After introductions, we considered the type of questions that will be addressed in MS 2001. Some examples were:

1. What is the slope of a function $f(x)$?
2. How can we find the local maxima and minima of $f(x)$?
3. Which two numbers add to 20 and have greatest product?
4. If a farmer has 100 m of fencing and wants to fence a rectangular field beside a river; what dimensions confer the largest area?
5. Which rectangle of a given area (permimeter) has the smallest perimeter (greatest area)?
6. What is the distance from a point to a line?
7. Sketch

$\frac{2x^2+x+1}{x+1}$

8.   What is $\sqrt{5}$? What are the roots of $x^3-x+1$

We stated that we knew how to answer these questions from MS 1001 but that we were going to develop the axiomatic and rigourous theory behind these answers. We discussed the axiomatic model. We proceed by writing down a set of axioms and deducing the conclusions from these.

As an example we wrote down the field axioms of the real numbers. From these we deduced that $x\times 0=0$ for all $x\in\mathbb{R}$. This was the first example of a result that we don’t have to assume: it follows from the axioms. We stated that the “rules” of algebra are the consequences of these axioms. We wouldn’t revisit this ground and the “rules” of algebra are assumed in this module.

We wrote down the axioms of inequality and derived some familiar properties of inequalities from these; including $1>0$. We introduced some notation for intervals. Finally we defined what an even and an odd function is. We also defined what it means for a function to be increasing or decreasing. Finally we gave the example of constant function – a function that is even and both increasing and decreasing.

Exercises

From the notes:

Let $a\in\mathbb{R}$, $a\neq 0$. Prove that if $a>0$  then $1/a>0$.