )
M
Exam Revision Worksheet
1
University of Cape Town
June 2017
Financial Mathematics
(a) Rx is invested into a savings account which offers an interest rate of 7, 5% per annum
compounded quarterly. After 4 years, a further R2x is deposited into the account.
MAM1010F Bootcamp
Term 2 : Skills Test 1 - Page 1 of 2
19 May 2017
1. (9 points) Find dy/dx if (i) ey cos x = 1 + sin (xy) and (ii) y = (tan x)2x showing all your
working.
2. (7 points) Find the derivative of the function f (x) = 3arctan x + sec (xe2x )
M
First Application: Marginal
Analysis
1. Suppose the total cost in dollars incurred each week by Polaraire for
manufacturing refrigerators is given by the total cost function
C x 8000 200 x 0.2 x 2
(0 x 400)
where x is the number of refrigerators manufactu
MAM1010F CT2 Revision Worksheet
1
Derivatives
Find
dy
dx
Wed 31 May 2017
if:
(a) y = 2 ln (sin x)
(b) y = log2 (sin x)
(c) y = 2x tan x
(d) y = sec (ex ) + arccos (2x )
(e) y = arctan (ln x)
(f) y = x arcsin (x2 )
(g) y = cos2 (x3 )
2
Implicit Differentia
MAM 1010F, 2017
Tutorial 15
This is a short Tutorial to give you some practice on the two topics covered in the last
week of lectures.
Integration by Parts
1
Evaluate:
1.1
(3x 4)e3x dx
1.2
arctan x dx
1.3
x4 ln x dx
1.4
x2 sin x dx
1.5
1.6
1.7
dx
arccos x
If you would like to access your examination script(s) please see the notice
board in the Mathematics Building for application details or alternatively go
to http:/www. mth. uct. ac. za .
Fill in your student number in the box below.
W
Department of Mathe
If'you would like to access your examination scripts) please see the notice
board in the Mathematics Building for application details or alternatively go
to http:/www.mth.uct.aoza . I
Fill in your student number in the box below.
Fill in your surname and
University of Cape Town
Department of Mathematics and Applied Mathematics
MAM1012S/ MAM1112S Class Test 1
13 August 2014
Time: 75 minutes
Total marks: 51
Full Marks: 50
Answer the questions in the space provided on the Question sheets. If you run out of s
The Not So Short
Introduction to LATEX 2
Or LATEX 2 in 139 minutes
by Tobias Oetiker
Hubert Partl, Irene Hyna and Elisabeth Schlegl
Version 4.20, May 31, 2006
ii
Copyright 1995-2005 Tobias Oetiker and Contributers. All rights reserved.
This document is fr
ALGEBRA QUALIFYING EXAM PROBLEMS
GROUP THEORY
Kent State University
Department of Mathematical Sciences
Compiled and Maintained
by
Donald L. White
Version: August 29, 2012
CONTENTS
GROUP THEORY
General Group Theory . . . . . . . . . . . . . . . . . . . .
CHAPTER 7
Cosets and Lagranges Theorem
Properties of Cosets
Definition (Coset of H in G).
Let G be a group and H G. For all a 2 G, the set cfw_ah|h 2 H is denoted
by aH. Analagously, Ha = cfw_ha|h 2 H and aHa 1 = cfw_aha 1|h 2 H.
When H G, aH is called th
M3210 Supplemental Notes: Basic Logic Concepts
In this course we will examine statements about mathematical concepts and relationships between these concepts (definitions, theorems). We will also consider
ways to determine whether certain statements are t
Closure operators on sets and algebraic lattices
Sergiu Rudeanu
University of Bucharest
Romania
Closure operators are abundant in mathematics; here are a few examples.
Given an algebraic structure, such as group, ring, field, lattice, vector space, etc.,
University of Cape Town
Department of Mathematics and Applied Mathematics
MAM1044H - Mathematical Methods for Dynamics (2013)
Final Examination
Time: 2 hours
Instructions
6 Read all questions carefully.
in This paper consists of 5 fully written an
16. Subspaces and Spanning Sets
It is time to study vector spaces more carefully and answer some fundamental
questions.
1. Subspaces: When is a subset of a vector space itself a vector space?
(This is the notion of a subspace.)
2. Linear Independence: Giv
Department of Mathematics and Applied Mathematics
MAM1000W
Tutorial 7
Answers and some solutions
April 2015
Extra Problems 5
2. (a) Between x = 3 and x = 5 the graph is close to a straight line with slope
P (5) P (3)
1
.
2
2
0
(b) P appears to be constan
Chapter 1
A survival kit of linear algebra
In this chapter we recall some elementary facts of linear algebra, which are needed throughout the
course, in particular to set up notation.
1
Vector spaces and subspaces
Reminder 1.1 (Vector space)
Let F be a fi
Equivalence relations
1. Binary relations
Definition 1.1 (Relation between two sets). If X and Y are sets, a relation between
X and Y is a subset R X Y . For a relation R X Y and x X, y Y if
(x, y) R, we write xRy and if (x, y) 6 R, we write x6Ry.
If xRy,
Chapter 7
The Peano Axioms
7.1
An Axiomatic Approach to Mathematics
In our previous chapters, we were very careful when proving our various propositions and theorems to only use results we knew to be true. However, many of
the statements that we take to b
Triangle inequality for complex numbers
This will be a sample formal proof write-up to use as a guide for doing
proof write-ups for the class. I went over a proof in the lecture for this
material, but a proof during a lecture looks different than a writte
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
MAM1043H/MAM1044H - M A T L A B P ROJECT
Hand-in Instructions
All work is required to be written up in LaTeX.
This project writeup is due at 12pm on Friday 2nd October
1. Easy:
The arrangement of seeds in
Order Relations
A relation R between two sets A and B is a subset of the Cartesian product A B.
If R is a relation between A and A, then R is said to be a relation on A (or in A). The set of
all first members of a relation R is its domain, and the set of
C
ankaya University
Department of Mathematics and Computer Science
2011 - 2012 Spring Semester
MCS 156 - Calculus for Engineering II
Final Examination
1) a) Find the angle between the planes x y = 4 and 2x y + z = 5.
y+1
z
x2
=
= and the plane 3x y 2z +
Dot Products and Projections
The Dot Product (Inner Product)
There is a natural way of adding vectors and multiplying vectors by scalars. Is there also a way to
multiply two vectors and get a useful result? It turns out there are two; one type produces a
DEPARTMENT OF MATHEMATICS AND APPLIED MATHEMATICS
MAM1044H - M ATHEMATICAL M ETHODS FOR DYNAMICS
S EMESTER 1 T UTORIAL 4
Vector Properties of Complex Numbers
1. Given two complex numbers 3 + 4i and 5 + 2i, add the two numbers graphically using the triang
ENGINE Workshops, 2014:
Complex Numbers
Getting the Motor Running.
1) MAM 1018S Nov 2013 Final Exam
3
a) Solve the equation z =1+ 3 i . List your solutions in mod-arg form, and sketch them
on an Argand diagram.
iz
z
b) Solve the equation e =e , and plot y