Jonathan Shock – Page 13

Arbitrary functions as the sum of odd and even functions

Let’s take a function h(x), whose domain is the real numbers. We are simply going to start by writing h(x) in a slightly strange way. We will write it as:

$h(x)=\dfrac{h(x)+h(-x)+h(x)-h(-x)}{2}$

This might seem an odd thing to do – we have essentially added zero to the original function (in the form h(-x)-h(-x)). However, we can see that we can split this as:

$h(x)=\dfrac{h(x)+h(-x)}{2}+\dfrac{h(x)-h(-x)}{2}$

It’s exactly the same thing we started with, right? But now it’s written in a peculiar way. Now let’s call the two fractions f(x) and g(x) respectively. So:

$f(x)=\dfrac{h(x)+h(-x)}{2}$

and

$g(x)=\dfrac{h(x)-h(-x)}{2}$

So our original function can be written as h(x)=f(x)+g(x). If you plug in f(x) and g(x) above you will see that we have said nothing which is not trivial in any of this. However, the interesting part comes when we look at the properties of f(x) and g(x). What is f(-x)?

$f(-x)=\dfrac{h(-x)+h(x)}{2}=\dfrac{h(x)+h(-x)}{2}=f(x)$

But this is just the defining property of an even function, so f(x) is even.…

By Jonathan Shock| March 10th, 2016|Courses, First year, MAM1000, Undergraduate|1 Comment

Mathematical induction with an inequality

In the tutorial sessions it was clear that one question in particular was causing problems. This is an induction proof with an inequality. The one which we will look at is the inequality:

$2^n>n^3$ for $n\ge 10$

I am going to talk you through it in more detail than would be needed for the formal proof but I want to give some intuition along the way.

So, we start off, as always with the base case. The base case is always the first integer for which the statement is claimed to be true. In this case it is for n=10. Let’s check for n=10. Is it true that?

$2^{10}>10^3$ ?

Well this is:

$1024>1000$

and so we should be happy with that. We’ve proved the base case. Note that you do not then need to check for n=11, or n=12. I have seen many students check a base case for n=1, and then also check for n=2 and n=3.…

By Jonathan Shock| March 9th, 2016|Courses, First year, MAM1000, Undergraduate|12 Comments

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

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Positions available at The UCT Department of Maths and Applied Maths

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Positions available at The UCT Department of Maths and Applied Maths

Two positions are currently open at the UCT department of Maths and Applied Maths: One as a lecturer or senior lecturer, and the other is a teaching position. Please share. PDFs are here: L/SL for the lecturer/senior lecturer position and here: Teaching for the teaching position.

How clear is this post?

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By Jonathan Shock| March 6th, 2016|Job advert|0 Comments

How to reduce the fear of mathematics

I sat this morning reading a little of The Book of Life, by Krishnamurti – something which I like to browse through and ponder from time to time. This morning’s meditation somehow felt very apt as I attempt to get almost 800 students to enjoy mathematics, and learn its techniques as well as its beauty. The meditation was the following:

How is the state of attention to be brought about? It cannot be cultivated through persuasion, comparison, reward or punishment, all of which are forms of coercion. The elimination of fear is the beginning of attention. Fear must exist as long as there is an urge to be or to become, which is the pursuit of success, with all its frustrations and tortuous contradictions. You can’t teach concentration, but attention cannot be taught just as you cannot possibly teach freedom from fear; but we can begin to discover the causes that produce fear, and in understanding these causes there is the elimination of fear.

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By Jonathan Shock| February 20th, 2016|Uncategorized|3 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!

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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.

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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.

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

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