8 Introduction to calculus Differentiate functions with one unknown variable. Find the slope of a function using differentiation. Derive marginal revenue and marginal cost functions using differentiation and relate them to the slopes of the corresponding total revenue and cost functions. | 8 Introduction to calculus Learning objectives After completing this chapter students should be able to Differentiate functions with one unknown variable. Find the slope of a function using differentiation. Derive marginal revenue and marginal cost functions using differentiation and relate them to the slopes of the corresponding total revenue and cost functions. Calculate point elasticity for non-linear demand functions. Use calculus to find the sales tax that will maximize tax yield. Derive the Keynesian multiplier using differentiation. The differential calculus This chapter introduces some of the basic techniques of calculus and their application to economic problems. We shall be concerned here with what is known as the differential calculus . Differentiation is a method used to find the slope of a function at any point. Although this is a useful tool in itself it also forms the basis for some very powerful techniques for solving optimization problems which are explained in this and the following chapters. The basic technique of differentiation is quite straightforward and easy to apply. Consider the simple function that has only one term y 6x2 To derive an expression for the slope of this function for any value of x the basic rules of differentiation require you to a multiply the whole term by the value of the power of x and b deduct 1 from the power of x. In this example there is a term in x2 and so the power of x is reduced from 2 to 1. Using the above rule the expression for the slope of this function therefore becomes 2 x 6x2-1 12x This is known as the derivative of y with respect to x and is usually written as dy dx which is read as dy by dx . 1993 2003 Mike Rosser Figure We can check that this is approximately correct by looking at the graph of the function y 6x2 in Figure . Any term in x2 will rise at an ever increasing rate as x is increased. In other words the slope of this function must increase as x increases. The slope is the derivative
