Given a function \(f(x)\), a quick way to get an idea of its behaviour is to sketch the graph of \(y=f(x)\).

This is often done by plotting discrete points and then connecting the points with a smooth curve.

The uncertainty in this approach is clear if one asks the following reverse question.

Given a discrete set of points \((x,y)\), what might be the behaviour of a corresponding function \(f(x)\)?

Is it this?Or this?

There are many possibilities!

Even if we know the function \(f(x)\) and use a sophisticated computer graphing tool, there is no guarantee that the resulting graph will reveal all the essential features of the function.

Consider these computer generated plots of the function

The function is actually undefined at \(x=3\), where it has the indeterminate form \(\frac{0}{0}\).

Yet even zooming in on a small interval about that point does not reveal this fact.

One must always keep in mind that computers plot a discrete set of points and make assumptions about the function's behaviour between these points.

Thus, caution is advisable with curve sketching.

We need the methods of calculus to analyze \(f(x)\) first and to generate a sketch of \(y=f(x)\) which accurately portrays the essential qualitative features of \(f\).

This process will be the focus of this unit.

Then, if needed, we can confidently use computers to generate quantitative information, knowing the relevant intervals of \(x\) and \(y\) and the points where special care is needed.