Tools for Mathematical Modeling

Transcription

1 Tools for Mathematical Modeling Dan Flath November 3, Modeling The world is a complicated place. There are so many more details than humans can assimilate that you might wonder how we can function at all. It is a danger, conveyed in the phrase lost in the details and in the old saying that you can t see the forest for the trees. And yet we do function quite well. We navigate through the details of our world by making models, by creating artificial forests from tiny selections of the trees that we see. A model is a useful description of a phenomenon made for a purpose. The purpose can be to aid understanding, or to help with decision making. The most classic model may be a road map. The lines on the map do not look at all like actual streets and the map does not show any of the trees or cars on the roads or houses and buildings, but it does contain all the information we need to drive from point A to point B. Isaac Newton s Roy G Biv description of the rainbow as a collection of seven colors, Red, Orange, Yellow, Green, Blue, Indigo and Violet is a model. In reality, there is an infinite number of color shades in the rainbow with no sharp boundaries between them. And yet, the model tells us that all rainbows contain the same colors in the same order and helps us to remember their general appearance. Another scientist (or even you) might make a model rainbow with a list of more or fewer or different colors, and that could be a perfectly fine model, as good as Newton s. A city may decide whether to build an expensive new bridge after making a model called a Cost-Benefit analysis. Even a simple rule-of-thumb is a type of model. Humans are model builders. Applied mathematicians and statisticians build mathematical models to make quantitative sense of the world. The United Nations projection of world population for the rest of the 21st century is based on a mathematical model. Several large teams of scientists have produced (different) models of global climate in an effort to understand observed climate change. In this document we will survey a few of the 21st Century Modeling Tools of the Trade: Dimensions and units 1

2 Energy and power Quantifying change Functions Derivatives Integrals 2 Dimensions and Units Driving down the road your speed can be 50 miles per hour or 50 kilometers per hour, but it is not just 50. Similarly, your height might be 6 feet or it might be 180 centimeters, but it is not simply 6 or 180. The numbers that describe the world we live in have units of measurement (units) attached to them that we must keep track of. Failure to do this properly can be costly. For example, on September 23, 1999 NASA lost a $125 million dollar Mars Climate Orbiter due to lack of clarity about whether calculations were done with metric system or English system units. Units of measurement themselves are grouped into classes, called dimensions. Roughly speaking, the dimension describes what type of quantity is measured and a unit provides a scale. A few frequently occurring dimensions, with some of their associated units, are shown in the table; Dimension Length Time Mass Velocity Area Energy Power Pure number Units meter, km, cm, foot, inch, mile second, minute, hour, year kilogram, gram meter/second, km/hour, miles per hour (mph) meter 2, sq inch, hectare, acre kilowatt-hour (kwh), joule, calorie watt, kilowatt (kw), horsepower one, dozen, thousand, million, percent (%), π The pure number dimension may seem like a special case, but it really is not. It differs from the others only in that a pure number can be shown without an explicit unit, in which case the unit is assumed to be one. Thus we can write: 1 dozen = 12 ones = 12 and percent = % = Pure numbers are used for counting. If counting people, for example, we can define the pure number unit, person : 1 person = 1. A quantity of a given dimension can be expressed using any of its units. For example: 2 hours = 120 minutes. The ratio of equal quantities is the pure number 1, so we can write 2 hour/(120 minute) = 1. Changing from one unit to another within a dimension is called unit conversion. The best way to do this is by multiplying the quantity expressed in one unit by fractions equal to 1, which does not change the value of the product. 2

3 The units themselves can be treated algebraically, canceling when the same unit appears once in the numerator and once in the denominator. For example, you can change 10 meters per second to kilometers per hour as follows: 10 m s = 10 m s 1 1 = 10 m s 1 km 1000 m 3600 s 1 hour = 36 km hour. Because they may be less familiar but are of vital importance in public policy, we will devote a separate section to energy and power and their units. We will not discuss any of the electrical dimensions (current, charge, voltage). You can have some fun browsing thousands of units in the Dictionary of Units of Measurement: 3 Energy and Power Mankind s ever increasing consumption of energy is perhaps the greatest issue facing us today. Most of the energy we now use comes from fossil fuels, and that seems to be behind a very frightening climate change. The search is on for alternative energy sources, including wind, solar and nuclear. To understand this issue, you must form a clear idea of the difference between energy and power and become comfortable with their standard scientific units. For example, when driving a car, think of the energy as the quantity of gas you use, a number of gallons. think of power as the rate at which you use the gas, a number of gallons per hour. You buy and pay for energy. You use up the energy faster or slower depending on the power. Energy and power are related by the equations Power = Energy Time, Energy = Power Time. For example, if your power is 3 gallons of gas per hour and you drive for 5 hours, then the energy you use is 3 gallons/hour 5 hours or 15 gallons. It would be possible to convert all energies into gallons-of-gas equivalents, but there is an internationally agreed-upon scale of units that is preferred. Power: measured in watts, or kilowatts, or megawatts,... Note that a watt is a rate of energy usage, not an amount of energy. (To be precise, 1 watt equals 1 joule of energy per second.) It is WRONG to think that power is watts per second or watts per hour. The per unit time is already built into the watt itself. 3

4 Energy: measured in watt-hours, kilowatt-hours, megawatt-hours,... By definition, 1 watt hour is the amount of energy consumed if you use up energy at a rate of 1 watt for an hour. Your monthly electric bill reports energy usage in kilowatt hours, which is what you pay for. Knowing the names of energy and power units is not really enough. You also want to get some idea of how big they are by relating them to things you are familiar with. For example, average human power expenditure is about 120 watts. In a day our energy requirement is about Energy = Power Time = 120 watt 24 hour = 2880 watt-hours = 2.9 kilowatt-hours. The 2.9 kwh of energy you burn is contained in the food you eat, approximately 2500 food calories, another unit of energy. (One watt-hour is about 0.86 food calories, so you can be comfortable thinking of a watt-hour as just a little less than a food calorie.) One megawatt would power about 400 US houses. 4 Quantifying Change Life is interesting and challenging because we are surrounded by change. The population of our city increases or decreases. Our incomes and the cost of living do not remain constant. Average global temperature is rising and bringing sea level up with it. If we are to plan for the future we must measure the changes we face. For example, the city manager of a growing city needs to know how many new roads and schools will be required during the next few years and to figure out how to pay for them. Suppose the population of a city increases by 2000 people during one year. Is that a big change or a small change? It all depends! For a town of 4000 people an influx of 2000 might seem overwhelming; it s a 50 percent increase. For a city of 4 million, 2000 new residents might be barely noticed; it s only a 0.05 percent increase. On the other hand, 2000 new people need the same amount of housing regardless of the city size. We measure the change in a quantity two ways: 1. Absolute change, the actual change 2. Percent change Percent change is also called relative change. It evaluates change taking into account the initial size of a quantity. It is useful for comparing changes occurring at different scales where it makes little sense to compare absolute changes, such as population changes in big and small cities or price changes of expensive and inexpensive items. 4

5 To plan several years into the future requires making some (educated) guesses about future changes. Two simple models assume constant change. 1. Linear growth Assumes that the absolute change is the same every year. 2. Exponential growth (also called geometric growth) Assumes that the percent change is the same every year. To get a feel for the difference between linear and exponential models consider a city of 100,000 people that increases by 2,000 people, 2%, in its first year. The table shows the model projections for future years. Year Population linear Population exponential Year Population linear Population exponential In the linear growth model, the population increases by 2000 every year forever, and by 10,000 every 5 years, forever. In the exponential growth model,, the population increases more and more every year, because this year s new residents contribute to the percentage increase next year. The difference between the linear and exponential models may seem small in the early years, but over time grows to be vast. Whatever model you design, linear, exponential, or other, always remember that it is only a model and is no guarantee for the future. Predicting the exact population of a city one year in advance is impossible. Getting close twenty years in advance would be amazing. The farther into the future you project, the less confident you should be. So many things can happen that there may well be no value at all in even the most sophisticated projections 35 years in advance. 5 Functions The world is complex because things are linked. Things depend on things that depend on other things, often in complicated ways. Everything depends on something else. The temperature outside your home on a given day depends on the time of day. The tax revenue for your city depends on its tax rate. 5

6 The number of sales a company makes depends on the quantity of advertising bought. Global average temperature depends on the quantity of greenhouse gasses in the atmosphere. Much of contemporary applied mathematics is organized around the concept of dependency, which is formalized in the concept of a function. In a functional relationship, there are outputs which depend on inputs, both of which have units attached. The relation can be displayed in a spreadsheet, called a table of values, or a table of data. For example, the annual CO 2 savings from producing solar energy in Germany depends on the year, and has been increasing. The year is the input, and the CO 2 savings is the output. Year CO 2 saved (millions of tons) Function notation assigns names to the input and output variables and to the function itself. In our CO 2 example, we might let t represent the year, s represent the CO 2 savings (in millions of tons), and h represent the function. We write s = h(t) and say that s is a function h of t. From the table we see, for example, that h(2012) = 18.9 million tons. As in our example, evaluating a real world function to create a table of values often amounts to data collection and may involve scientific research. It can be very expensive. It is usually necessary to be satisfied with approximate values. Humans do not easily assimilate lists of numbers, so visual ways to represent functions have been devised. The graph of a function is created by plotting the inputs on a horizontal axis and the outputs on a vertical axes. Doing this for the CO 2 data, as in Figure 1, makes the trend toward increased carbon dioxide savings clearly visible. The same function could also be displayed with a bar graph. Some functions can be given by formulas. For example, the function y = f(x) = 10x + 5 has a straight line graph and y = g(x) = x 2 has a graph that is a parabola. It is so easy to work with formulas that messy real world functions are often modeled by formulas that give useful approximations. In fact, finding such formulas is one of the jobs of applied mathematicians and statisticians. But you must always bear in mind that formulas are just models, not the real world itself, so they must be used with care. More complex situations require description with multivariable functions, where the output depends on multiple inputs. 6 Derivatives The derivative is the first tool from calculus most students encounter. It answers a specific type of question. Here are three examples. 6

7 CO 2 Saved Using Solar Power in Germany Millions of tons of CO Year Figure 1: Using solar power keeps carbon dioxide out of the atmosphere. You are driving around 50 mph. How much would driving 3 miles per hour faster increase the distance you need to bring your car to a complete stop? You have been selling gas at $2.50 per gallon. If you increase the price by $0.02 per gallon, how much less gas would you sell in a week? If one horizontal wooden beam is 1 cm thicker than another, how much more weight can it support? All three questions have the same form: when you make a small change in one variable, how much does a related variable change? In the language of functions, the question can be formulated as follows: Basic Question: You are given a function y = f(x), so that the value of variable y is determined by the value of x. When you make small changes in the x value, how big (approximately) are the resulting changes in the y value? There s a simple answer to the Basic Question in the case where all x values are near a single value x = a. If the changes in x are small enough, then the changes in y are approximately proportional to the changes in x. Written as an equation, the most important equation in differential calculus, we have The Multiplier Equation If y is a function of x, and x is near a, there is a constant m, called the derivative, such that Change in y m Small change in x. The derivative, m, is a number with units. It s a proportionality constant relating small changes in x to changes in y. You can think of the derivative as a sort of conversion factor that converts small changes in one variable to approximate changes in another variable. The standard notation for the derivative, m, of a function y of x is: m = dy/dx. 7

8 Example 1. The fuel efficiency, f, of your car in miles per gallon (mpg) depends on your speed, s, in miles per hour (mph). At 55 mph we have m = df/dv = 0.54 mpg/mph. About how much will your fuel efficiency change if you speed up from 55 mph to 57 mph? Solution. Using the multiplier equation we have Change in mpg m Change in mph = 0.54 mpg 2 mph = 1.08 mpg. mph Speeding up the two miles per hour reduces your fuel efficiency by about 1.08 miles per gallon. Notice that the derivative carries no information about the actual fuel efficiency, only about the amount of change in the efficiency. Always remember that the change in x must be small for the multiplier equation to give a good approximation for the change in y. The change is small enough if it takes place on a part of the graph of y vs x that is approximately straight. 7 Integrals Some questions can in principle be solved by dividing them into smaller questions, answering the small questions, and adding the results. For example: The population of New York state is the sum of the population of each of its counties. The number of cars on a 100 mile stretch of road is the sum of the number of cars in each 1/4 mile segment. The volume of oil spilled by a leaking oil tanker during a day is the sum of the volume leaked during the 48 half hour periods. An integral is a tool designed to answer this type of question. Let s illustrate with two simple examples. Example 1. On a trip you drove 30 miles/hour for 1/2 hour, then you drove 60 mile/hour for two hours, and then drove 40 miles/hour for 1 hour. How far did you drive? Distance = Distance in timel 1 + Distance in time 2 + Distance in time 3 = 30 mile hour 1 mile mile hour hour hour hour 1 hour = 15 mile mile + 40 mile = 175 miles. Using the variable t for time, we write dt for differential time, which in this example represents the lengths of the three time subintervals. We let s 8

9 3 g/cm 2 5 cm 4 g/cm 2 2 g/cm 2 5 cm 5 cm 5 cm Figure 2: A 10 cm 10 cm block made of materials with three different densities. represent the speed you are driving, and we were given an s value for each of the subintervals. The distance we want is the sum of three terms of the form s dt. Integral notation is a shorthand notation to describe the process, where the integral sign stands for sum. We write Distance = Sum of s dt = s dt = 175 miles. Example 2. We want to know the mass of the 10 cm 10 cm block shown in Figure 2. Since we know the area and the density of each region, we compute Mass = Mass of region 1 + Mass of region 2 + Mass of region 3 = g 4 cm 2 50 g cm2 + 3 cm 2 25 g cm cm2 cm2 = 200 gram + 75 gram + 50 gram = 325 grams. Writing A for area, da for the differential area (the areas of the subregions), and ρ for the density, we can write Mass = ρ da = 325 grams. In the examples you see that an integral is a sum of products. To define and evaluate an integral you need three things. 1. A way to break the big question into small questions. It may be helpful to think of breaking some region implicit in the problem into subregions. There is one term in the integral sum for each subregion. In Example 1 we subdivided the 3.5 hour time of the trip into three time subintervals. In Example 2 we subdivided the area of the block. 9

10 2. A measure of the size of each subregion. This contributes the differential factor to each term. In Example 1 this is the length of each time subinterval, in hours. Example 2 it is the area in square centimeters. 3. A density value for each subregion. This was miles per hour in Example 1 and grams per square centimeter in Example 2. It is a measure of how thickly or thinly the quantity you want to calculate (distance traveled or mass) is spread over the subregion. Part of the point of breaking the big problem into smaller problems is that the density probably varies less within a small subregion than within a large one so a single value is more representative of the whole region. If the density were constant, not varying at all over the entire region, there would be no reason to subdivide. Even the most complex integrals are constructed in just the simple way shown above. It can be hard, expensive work to collect the necessary differential size and density data, and as always in the real world you must be satisfied with approximations. In 10