QMI Lesson 3: Modeling with Functions
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1 QMI Lesson 3: Modeling with Functions C C Moxley Samford University Brock School of Business
2 Mathematical Models A mathematical model helps answer problems that arise from real-world situations. Creating and using mathematical models involves four general steps.
3 Mathematical Models A mathematical model helps answer problems that arise from real-world situations. Creating and using mathematical models involves four general steps. 1 Formulate: From the real-world problem, derive a mathematical statement which captures the way one (dependent) variable relies on another (independent) variable. This is perhaps the hardest step.
4 Mathematical Models A mathematical model helps answer problems that arise from real-world situations. Creating and using mathematical models involves four general steps. 1 Formulate: From the real-world problem, derive a mathematical statement which captures the way one (dependent) variable relies on another (independent) variable. This is perhaps the hardest step. 2 Solve: Use the techniques taught in this course to solve the mathematical problem arising from the real-world problem.
5 Mathematical Models A mathematical model helps answer problems that arise from real-world situations. Creating and using mathematical models involves four general steps. 1 Formulate: From the real-world problem, derive a mathematical statement which captures the way one (dependent) variable relies on another (independent) variable. This is perhaps the hardest step. 2 Solve: Use the techniques taught in this course to solve the mathematical problem arising from the real-world problem. 3 Interpret: Translate the solution into its real-world meaning.
6 Mathematical Models A mathematical model helps answer problems that arise from real-world situations. Creating and using mathematical models involves four general steps. 1 Formulate: From the real-world problem, derive a mathematical statement which captures the way one (dependent) variable relies on another (independent) variable. This is perhaps the hardest step. 2 Solve: Use the techniques taught in this course to solve the mathematical problem arising from the real-world problem. 3 Interpret: Translate the solution into its real-world meaning. 4 Test: Ensure that your real-world solution is an appropriate answer to your original real-world problem.
7 Mathematical Models Definition (Polynomial Function) A polynomial function of degree n is one which can be written in the form
8 Mathematical Models Definition (Polynomial Function) A polynomial function of degree n is one which can be written in the form f (x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 (a n 0 ).
9 Mathematical Models Definition (Polynomial Function) A polynomial function of degree n is one which can be written in the form f (x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 (a n 0 ). Here, n must be a non-negative integer.
10 Mathematical Models Definition (Polynomial Function) A polynomial function of degree n is one which can be written in the form f (x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 (a n 0 ). Here, n must be a non-negative integer. The numbers a n, a n 1,..., a 1, a 0 are called coefficients.
11 Mathematical Models Definition (Polynomial Function) A polynomial function of degree n is one which can be written in the form f (x) = a n x n + a n 1 x n a 2 x 2 + a 1 x + a 0 (a n 0 ). Here, n must be a non-negative integer. The numbers a n, a n 1,..., a 1, a 0 are called coefficients. Definition (Linear Function) A degree one polynomial is a linear function, i.e. it can be written f (x) = a 1 x + a 0 with a 1 0.
12 Example Overdraft charges are a major source of revenue for some banks. The following gives the revenue from overdraft fees in billions of dollars from 2004 to Year (t) Revenue (y)
13 Example Overdraft charges are a major source of revenue for some banks. The following gives the revenue from overdraft fees in billions of dollars from 2004 to Year (t) Revenue (y) Here, t is in years, and t = 0 corresponds to year A linear model giving the approximate projected revenue from overdraft fees is given by f (t) = 2.19t (0 t 5).
14 Example Plot the points from the previous problem. And plot the graph of f over those points.
15 Example Plot the points from the previous problem. And plot the graph of f over those points.
16 Example Plot the points from the previous problem. And plot the graph of f over those points
17 Example, Recall: f (t) = 2.19t Assuming that the trend continues, what is the projected revenue in 2012?
18 Example, Recall: f (t) = 2.19t Assuming that the trend continues, what is the projected revenue in 2012? Well, in 2012, t = 8. So f (8) = 2.19(8) = The projected revenue is $44.64 billion.
19 Example, Recall: f (t) = 2.19t Assuming that the trend continues, what is the projected revenue in 2012? Well, in 2012, t = 8. So f (8) = 2.19(8) = The projected revenue is $44.64 billion. At what rate did revenue increase from years 2004 to 2009?
20 Example, Recall: f (t) = 2.19t Assuming that the trend continues, what is the projected revenue in 2012? Well, in 2012, t = 8. So f (8) = 2.19(8) = The projected revenue is $44.64 billion. At what rate did revenue increase from years 2004 to 2009? The real rate was = 2.1, but the modeled rate was 2.19 (in billions of dollars).
21 Quadratic Functions Definition (Quadratic Function) A degree two polynomial is a quadratic (parabolic) function, i.e. it can be written in the form f (x) = a 2 x 2 + a 1 x + a 0 with a 2 0.
22 Quadratic Functions Definition (Quadratic Function) A degree two polynomial is a quadratic (parabolic) function, i.e. it can be written in the form f (x) = a 2 x 2 + a 1 x + a 0 with a 2 0. The graph of a quadratic function is a parabola which opens up if a 2 > 0 or down if a 2 < 0.
23 Rational and Power Functions Definition (Rational Function) A rational function is the quotient of two polynomial functions.
24 Rational and Power Functions Definition (Rational Function) A rational function is the quotient of two polynomial functions. Definition (Power Function) A power function is a function of the form f (x) = x r, where r is any non-zero number.
25 Rational and Power Functions Definition (Rational Function) A rational function is the quotient of two polynomial functions. Definition (Power Function) A power function is a function of the form f (x) = x r, where r is any non-zero number. Is f (x) = x2 3x x 1 +x a rational function?
26 Rational and Power Functions Definition (Rational Function) A rational function is the quotient of two polynomial functions. Definition (Power Function) A power function is a function of the form f (x) = x r, where r is any non-zero number. Is f (x) = x2 3x a rational function? Yes! x 1 +x Notice: f (x) = f (x) x x = x3 3x 2. 1+x 2
27 Rational and Power Functions Definition (Rational Function) A rational function is the quotient of two polynomial functions. Definition (Power Function) A power function is a function of the form f (x) = x r, where r is any non-zero number. Is f (x) = x2 3x a rational function? Yes! x 1 +x Notice: f (x) = f (x) x x = x3 3x 2. 1+x 2 Is f (x) = x x a power function?
28 Rational and Power Functions Definition (Rational Function) A rational function is the quotient of two polynomial functions. Definition (Power Function) A power function is a function of the form f (x) = x r, where r is any non-zero number. Is f (x) = x2 3x a rational function? Yes! x 1 +x Notice: f (x) = f (x) x x = x3 3x 2. 1+x 2 Is f (x) = x x a power function? No!
29 Supply Definition (Supply Equation) This equation expresses the relationship between unit price and the quantity supplied. The graph of this equation is called the supply curve. Often, the independent variable in the supply function is the quantity supplied.
30 Supply Definition (Supply Equation) This equation expresses the relationship between unit price and the quantity supplied. The graph of this equation is called the supply curve. Often, the independent variable in the supply function is the quantity supplied. It is characteristic of the supply curve to be an increasing function.
31 Demand Definition (Demand Equation) This equation expresses the relationship between unit price and the quantity demanded. The graph of this equation is called the demand curve. Often, the independent variable in the demand function is the quantity demanded.
32 Demand Definition (Demand Equation) This equation expresses the relationship between unit price and the quantity demanded. The graph of this equation is called the demand curve. Often, the independent variable in the demand function is the quantity demanded. It is characteristic of the demand curve to be a decreasing function.
33 Equilibrium Definition (Market Equilibrium) Market equilibrium occurs when supply and demand are equal, i.e. it is when the the supply and demand curves intersect. It corresponds to an equilibrium quantity and price.
34 Supply, Demand, and Equilibrium
35 Supply, Demand, and Equilibrium Which curve is the supply curve?
36 Supply, Demand, and Equilibrium Which curve is the supply curve? The demand curve?
37 Example The demand function for a certain Bluetooth device is given by p = d(x) = 0.025x 2 0.5x + 60, and its supply function is p = s(x) = 0.02x x + 20, where p is expressed in dollars and x is measured in the thousands of units. Find the equilibrium price and quantity.
38 Example The demand function for a certain Bluetooth device is given by p = d(x) = 0.025x 2 0.5x + 60, and its supply function is p = s(x) = 0.02x x + 20, where p is expressed in dollars and x is measured in the thousands of units. Find the equilibrium price and quantity. We must solve the system of equations p = 0.025x 2 05x + 60 p = 0.02x x + 20.
39 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 =
40 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 =
41 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 =
42 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 = 9x x 8000 = 0 =
43 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 = 9x x 8000 = 0 = (9x + 400)(x 20) = 0 =
44 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 = 9x x 8000 = 0 = (9x + 400)(x 20) = 0 = x = 20 is the solution,
45 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 = 9x x 8000 = 0 = (9x + 400)(x 20) = 0 = x = 20 is the solution, i.e. the equilibrium quantity.
46 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 = 9x x 8000 = 0 = (9x + 400)(x 20) = 0 = x = 20 is the solution, i.e. the equilibrium quantity. Thus, p = s(20) = d(20) = 0.025(20) 2 0.5(20) + 60 = 40.
47 Example Thus, we have 0.025x 2 05x + 60 = 0.02x x + 20 = 0.045x x 40 = 0 = 45x x = 0 = 9x x 8000 = 0 = (9x + 400)(x 20) = 0 = x = 20 is the solution, i.e. the equilibrium quantity. Thus, p = s(20) = d(20) = 0.025(20) 2 0.5(20) + 60 = 40. Thus, the equilibrium price is $40 per headset.
48 Graph of Example
49 Graph of Example
50 How to Construct Mathematical Models These three steps can help you construct a mathematical model.
51 How to Construct Mathematical Models These three steps can help you construct a mathematical model. 1 Label each variable in the problem and draw a figure or diagram.
52 How to Construct Mathematical Models These three steps can help you construct a mathematical model. 1 Label each variable in the problem and draw a figure or diagram. 2 Find ways of representing one variable in terms of the others. There may be multiple ways of doing this. Pick the most useful one.
53 How to Construct Mathematical Models These three steps can help you construct a mathematical model. 1 Label each variable in the problem and draw a figure or diagram. 2 Find ways of representing one variable in terms of the others. There may be multiple ways of doing this. Pick the most useful one. 3 Write a function for the quantity sought. Pay careful attention to the domain of this function.
54 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight.
55 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price.
56 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price. Then, in terms of x, we have p = (300 (x 200))
57 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price. Then, in terms of x, we have p = (300 (x 200)) = 500 x.
58 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price. Then, in terms of x, we have p = (300 (x 200)) = 500 x. So, the total revenue R(x) is given by
59 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price. Then, in terms of x, we have p = (300 (x 200)) = 500 x. So, the total revenue R(x) is given by R(x) = x(500 x).
60 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price. Then, in terms of x, we have p = (300 (x 200)) = 500 x. So, the total revenue R(x) is given by R(x) = x(500 x). Domain =
61 Example PlaneX charges $300 per person if exactly 200 people sign of for a group flight. If more than sign up, the per-person fare is reduced by $1 for each person. Say x people (where x 200) people booked a group flight. Write a function for the total revenue realized by PlaneX when x people booked a group flight. Let p be the per-person price. Then, in terms of x, we have p = (300 (x 200)) = 500 x. So, the total revenue R(x) is given by R(x) = x(500 x). Domain = [200, 500].
62 Examples Determine if the following are true or false.
63 Examples Determine if the following are true or false. Any polynomial function can be written as a sum of constant multiples of power functions.
64 Examples Determine if the following are true or false. Any polynomial function can be written as a sum of constant multiples of power functions. True! If r > 0, then the power function f (x) = x r is defined for all values of x.
65 Examples Determine if the following are true or false. Any polynomial function can be written as a sum of constant multiples of power functions. True! If r > 0, then the power function f (x) = x r is defined for all values of x. False. Take, for instance r = 1/2.
66 Assignment Read 2.4. Do problems 6, 12, 18, 26, 38, 48, 62, 66, 74, 80, 82 in 2.3.
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