MAM1000W – Page 4 – Mathemafrica

Where did that substitution come from?

If you want to understand maths, you really have to do it. I recommend going through these examples and using the substitutions given here as hints. Get a blank piece of paper, put your notes away and try to do these examples and see if you get the same answers as in class. If you don’t, write in the comments, and we can see where things may have gone astray.

I’ve been teaching integration by substitution, including by trig substitutions over the last few days, and a frequent question which a newbie substituter will ask is “how did you know to make that substitution?”. It’s a very reasonable question, and one that takes practice to build the correct intuition, but I’ll do my best to give some motivation now as to why we made some of the substitutions we made. We won’t solve the integrals, but we will motivate here why we make particular choices for substitutions.…

By Jonathan Shock| July 22nd, 2016|Courses, English, First year, MAM1000, Undergraduate|2 Comments

Down the rabbit hole

The following has been a rather interesting journey – from a test question which seemed fine, to a subtlety which seemed easy, to a discussion with a number of different mathematicians about the nature of distributions, measure theory and regularisation. I will try and make it as clear as possible in the post below. Note, as mentioned in the comments, we have actually only found the solution to this problem for a constrained range of x, and not x $\in \mathbb{R}$ . I didn’t want to complicate things any more than necessary here for first year students, but the comments are very important too.

In a recent class test, there was a question, written by me, which was not quite the question that I wanted to ask. It turns out that it does have an answer, but it’s not an answer that can yet be found by the means at the class’s disposal.…

By Jonathan Shock| May 27th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|7 Comments

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The Newton-Raphson Method

Courses, First year, MAM1000, Uncategorized, Undergraduate

The Newton-Raphson Method

How would you go about finding the value of $\sqrt{3}$ if you didn’t have a square root button on your calculator? Well, the most obvious thing might be to try some values, based on your knowledge of the square root function. You are being asked to find that x for which:

$x=\sqrt{3}$

or, in other words, that x, which, when squared gives 3. We have to be a little careful here because we know that there will actually be two numbers which satisfy this (one positive, one negative), and we are interested in the positive one only.

So, we try some values, but we don’t do it randomly, we can see that because $1^2=1$ and $2^2=4$ that whatever number squared gives 3 must be between 1 and 2. We can try something called a binary intersection. This just means taking the values which we know bound the right answer (ie. we know that 1<x<2), and trying the number in the middle.…

By Jonathan Shock| May 15th, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|4 Comments

Proof by induction winner: Gianluca Truda

With many congratulations for the winning entry, as voted for by mostly MAM1000W students!

At the end of every year, many families celebrate the holiday season by decorating a Christmas tree, and almost all of them will use some form of lights. The kind of cheap lights that sit on a long wire that gets wrapped around a tree, and then sparkle in a whole lot of awesome colours when you plug them into the power socket.

In my family, we have a whole lot of these kinds of cheap, fragile lights in a big box. Every year, when we decorate the tree, the box gets opened and is full of hopelessly tangled wires which were hastily shoved in the year before. It’s always a bit of a pain untangling the wires and it can get really
boring testing each string of lights to see if they are still working.…

By Jonathan Shock| April 23rd, 2016|Courses, First year, MAM1000, Undergraduate|3 Comments

Logical implications and the structure of if and only if statements

We had a homework assignment a couple of weeks back. It was looking at mathematics in a very different way from how many had seen it before, and it caused a lot of confusion. I would like to try and add some clarity to what we were doing. My thought was, rather than going through the questions themselves, I would like to add annotations to the proof itself. Let’s see how this works. The proof that you were given is in black, the annotations are in blue, and after I’ve been through the proof, I will expand on it in a simplified form.

Theorem: The function f is differentiable at x=a if and only if there is a constant m and a function E of x, defined for all $x \not = a$ , such that

$f(x)=f(a)+m(x-a)+E(x)(x-a)$ for all $x \not = a$ – (eq 1)

and,

$\lim\limits_{x\to a}E(x)=0$ .

(If both these conditions are satisfied, then $f'(a)=m$ .)

What we are doing here is giving another definition of differentiability (at a particular point, a).…

By Jonathan Shock| April 22nd, 2016|Courses, First year, MAM1000, Uncategorized, Undergraduate|1 Comment

The squeeze (or sandwich) theorem

Let’s say I ask you how tall Craig is going to be when he’s 15 years old, and let’s say, given his genetic information, you just can’t tell what height he will be at that age. However, you do know that he’s going to be taller than Lisa, and shorter than Khangelani up until the age of 15 (they are all the same age). With this information alone, you haven’t learnt much about the height of Craig when he’s 15. However, what if you can figure out, using your clever genetic detective work, that at the age of 15 Lisa and Khangelani will be the same height? The only way for this to be true, is if Craig (whose height is sandwiched in between that of Lisa and Khangelani) is also the same height at that age.

That’s really all the sandwich theorem is. Let’s look at a mathematical example.…

By Jonathan Shock| April 9th, 2016|Courses, First year, MAM1000, Undergraduate|4 Comments

Lecture 2: Sets

List of the things I learnt today

Definition of a set
Different ways of representing a set
Different kinds of sets
Intervals
Combinations of sets
Set notation in functions

Definition of a set

A set is a collection of objects

Different ways of representing a set

Sets of objects are usually denoted by uppercase letters e.g.

let $\mathit{A}$ be the set of all odd numbers.The objects contained in a set are called elements which are denoted by lowercase letters e.g. $\mathit{a}$ is an element of $\mathit{A}$ .

A set can be represented in two ways:

1.List

$\mathit{A}=\{a, b, c, d\}$

The set above is an finite list, you can count the number of elements.

$\mathit{B}=\{a, b, c, d,... t\}$

Is also a finite list. It’s important to present enough elements to produce a sequence if the use of ellipses is implemented for shorthand purposes.

$\mathit{C}=\{1, 2, 3, 5, 8,... \}$

This is an infinite set, you cannot count the number of elements in the set.

2. Set Builder Notation

$\mathit{D}=\{e \in \mathbb{R} : \mathit{C} (e) \}$

In the example above $e$ is an element in the set $\mathbb{R}$ where $\mathit{C} (e)$ is a characteristic of the elements which are specific to set $\mathit{D}$ or you can replace “:” with”|”.…

By Tumelo Lephadi| February 17th, 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!

…

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?

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

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