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Unformatted text preview: CHAPTER 6 More Differentation, and Optimisation In the previous chapter we differentiated simple functions: powers of x , and polynomials . Now we introduce further techniques of differenti- ation, such as the product, quotient and chain rules , and the rules for differentiating exponential and logarithmic functions . We use differentiation in a variety of economic applications: for example, to find the profit-maximising output for a firm, the growth rate of GDP , the elasticity of demand and the optimum time to sell an asset . ./ 1. Graph Sketching The first step towards understanding an economic model is often to draw pictures: graphs of supply and demand functions, utility functions, production functions, cost functions, and so on. In previous chapters we have used a variety of methods, including differentiation, to work out the shape of a function. Here it is useful to summarise them, so that you can use them in the economic applications in this chapter. 1.1. Guidelines for Sketching the Graph of a Function y ( x ) Look at the function y ( x ) itself: to find where it crosses the x- and y-axes (if at all), and whether it is positive or negative elsewhere; to see what happens to y when x goes towards ; to see whether there are any values of x for which y goes to . Look at the first derivative y ( x ) to find the gradient: It may be positive for all values of x , so the function is increasing; or negative for all values of x , so the function is decreasing. Otherwise, find the stationary points where y ( x ) = 0, and check whether the graph slopes up or down elsewhere. Look at the second derivative y 00 ( x ) to see which way the function curves: Use it to check the types of the stationary points. It may be positive for all values of x , so the function is convex; or negative for all values of x , so the function is concave. Otherwise, it tells you whether the gradient is increasing or decreasing in dif- ferent parts of the graph. Depending on the function, you may not need to do all of these things. Sometimes some of them are too difficult for example, you may try to find where the function crosses the 97 98 6. MORE DIFFERENTATION, AND OPTIMISATION x-axis, but end up with an equation that you cant solve. But if you miss out some steps, the others may still give you enough information to sketch the function. The following example illustrates the procedure. Examples 1.1 : Sketch the graph of the function y ( x ) = 2 x 3- 3 x 2- 12 x (i) y (0) = 0, so it crosses the y-axis at y = 0 (ii) To find where it crosses the x-axis, we need to solve the equation y ( x ) = 0: 2 x 3- 3 x 2- 12 x = 0 x (2 x 2- 3 x- 12) = 0 One solution of this equation is x = 0. To find the other solutions we need to solve the equation 2 x 2- 3 x- 12 = 0. There are no obvious factors, so we use the quadratic formula: 2 x 2- 3 x- 12 = 0 x = 3 105 4 x 3 . 31 or x - 1 . 81 So the function crosses the...
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- Spring '10