Chapter 14Optimization and Analysis of Models1This chapter is designed to help you take you knowledge of building models to the nextlevel - applying them to solve problems involving questions about optimization. In general,optimization is the process of trying to make something as efficient as possible, or as large aspossible or as cheap as possible. It’s the study of minimizing or maximizing a quantity, likeprofit, as a function of some other quantity, like production. In order to optimize a quantity,though, we need a few things. The first is a skill you already have - the ability to create amodel equation that represents how the quantity to be optimized varies as a function of someother quantity. For example, we might produce a model equation describing how the profitsof a company depend on the number of items they produce, since the more you produce:(a) the more you can sell, generating more revenue but (b) the more it costs, in labor andmaterials. The other tool that you need is a knowledge of marginal analysis, which measureshow a change in the independent variable will cause a change in the dependent variable ina model. We will focus our study on the marginal analysis and optimization of polynomialmodels, although this is only the tip of the iceberg.• As a result of this chapter, students will learn√What marginal analysis is√How to interpret the results of marginal analysis√What the derivative of a power function is√What the derivative of a polynomial function is• As a result of this chapter, students will be able to√Compute the derivative of a power function√Compute the derivative of a polynomial√Maximize or minimize a polynomial, using both algebra and Excel1c2011 Kris H. Green and W. Allen Emerson415416 CHAPTER 14. OPTIMIZATION14.1 Calculus with Powers and PolynomialsWe have spent some time discussing the basic families of functions. These functions can beused to model the behavior of various real-world business situations. For example, supposewe have data based on the total cost of paying back a loan (for a fixed principal and fixedpayback period). We can use this data to develop a function, call it C(r), which representsthis cost as a function of different interest rates on the loan. Suppose interest rates areincreasing. How will this affect the cost of paying back the loan?This question really centers on how the function C changes as the interest rate r increases.To answer this question, we will turn to our knowledge of families of functions. In particular,we will use what we know about the parameter A in the general formula for a linear function,y = A + Bx.Figure 14.1: Slope between two points.Look at the graph of the linear function shown in figure 14.1. Also shown on the graphare two points. These points are labeled with the coordinates (x1, y1) and (x2, y2). What isthe total change in the linear function between the two points?Between these points, there is a change of y2− y1. This is just the vertical separationbetween the two points. Now, how quickly is the function changing at the first point? Thisis not a question of total change, but of the rate of change of the function. Another wayof asking this question is ”If I make a small change in x from x1to x2, how much will thefunction change?” To answer this question, we look at the slope of the line. As you mayrecall, the slope of a line can be calculated from the formulaslope = A =y2− y1x2− x1.For the function above, we see that the two points have coordinates (1, 3) and (7, 1).Thus, the slope of the line is (1 − 3)/(7 − 1) = −2/6 = −1/3. The negative tells us thatthe function (in this case a straight line) is decreasing. This means that, as we move fromleft to right, the value y of the function gets smaller. There are several nice things aboutstraight lines that we can see from this example. First, unlike nonlinear functions, the slopeof a straight line is exactly the same at every single value of x. This means that the slopeof the function at the first point is −1/3 and slope at the second point is also −1/3 and theslope at x = 249 is also −1/3. Second, it is easy to calculate the slope of a straight line. We14.1. CALCULUS WITH POWERS AND POLYNOMIALS 417simply look at the change in the values of the function (the y values) and divide this by thechange in the x values between the two points. This will not hold for any other family offunctions.To find the slope of a nonlinear function, we take advantage of a property of smoothfunctions. As illustrated in the graphs in figure 14.2, if we have the graph of a nonlinearfunction, and we zoom in on the graph, it begins to look linear.Figure 14.2: Series of graphs showing how the function changes as we zoom in on x = 1.For some functions, we need to zoom in more, and for others we zoom in less to see thislinear-like appearance. In order to calculate the slope, we will use this feature, called locallinearity, to determine the slope of a functions at any point. Specifically, if we pick twopoints on the function, and draw a line between them, we will call the slope of this line theaverage rate of change of the function. If we call these two points (x1, f(x1)) and (x2, f(x2)),then the average rate of change between the points isaverage rate of change =f(x2) − f(x1)x2− x1.Notice that the graph in figure 14.3 shows how the average rate of change can be quitedifferent from the actual rate of change (called the instantaneous rate of change or derivative).Figure 14.3: Average slope between two points.However, if we move the second point closer to the first, we can get a more accurateapproximation to the instantaneous rate of change of the function near the first point. If the418 CHAPTER 14. OPTIMIZATIONtwo points are close enough, the average rate of change will be a very good approximationto the instantaneous rate of change. This fact will help us in many cases where we only havedata, instead of an actual function.14.1.1 Definitions and FormulasQuotient A quotient is simply the result of dividing one quantity by another quantity.Average slope The average slope between two points on a function is what you get whenyou start with a function (f), evaluate it at two points (say x1and x2) and then takethe difference of these values, f (x2) −f(x1) and divide it by the distance between thetwo x-values (x2− x1). Thus,average slope =f(x2) − f(x1)x2− x1.Note that the order is important! If you start with x2first in the