How does analysis build on my earlier studies?

Once we have a good understanding of the number line, we use ideas from analysis to study more complicated geometric objects such as curves. In particular, we can consider graphs of functions $f\colon\mathbb{R}\to\mathbb{R}$. The ideas and techniques used to study real numbers can also be used to study the curves formed by such graphs. This is the subject of Chapters 4 and 5.

Before coming to university, you will have spent a lot of time studying graphs of functions, becoming expert at computing derivatives and integrals. These are important skills. Such computations give information about the geometry of the graph: the derivative tells you the slope (or gradient) of the graph at a given point, and the integral tells you the area under the graph.

Analysis provides the foundations for differential and integral calculus. In the latter half of this course, we shall use analysis to complement your important computational calculus skills with equally important (but somewhat deeper) theoretical understanding of how and why differentiation really works. For instance, you will have learned a lot of formulæ  for computing derivatives. But have you paused to ask, for instance:

Or why is $\frac{\mathrm{d}}{\mathrm{d}x}x^{3}=3x^{2}$? Or why is $\frac{\mathrm{d}}{\mathrm{d}x}\arcsin x=(1-x^{2})^{-1/2}$? Why does the chain rule hold? Where do all these formulæ  actually come from? If you stop and think about it, it is almost magical that the slope of the sine graph is given precisely by the cosine function. What is ‘behind the magic’?

For that matter, is it even clear what the ‘slope’ of a graph really means? For a straight line, we have a simple notion of slope given by the ratio of ‘height’ vs ‘length’. However, if the graph is curved, then it is not at all obvious what ‘slope’ refers to or how to compute it. So we could ask a more fundamental questions:

Analysis lies at the heart of all of these questions. The key notion of a limit (this time the limit of a function rather than a sequence) provides the perfect tool for making sense of derivatives and understanding where all the familiar formulæ  from calculus come from.

Thus, this course builds on your calculus skills, but is of a different flavour to that of your previous experience of calculus. Here we are interested in getting ‘under the hood’ and understanding the reason behind a lot of the rules and formulæ  you have learned so far. Much of what you have learned in Introduction to University Mathematics will help us in this task. We shall make frequent use of concepts, results, formalism and skills developed there.