CLASS NOTES: BUSINESS CALCULUS

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1 CLASS NOTES: BUSINESS CALCULUS These notes can be thought of as the logical skeleton of my lectures, although they will generally contain a fuller exposition of concepts but fewer examples than my lectures. Since we will be following the book closely in this class all of the sections of these notes will correspond to sections in the book (not lecture days). The style is more conceptual than the textbook (or my lectures), especially in the beginning, so you will have to read these notes more slowly in order to understand them well. However, for the kind of student who wants to know what is really going on, these notes give the precise logical picture as directly as possible, and can be thought of as a supplement to the more examples-driven textbook. Moreover, at the end of each section I give a brief Takeaway which tells you what I think is really important about what was covered in the section, and this should be valuable to everyone when studying for quizzes and exams. 1. Functions and Change The point of this Chapter is to discuss the mathematical concept of a function and its relation to real world quantities. In later Chapters we will use the tools of Calculus to analyze functions more closely What is a function? Here is the definition of a function of single variable: Definition 1.1. Suppose A and B are collections of numbers. A function f from A to B is a correspondence which assigns one element of B to each element of A. If f assigns the number b in B to the number a in A, we write f(a) = b. The set A is called the domain of f, and the set B is called the range of f. This definition is deliberately abstract because it is meant to capture a wide range of phenomena. For a more concrete understanding of the concept, you can think of a function f as a kind of machine which takes elements of the set A as input and outputs elements of the set B. In mathematics, the most common and powerful ways of describing functions are by means of algebraic formulas and graphs. In the following example, the symbol R stands for the collection of all real numbers. 1 Example 1.2. The equation f(x) = 2x 1 defines a function f : R R, because it tells us what f outputs for every number x. For example, f(1) = = 1, so f outputs 1 when given 1, f(0) = = 1, so f outputs 1 when given 0, 1 Recall that a number is said to be real if it can be expressed by a (possibly infinite) decimal expression, which includes every kind of number that you will encounter in real life. For example, 0, 1, π = , and 3 4 =.75 all count as real numbers. The only numbers that are not in the set R are those involving the imaginary number i = 1 (and you need not worry about these for now). 1

2 CLASS NOTES: BUSINESS CALCULUS 2 f(4) = = 7, so f outputs 7 when given 4. When we define a function f(x) using an algebraic formula, we usually take the domain to be the set of all numbers x for which the algebraic formula is defined. For example, The function defined by the formula f(x) = 2x 1 is defined for every real number, so its domain is the set R. The function defined by the formula f(x) = x2 +1 makes sense for all real numbers x 1 except x = 1, because division by 0 is undefined. Therefore the domain of f is the set of all real numbers except 1. This set is symbolically denoted R \ {1}, 2 or in interval notation as (, 1) (1, ). 3 The function defined by the formula f(x) = x 1 makes sense for all real numbers x 1, but it does not make sense when x < 1 because you cannot take the square root of a negative number. Therefore the domain is the set of all real numbers x 1, which in interval notation is denoted by [1, ). It is important to remember that an algebraic formula is just a way of defining a function, but is not the same thing as the function itself (which is just the abstract correspondence). For example, the function defined by the formula f(x) = 2x 1 can also be defined using the graph of Figure 1 below. Definition 1.3. The graph of a function f : A B is the set of all points of the form (x, f(x)) in the Cartesian plane, where x is a number in the set A. More concretely, the graph of a function is built by placing a point at height y = f(x) above each point x on the x-axis. Since there is at most one y-value f(x) above or below each value of x on the x-axis, the graph of a function must always pass the vertical line test. The vertical line test says that a curve in the Cartesian plane defines a function if and only if every vertical line in the plane meets it at most one time. The mathematical concept of a function is used to describe the way that real world quantities depend on one another. For example: The volume of a ball depends on the length of its diameter, 2 In general, {a, b, c} denotes the set containing the numbers a, b, and c, and A \ B denotes the set of all elements in A which are not elements of B, therefore R \ {1} denotes the set of all real numbers except for the number 1. 3 Recall how interval notation works: The set of all numbers x such that a < x < b is denoted by (a, b), the set of all numbers x such that a x < b is denoted by [a, b), the set of x such that a x b is denoted by [a, b], and so on. In other words, a square bracket next to an endpoint indicates that the endpoint is included in the interval, whereas a round bracket indicates that every number up to the endpoint, but not the endpoint itself, is included in the interval. We denote the set of all x < a by (, a) and the set of all x a by [a, ). The idea of this notation is that the set contains all x up to, but not including, ±, which is why the brackets next to infinity are always round. With interval notation, the smaller number must always be to the left of the larger number just as it appears on the number line, so, for example, the notation (3, 1) makes no sense, it must be written as ( 1, 3). Finally, given any sets A and B, A B denotes the union of the two sets A and B, which contains every element occuring in either set.

3 CLASS NOTES: BUSINESS CALCULUS 3 Figure 1. The graph of the function f : R R defined by the expression f(x) = 2x 1. The energy with which a car collides with a wall depends on its speed and mass at the time of collision, Your height depends on your age. All of these dependencies can be described by functions (sometimes involving more than one variable). In some cases, the function can be defined algebraically. Thus, the volume V (in square inches) of a ball of diameter D (in inches) can be described by the formula V = f(d) = π 6 D3. Similarly, the kinetic energy K of the car (or any object) is given in terms of its mass m and velocity v by the formula K = f(m, v) = 1 2 mv2. Your height is likely difficult to define algebraically as a function of your age, but it is nevertheless a function: at any given time t after your birth, you will have exactly one height h. Many real world quantities which depend on one another do not have perfectly accurate algebraic formulations. To describe such real world functions we resort to tables or ordinary language, and then try to find algebraic formulas or graphs which fit the information as best as possible. In all cases, the key to applying mathematics to the real world is to describe, as best as possible, real world functions by means of algebraic expressions and graphs. These mathematical descriptions are called mathematical models of the reality. You can then use powerful techniques to analyze the mathematical models to make predictions that are far deeper and more precise than you could make using every day reasoning. Of course, the predictions will only come true if the mathematical model really does match the reality, so the process of building a good model is just as important as the mathematics you use to analyze it.

4 CLASS NOTES: BUSINESS CALCULUS TAKEAWAY The mathematical concept of a function is useful for modeling the relationship between quantities in the real world which depend on one another. The primary mathematical ways of describing them are by means of algebraic expressions and graphs, the primary ways in which we describe functions in the real world is by means of tables (data collection) and ordinary language. The homework problems of Section 1.1 are meant to give you practice in passing between the various ways of representing a function. This skill will prove to be essential not only to understand the mathematical analysis involved in calculus, but also in being able to apply mathematics to real world problems Linear Functions. Definition 1.4. A linear function is any function which can be described by an algebraic formula of the form f(x) = mx + b, where m and b are constants. The number m is called the slope of the function, and the number b is called the y-intercept of the function. Linear functions are called linear functions because their graphs are straight lines in the plane. The constant b is called the y-intercept because the line crosses the y axis at height y = b. The constant m is called the slope because it tells you how much the graph of the function moves vertically as you move one unit to the right. Equivalently, if m is a fraction p, then this tells us that m rises p units for every q units you move to the q right. The fact that linear functions have a constant slope (i.e. always rise or fall at the same rate as you move right) is a defining property. Wavy graphs rise and fall at different rates as you move right. Example 1.5. The following are all linear functions, together with their graphs.

5 CLASS NOTES: BUSINESS CALCULUS 5 Given the equation of a line in standard form, you should be able to draw its graph easily. Going the other direction, every line in the plane which is not vertical 4 is the graph of a function in standard form y = mx + b. If the line passes through two points (x 0, y 0 ) and (x, y), then the rise of the line between x 0 and x is y y 0, and the run of line is given by x x 0 (see Figure 2). Hence the slope of the line is y y 0 x x 0 = m. Using this formula, we can find the slope of a line passing through two given points (x, y) and (x 0, y 0 ). Moreover, if we multiply both sides of the slope formula by (x x 0 ), we obtain the point-slope formula y y 0 = m(x x 0 ). In this formula we treat x and y as variables, and can use it to find the equation of the line if we are given its slope m and one point (x 0, y 0 ) which the line passes through. Figure 2. Rise and run in the slope formula. Example 1.6. To find the standard equation y = mx + b of the line passing through the points (1, 2) and (3, 0), we start by finding its slope using the slope formula: m = = 1. Now we can plug m and either of the two points above into the point-slope formula, in this case we ll use (x 0, y 0 ) = (1, 2) to get: y 2 = 1(x 1). We re not quite done, because we still need to simplify this down to standard form. Using basic algebra we get y = x Vertical lines obviously fail the vertical line test, all other lines pass the vertical line test.

6 CLASS NOTES: BUSINESS CALCULUS 6 There are other methods of finding the equation of a line which passes through two points, but the method of the example above is the best one for our purposes because it reinforces your understanding of the concept of slope, which will be central to us when it comes time to understand calculus. As I mentioned above, the defining characteristic of a linear function is that its slope is constant. When there is not a graph present, it is easier to think of the rise between a pair of points (x 0, f(x 0 )) and (x 1, f(x 1 )) 5 as simply the change in y = f(x), given by = f(x 1 ) f(x 0 ). Similarly, the run is just the change in x, given by x = x 1 x 0. In these terms, the slope of f(x) between x 1 and x 0 is defined by the ratio. If this ratio x is the roughly the same for every pair of values (x 0, f(x 0 )) and (x 1, f(x 1 )) occuring in a given table of data, we can infer that the real world function is roughly linear. Moreover, we can even model the real world data with an algebraic expression. Example 1.7. The following tables of values describes the distance d (in miles) that a traveler is from home t hours after leaving his house. d t In this table, distance d is described as a function of time t. In the top table, the change in distance between time t = 1 and t = 2 is given by d = = 40 miles, and the change in time is t = 2 1 = 1 hour. Thus we have a slope of 40 = 40 over this time 1 period. On the other hand, between time t = 2 and t = 4, the change in distance is given by d = = 30 miles, and the change in time is t = 4 2 = 2 hours. Hence the slope of the distance function between time t = 2 and t = 4 is only 15. Since the slope between times t = 1 and t = 2 does not match the slope between t = 2 and t = 4, the function f(t) = d is not linear. Now suppose the traveler s distance from home is described by the following table: d t In this case, it is easy to verify that the slope of the distance function is 30 between every pair of t-values. This real life function therefore appears to be linear, and can be modeled by a function of the form d = mt + b. 6 We know that the slope of this function is m = 30, and using the point-slope formula with the point (t 0, d 0 ) = (1, 20), we deduce that distance function has the form d 20 = 30(t 1), which can be re-rewritten in the standard form d = 30t Remember that, in the graph of a function, f(x) = y for all x, by definition. Therefore, setting f(x 0 ) = y 0 and f(x 1 ) = y 1, we can think of f(x 1 ) f(x 0 ) = y 1 y 0 as the change in y, denoted y. I will often switch between these two equivalent ways of formulating things. 6 here d is playing the role of y and t is playing the role of x. Likewise t 0 will be playing the role of x 0 and d 0 the role of y 0 in the point-slope formula.

7 CLASS NOTES: BUSINESS CALCULUS 7 Using common sense, we see there must be something wrong with the algebraic formula d = 30t 10 we have used to model the data in the table, because at time t = 0 it predicts that the traveler will be a distance of d = = 10 miles away from home, which makes no sense whatsoever. Moreover, common sense tells us that the driver must stop driving some time, so our linear model d = 30t 10 cannot stay valid forever. However, our model appears to be valid between time t = 1 and t = 5, so we can predict with some confidence that the driver was about d = = 35 miles away from home 1.5 miles after leaving the house. Of course, our model could be way off, and the only way to figure out whether or not it really is correct (or close to correct) is to find out more about the individual driver and his habits. Does he avoid using the bathroom and like using cruise-control? Then the model is more likely to be right. Does he like to drive like a maniac and needs to stop and pee every 30 minutes? Then the model is likely incorrect. Purely mathematical analysis only tells us what must be true if our model is correct, it takes logic and common sense to determine whether the model is correct in the first place. Keep this in mind at all times, especially if you become an expert someday. 1.2 TAKEAWAY Linear functions can be written in the standard form y = f(x) = mx + b, and given a function written in this form you should be able to quickly draw its graph. Conversely, given the graph of a line, or even just two points on the line, you should be able to find the standard form of the line using the slope formula and the point-slope formula. Moreover, using the fact that linear functions have constant slope, you should be able to identify whether or not a given table of data is (possibly) describing a linear function, and if it is, you should be able to find an explicit algebraic formula which models it. I expect that this section is largely just review for most of you, but you should make sure you understand every part of it extremely well, because your understanding of the geometric and algebraic properties of lines will play an essential role in your understanding of the derivative Average Rate of Change and Relative Change. The concepts of this section are easiest to understand if we think of our functions f as functions of time, so for the most part I will treat functions here as functions of time. However, the same concepts apply to any kind of function. The change in a function between time t 1 and t 2 (where t 2 is bigger than t 1 ) is = f(t 2 ) f(t 1 ). The order in which f(t 2 ) and f(t 1 ) appears does matter here, because > 0 means that f has increased between time t 1 and t 2, whereas < 0 means it decreased. Definition 1.8. The average rate of change of f between time t 1 and t 2 is given by the formula t = f(t 2) f(t 1 ). t 2 t 1 Example 1.9. So, for example, if t is the time, in hours, since you left your home, and f(t) = t 2 + 2t is your distance, in miles, from home at time t, then your average rate of

8 CLASS NOTES: BUSINESS CALCULUS 8 Figure 3. A secant line. change between t 1 = 1 and t 2 = 3 is given by t = f(3) f(1) 3 1 = ( ) ( ) 2 Your average rate of change between t 1 = 1 and t 2 = 4 is t = f(4) f(1) 4 1 = ( ) ( ) 3 Your average rate of change between t 1 = 2 and t 2 = 5 is given by t = f(5) f(2) 5 2 = ( ) ( ) 3 = 6mph. = 7mph. = 9mph. Observe that if f(t) is a linear function of the form f(t) = mt + b, then the average rate of change of f is the same between any two points t 1 and t 2, it just corresponds to the slope of f(t): t = f(t 2) f(t 1 ) = mt 2 + b (mt 1 + b) = m(t 2 t 1 ) = m. t 2 t 2 t 2 t 1 t 2 t 1 This is not true for any other kind of function. We now switch to the graphical perspective. The secant line connecting two points (t 1, f(t 1 )) (t 2, f(t 2 )) on the graph of f(t) is just the line which passes through (t 1, f(t 1 )) and (t 2, f(t 2 )) (see Figure 3). Proposition The average rate of change of f(t) between t 1 and t 2 is equal to the slope of the secant line connecting (t 1, f(t 1 )) to (t 2, f(t 2 )). Given the graph of f(t), you should be able to use this fact to roughly eyeball the average rate of change of f(t) between t 1 and t 2. You should also be able to eyeball whether or not a function is concave up or concave down over a region. Definition A function f(t) is concave up over the interval [t 1, t 2 ] if the slope of the secant line connecting (t 1, f(t 1 )) to (t, f(t)) increases as you move right from t = t 1 to t 2. It is concave down if the slope of the secant line connecting (t 1, f(t 1 )) to (t, f(t)) decreases as you move right from t = t 1 to t = t 2.

9 CLASS NOTES: BUSINESS CALCULUS 9 In other words, f(t) is concave up between t 1 and t 2 if its graph looks like part of a smile, and it is concave down if it looks like part of a frown (notice that f(t) itself can be increasing or decreasing over the interval). So, for example, the graph of Figure 3 appears to be concave up between t = 10 and t = 0, and concave down between t = 0 and t = 10. If the graph of f(t) is concave up between t 1 and t 2, this means that the average rate of change in f is increasing as time moves forward. In other words, f is accelerating between t 1 and t 2. On the other hand, if f(t) is concave down between t 1 and t 2, then f(t) is decelerating. That s a big part of the reason why we care about concavity. The last concept of importance in this section is the concept of relative change. This, in fact, really doesn t involve time at all. It merely tells us how large the change of a quantity is as a proportion of the original quantity. Definition The relative change in f(t) between t 1 and t 2 is given by f(t 1 ) = f(t 2) f(t 1 ). f(t 1 ) Notice the length of the time interval [t 1, t 2 ] does not enter in here. In some cases, the relative change in f is more important to us than the absolute change. For example, if someone says that they think the value of one share of ACME stock will increase by 1 dollar next year, that could be good or bad. If ACME stock already costs $1,000 a share, then the relative change in value is only =.001, 1000 which amounts to an increase of only.1%, in which case you might be better off just keeping your money in a savings account. On the other hand, if ACME stock is only priced at $2 a share, then the relative change will be 3 2 =.5, 2 which is a 50% increase, which is a California real estate bubble level of growth. 1.3 TAKEAWAY You must understand the real world meaning of average rate of change. You must also know the algebraic formula for it by heart and the graphical interpretation as the slope of the secant line. In particular, for homework purposes you should be able to compute average rates of change using algebraic formulas, graphs, or tables. In addition, you must be able to identify the concavity of a graph by looking at it, and understand how it corresponds to acceleration or deceleration. Finally, you must understand the real world meaning (and usefulness) of relative change, and you must know its formula by heart Applications of functions to economics. Pretend you are a business owner who produces (or in some way supplies) objects to people in exchange for money. Here are some of the relevant functions you want to understand when deciding on just how much of your product x you wish to supply. Our unit of currency will be dollars throughout. Definition The cost function C(x) gives the cost of producing x objects. The revenue function R(x) gives the total amount of money people will pay you for x objects.

10 CLASS NOTES: BUSINESS CALCULUS 10 The profit function P r(x) (or π(x), according to the book), is the total profit you make from selling x objects, and is given by the formula P r(x) = R(x) C(x). First let s analyze the cost function. Typically the cost function is made up of two parts: fixed costs, which you will have to pay no matter how many objects x that you produce, and variable costs, which are directly related to how many objects x that you produce. So, for example, if your business is a bakery then your fixed costs will include things like rent, phone bills, etc. Your variable costs will include the cost of raw materials for making cakes, as well as the salary you pay to your employees to spend time baking them. Fixed costs will appear as the constant in your cost function C(x), whereas your variable costs will make up the rest of C(x). Example Suppose C(x) = x + x 2. Then your fixed cost is $100, and the variable cost of producing x items is 5x + x 2. The fixed cost will always be equal to C(0), the cost of producing no objects. So, in particular, you can always find the fixed cost part of C(x) from its graph by looking at its y-intercept. Notice also that C(x) should always be increasing as x gets larger, because the cost of making x+1 objects should always be more than the cost of making x objects. Now let s analyze revenue. If the price p of the object is fixed, then the revenue from selling x objects is easy to find, it is just R(x) = px. This formula works well enough for most situations, but be aware that in reality you might need to make a a more careful analysis. It is indeed true that if you sell x objects at a price of p, then you will make px dollars, but it is not necessarily true that you actually can sell x objects at a given price p. In general, the higher your price, the fewer objects you will sell. In other words, the price p which you must set in order to sell x objects is usually a decreasing function p(x) of x. Taking this into account, your revenue function will be R(x) = p(x) x. Definition The marginal cost at x is C(x + 1) C(x), it is the cost of producing the (x + 1) st object once you have already produced x objects. Likewise, the marginal revenue is R(x + 1) R(x) and the marginal profit is P r(x + 1) P r(x). We generally expect marginal cost to go down as x gets large, because one can often produce a large number of items more efficiently than a small number (although this need not always be true, especially if a somewhat rare resource is required to make the objects). We likewise expect marginal revenue to decrease as x increases. Finally we discuss the concept of supply and demand. Definition The supply function S(q) outputs the price required to get suppliers to sell q items. The demand function D(q) outputs the price required for consumers to purchase q items. In real life we expect S(q) to be an increasing function and D(q) to be a decreasing function (although there can be odd exceptions here and there). Under this assumption S(q) and D(q) will take on the same value for only one quantity q.

11 CLASS NOTES: BUSINESS CALCULUS 11 Definition The equilibrium price is the price p at which suppliers are willing to sell the same number of items as buyers are willing buy. Keep in mind that price p is the output of the supply and demand functions, so the equilibrium price is the value of p = S(q) = D(q) for the quantity q where S(q) = D(q). Theoretically, the equilibrium price is going to be the actual price of the object that we see in real life. Example Suppose that the supply curve for q tons of sand is given by S(q) = q 2, and the demand curve for sand is given by D(q) = To find the equilibrium price we q set S(q) = D(q), solve for q to find the equilibrium quantity, then plug q back into S(q) (or D(q)) to find the equilibrium price. So to begin with we have 1000 = q 2. q Multiplying both sides by q gives 1000 = q 3. So the equilibrium occurs at q = 10 tons of sand, and the equilibrium price is S(10) = 100 dollars per ton. Now suppose a $10 tax per ton of sand is placed on widget suppliers. Since their other underlying costs are the same, this implies that the market price of widgets must by $10 greater per ton to entice suppliers to produce the same amount of sand. Hence the new supply curve becomes S taxed (q) = S(q) + 10 = q To find the new equilibrium we set S taxed (q) = D(q) as before and solve for q. This is difficult to do (don t even bother trying without a computer), but we get q 9.6 tons of sand. The new equilibrium price becomes S taxed (9.6) = dollars per ton. Hence, the total price increase passed on to the consumer is $2.16 per ton, and.4 fewer tons of sand are produced for everyone. The government makes a total of = 96 dollars on the tax. 1.4 TAKEAWAY You need to understand the relationship between the cost function, the revenue function, and the profit function. You also need to know marginal cost. Finally, you need to understand the supply and demand functions, and be able to use them to find equilibrium prices. You should also know how to determine the exact effect of a tax on equilibrium price and overall supply. address: trent.schirmer@okstate.edu