As foreshadowed in the previous chapter, we will learn curve sketching technique in this chapter. Mathematical ideas related to this technique include: maxima and minima, a point of inflection, and curvature (or concavity). The question we did not answer in Chapter 2 – why does the quadratic function have to shape like a parabola? – will also become clear after you learn this technique.

In this chapter, we also introduce the idea of the differential of a function, which is related to the derivative we learnt in the previous chapter. They are similar but different notions and the distinction of them will become important especially in Chapter 6where we deal with multivariate calculus. We apply these two mathematical ideas to discuss elasticity in the last section of this chapter. The price elasticity of demand will be our particular focus.

Chapter goals By studying this chapter you will

1. (1) be able to use the differentiation technique to sketch various functions;

2. (2) be able to explain the difference between the derivative and the differential of a function; and

3. (3) be able to calculate the price elasticity of demand using differentiation and appreciate the economic information it carries.

Curve sketching

In the previous chapter, I emphasised the importance of drawing a diagram when we want to find the global maximum or the global minimum of a function. Here, we study the way to sketch a graph of a function that contains sufficient information of it. What we mean by sufficient information will become clear after you have read this section.