Differentiation - CHAPTER 5 Differentiation Differentiation...

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Unformatted text preview: CHAPTER 5 Differentiation Differentiation is a technique that enables us to find out how a function changes when its argument changes. It is an essential tool in economics. If you have done A-level maths, much of the maths in this chapter will be revision, but the economic applications may be new. We use the first and second derivatives to work out the shapes of simple functions, and find maximum and minimum points . These techniques are applied to cost, production and consumption functions . Concave and convex functions are important in economics. ./ 1. What is a Derivative? 1.1. Why We Are Interested in the Gradient of a Function In many economic applications we want to know how a function changes when its argument changes so we need to know its gradient. This graph shows the production function Y ( L ) of a firm: if it employs L workers it produces Y units of output. gradient = Y L = 200 10 = 20 The gradient represents the marginal product of labour- the firm produces 20 more units of out- put for each extra worker it employs. The steeper the production function, the greater is the marginal product of labour. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L Y Y ( L ) L Y 10 20 30 40 50 200 400 600 800 With this production function, the gradient changes as we move along the graph. The gradient of a curve at a particular point is the gradient of the tangent . The gradient, and hence the marginal product of labour, is higher when the firm employs 10 workers than when it employs 40 workers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L Y Y ( L ) 10 20 30 40 50 200 400 600 800 77 78 5. DIFFERENTIATION 1.2. Finding the Gradient of a Function If we have a linear function: y ( x ) = mx + c we know immediately that its graph is a straight line, with gradient equal to m (Chapter 2). But the graph of the function: y ( x ) = x 2 4 is a curve, so its gradient changes. To find the gradient at a particular point, we could try to draw the graph accurately, draw a tangent, and measure its gradient. But we are unlikely to get an accurate answer. Alternatively: Examples 1.1 : For the function y ( x ) = x 2 4 (i) What is the gradient at the point where x = 2?...
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Differentiation - CHAPTER 5 Differentiation Differentiation...

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