MAM1000 – Page 6 – Mathemafrica

Domain of a composite function – part 1

This post was written by Muhammad Azhar Rohiman, a first year student on MAM1000W at UCT. This post came about when he asked me a question related to domains of composite functions, and it was clear that on first learning about such topics, there are some simple misunderstandings. I suggested that he write a couple of paragraphs explaining what he had learnt, and the following is, I think, a very clear explanation of some of the ideas and pitfalls of this topic.

Consider the two functions below, from which we want to find the domain of $( f \circ g )(x)$

$f(x) = \frac{1}{x+2}$ , $g(x) = \frac{x}{x-3}$

We know that f(x) and g(x) cannot be defined at the values x = -2 and x = 3 respectively. This can be written as follows: f(-2) and g(3) are not defined. The domain of a composite function will not allow any values restricted by the domain of the starting function, which is g(x).…

By Jonathan Shock| March 7th, 2016|Courses, First year, MAM1000, Undergraduate|4 Comments

Mathematical induction

One of the concepts that most students seem to struggle with the most in the first year maths course is that of mathematical induction. It feels abstract, yet when you have to prove a concrete statement, it feels like all the assumptions, and cases you look at shouldn’t have any real impact on the thing that you’re trying to prove. I will now try and prove that this is not true (though not by mathematical induction!).

I’m going to start with a ladder brought from a magical ladder supplier. Steps on the ladder are labeled S(1), S(2), S(3) etc. The question is, are there infinitely many steps on the ladder? Well, with the information that I’ve given you so far, you just don’t know, but the manufacturer has given a little leaflet with the it. In the leaflet it says:

“As long as your ladder has a step S(n), we hereby guarantee that it will have a step S(n+1), for any integers n”.…

By Jonathan Shock| March 6th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments

Polynomial division

Note: In the following I use the words power, and order somewhat interchangeably, in relation to the exponent of x.

Please also forgive the rather bad formatting for some of the expressions in this text. WordPress and LaTeX are somewhat unforgiving.

One of the topics which seems to have caused the most problems in the assignments for MAM1000W so far this year is that of polynomial division. Thus I want to go through an example here to show you exactly what we’re doing when we think about performing such a calculation.

If you think you already know what to do, but want to have some more practice, plug in some random polynomials into this page and make sure that you get the same steps as them.

In fact, the first question might be: what’s the point? Often the expression looks more complicated in the end than it does at the beginning.…

By Jonathan Shock| March 6th, 2016|Courses, First year, MAM1000, Undergraduate|8 Comments

First year resources – part 4: revision

This is a continuation of the previous posts, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The following sections in the resource book are about mentors and the whiteboard workshop. They are really quite specific to the course, and more about the details than the philosophy of it, so I am not including them here.

The next section on the other hand is vital, as most students are never given much guidance in how best to revise, and it is one of the most important skills they can gain. I have written my thoughts from my own experience, but I am not trained specifically in education, so I am grateful for any additional thoughts.

How to revise

Revision is a bit like comedy: Timing is everything!

…

By Jonathan Shock| January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments

First year resources – part 3: Tutorials

This is a continuation of the previous post, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The next part of the resource book is written by a former first year who very helpfully wrote “How I achieved over 80% in MAM1000W”. I am not including this here as I shan’t be altering what he has written

The next section is entitled “Tutorials”, and helps to make the tutorial problem sets and tutorial sessions themselves as useful as possible.

The weekly tutorial questions which you will be given to help practice what you’ve learnt in class are renowned for being time-intensive. This is true, but on top of going to lectures, these are the key element of the course for helping you to really master its content.

…

By Jonathan Shock| January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments

First year resources – part 2: Why study mathematics?

This is a continuation of the previous post, essentially collecting thoughts for first year students. I am asking you, the reader to suggest what might be wrong, or missing from this, and anything else which will be helpful for a new first year who is just arriving at university to study maths…

The second part of the resource book is a “Meet the team” section, which includes photos and a short bio of the convenor (me), the lecturers, and the senior tutors for the course.

The next section is entitled “Why study mathematics”, and is in some ways the most controversial/important section in here.

For quite a few of you, the immediate answer you may have thought of to this question is not one that will make you happy, but I hope that this section will give you reasons to feel really positive about taking this course.

For some of you, the reason to study maths is because you have to study it as a prerequisite for your course.

…

By Jonathan Shock| January 26th, 2016|Courses, First year, MAM1000, Undergraduate|6 Comments

First year resources – part 1: Intro to first year mathematics

As mentioned in the previous post, I am writing a resource book for the first year course that I will be in charge of this year. I want to crowd-source ideas for this a little. By that I mean that I will put up a number of the sections that I’m writing, and I’m keen to know, from those who have been through the course whether you feel that there is anything wrong, or missing from this. It is hard to know for someone who has not been on the learning end of a course like this for a long time what makes for the most useful information.

The second reason for doing this is that I think that these thoughts can be used for a wide variety of courses, and so if anyone wants to take these resources and use them in other contexts, then I will be very happy for them to do so.…

By Jonathan Shock| January 26th, 2016|Courses, First year, MAM1000, Undergraduate|3 Comments

First year mathematics experience enhancement – a question for you!

I am coming to you today with questions. Well, questions based on some of my own ideas…

This year I will be not only teaching, but entirely in charge of the UCT first year mathematics for scientists courses, known as MAM1000W. I have a number of changes I plan on making, not so much to the syllabus, but to the extra activities associated with the course, in an attempt to make it as rich and deep a learning experience as I can.

The first step of this has been altering the structure of the resource book. The resource book is a PDF which will be sent to all first years taking the course. Historically, it contains a little about the course content, a bit about how your marks will be calculated, a bit about good practice in terms of how to work, and then the second half is filled with tutorial questions.…

By Jonathan Shock| January 25th, 2016|Courses, MAM1000|10 Comments

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UCT MAM1000 lecture notes part 50 – linear algebra part iii

Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate

UCT MAM1000 lecture notes part 50 – linear algebra part iii

Gauss reduction

So far we have seen that we have a way to translate a system of linear equations into a matrix. We can manipulate the matrix in ways which correspond to operations on the equations which keep the important information in the system of equations the same (ie. the solution of the equations before and after the operations is the same). We have seen a couple of examples of when we can read off the solution from the matrix having performed the operations. So far the order with which we perform the operations feels a bit arbitrary, although we know that we would like to get the matrix into reduced row echelon form. There is however a very systematic way of going about this, and the term for the process is called Gauss Reduction.

Here is a detailed view of what Gauss Reduction will give us:

Gauss Reduction:

To solve a system of linear equations:

1) First find the augmented coefficient matrix of the system of equations.…

By Jonathan Shock| October 4th, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|1 Comment

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UCT MAM1000 lecture notes part 49 – linear algebra part ii

Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate

UCT MAM1000 lecture notes part 49 – linear algebra part ii

Matrices

Solving a system of linear equations is not technically difficult: just eliminate the variables in a systematic fashion. When there are only two or three variables, this is easy to manage. But for a bigger system, things can quickly get confusing. We need to develop a systematic method.

The first thing to notice is that the names of the variables don’t matter. Consider, for example, the two systems

$\begin{array}{cc} x + y &=3\\ 2x-y &= 4 \end{array}$

and

$\begin{array}{cc} u + v &=3\\ 2u-v &= 4 \end{array}$

It’s clear that if we ignore the names of the variables — $x$ and $y$ versus $u$ and $v$ — these two systems are the same. The reason we can tell that they’re the same is because the {\em coefficients} of the variables are the same and the numbers on the right hand side are the same. These are really the only things about a system of linear equations that matter, and so what we can do is strip the system down to its bare bones and rewrite it like this:

$\left( \begin{array}{cc|c} 1&1&3\\ 2&-1&4 \end{array} \right)$

This is an augmented coefficient matrix (in general, a rectangular array of numbers, like the above, is called a matrix; a matrix with an additional vertical line, which plays the same role as the equals signs in the original equations, is augmented).…

By Jonathan Shock| October 3rd, 2015|Courses, English, First year, Level: intermediate, MAM1000, Uncategorized, Undergraduate|1 Comment