TA Handout 2 What is a Derivative and How Can We Make Use of It?

Transcription

1 TA Handout What is a Derivative and How Can We Make Use of It? 1 Definition and Intuition A simple wa to think of about a derivative is as a measure of the rate of change in a function That is, given a function of the form =f(x), the first derivative tells us the slope of the function at an given x The formal definition of a derivative is: f '( x) lim x 0 x The intuition behind this definition is that we can approximate the slope of a line tangent to an given point B on a curve =f(x) b taking the slope of two points, A and C, that lie to either side of it Furthermore, we can get better and better approximations of the slope at B the closer together we move A and C At the limit, when the difference between points A and C goes awa, we have the derivative of the function at point B Figure 1 illustrates this process graphicall using the example from Chapter of Samuelson and Marks To approximate the slope at point B, where quantit equals lots, A and C do a better job than A 1 and C 1 As A and C move closer to B, our estimated slope more closel approximates the true slope for x= Figure 1 Profit ($000) 00 Profit Curve from the Sale of Microchips (SM Figure 4) B C C A1 A Quantit (Lots) Some Rules for Taking a Derivative

2 The following are several useful rules for taking simple derivatives, including the ones given in the Appendix to Chapter of Samuelson and Marks If ou are interested in handling more complex derivatives in the future (ie outside this class), a useful reference is Fundamental Methods of Mathematical Economics b Alpha Chiang a The derivative of a constant is zero That is, if =a, then: = 0 b The derivative of a constant times a variable is just the constant That is, if =ax, where a is a constant, the derivative is: = a c The derivative of a function of the form =ax n, where a and n are constants, is: = nax n 1 d The derivative of a function of the form =f(x)+g(x), is: = df dg + e (Product Rule) The derivative of a function of the form =f(x)g(x), is: = df g( x) + dg f ( x) f (Quotient Rule) The derivative of a function of the form =f(x)/g(x), is: = df g x dg ( ) f ( x ) g( x) g (Chain Rule) If z is a function of, or z=f() and in turn is a function of another variable, sa =f(x), then the derivative of z with respect to x is equal to the derivative of z with respect to times the derivative of with respect to x: dz = dz

3 h (Second Derivative) The second derivative of a function is found b taking the derivative of the first derivative of a function Thus, if =ax n, then the second derivative of with respect to x is: d n nax n = ( 1) Note that the derivative of the functions we will stu is often itself a function The derivative of the function =1+7x+15x, for example, is: = x The fact that this derivative is a function of x just means that the slope of the curve changes over the length of the curve In Figure 1, for example, the slope of the curve at point A 1 is greater than that at B, which is greater than that at C 1 The next section will exploit this fact in order to find a local maximum, but noticing this characteristic about derivatives also helps us to see wh the derivative of a straight line (like =mx+b) is constant: its slope never changes 3 Maximization and Second Derivatives Derivatives are ver useful in helping us to find the maximum of a function This is because a maximum usuall occurs as the slope of the curve goes from positive to negative As a result, to determine the maximum algebraicall, we just have to set the first derivative equal to zero and solve for x Figure Maximum: Slope Decreasing d < 0 Figure 3 Minimum: Slope Increasing > 0 x x However, as indicated in Figures and 3 above, because the minimum of a curve will also usuall have a slope of zero, we can use the second derivative to determine whether we have a minimum or maximum This test is especiall useful for functions that are hard to draw As a general rule, we can sa that if the second derivative is negative at a point where the slope is zero, it is indeed a maximum, while if the second derivative is positive, it is a minimum This rule can be summarized as follows: 3

4 If If and d = 0 < 0 and d = 0 > 0 Maximum Minimum The intuition behind this rule is that the second derivative provides a measure of the rate of change of the slope of a curve A negative second derivative means that the slope is decreasing from left to right (ie turning from positive slope to negative slope), as shown in Figure below A positive second derivative, on the other hand, means that the slope increasing from left to right (ie turning from negative to positive), as shown in Figure 3 4 The Microchip Example The microchip profit curve in Figure 1 is given b the equation: π = q 0q To find the quantit which maximizes profit, we need to set the derivative of this equation equal to zero, or: dπ = q = 0 dq From this we obtain 40q=13, or q=33 lots To make sure it is a maximum, we need to evaluate the second derivative at q=33 lots: d π = 40 < 0 dq Since the second derivative is everwhere negative, we can be sure we have a maximum 5 A Check of Your Understanding Use the following sample problems to test our understanding a Find the first and second derivatives of π with respect to q of the following expressions: 1 π = 1q q π = 15q q 3q 7 3 π = q q 4

5 b Suppose our marketing director gives ou the following information about our firm and the product it sells: 1 Consumer demand can be modeled as p=150-5q Your compan faces fixed costs of $5 and variable costs of $10 per unit produced Derive expressions for total revenue, total cost, marginal revenue, marginal cost and profit What is the profit maximizing level of output? Demonstrate that our result is actuall a maximum 5