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1 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1

2 Equation of a Straight Line In general: y = mx + b where y is the dependent variable x is the independent variable b is the y intercept, or constant m is the slope (or gradient): slope = y/ x = rise/run 3 Application Demand functions are often written as linear Sales with lower price > Sales with higher price Suppose that the Husky Gear shop sells 200 footballs a month when they charge \$80, but sales go down to 160/month when they charge \$90 Can you graph this relationship? 4 2

3 Demand Graph 5 Demand Equation What is the equation of the demand function? 6 3

4 Quadratic Functions One of the most basic non linear relationships can be represented by a quadratic equation Quadratic equations have the general form y = ax 2 + bx + c This is also called a polynomial of the 2 nd degree 7 The Parabola The graphical representation of a quadratic equation is called a parabola y x Features of the parabola: U shaped (convex) if a > 0 shaped (concave) if a < 0 Top (or bottom) of function at x = b/(2a) Intercept is y = c (where x = 0) 8 4

5 Application We want to calculate the revenue as a function of the number of footballs sold Revenue = number of units*price/unit = n*p 9 Football Revenue Curve 10 5

6 What is an Equation? An equation is an algebraic expression with one or more unknown variables and one or more known coefficients that is equal to something If there is no = sign, then you do not have an equation Solving an equation means finding the value of the unknown variable(s) that satisfy the equation 11 One Equation, One Unknown If there is only one equation with one unknown, we can solve it: Objective: Isolate variable on one side, and on top Key #1: Add on one side, subtract on the other Key #2: Multiply on one side, divide on the other 12 6

7 Football Example How much should we charge? We determined that there was a relationship between the price of a Husky football and the number of footballs sold per week It was n = 520 4p If we want to sell 200 footballs per week, how much should we charge per football? 13 Verifying You should always double check your work You can do this by substituting the solution back into the original equation to make sure that the equation is satisfied. n = 520 4p n = 520 4*80 n = 200 Indeed! 14 7

8 Solving Linear Equations Graphically 15 Solving Quadratic Equations By examining the graph of the parabola, we can see that the parabola crosses the x axis twice These points, where y = 0, are called the roots or solutions of the equation At the roots: ax 2 +bx+c = 0 The two values of x that satisfy this equation are given by the following formula: x 2 b b 4ac 2a 16 8

9 Solving Quadratic Equations In general, a quadratic equation will have two solutions, because a parabola will tend to cross the x axis twice Not always the case! It may be tangent to the x axis (cross once) It may never cross the x axis Remember: Need a positive number under the square root (b 2 4ac 0) for one or two real solution(s) to exist 17 Putting it All Together Return to our Husky Football example Suppose that the cost of selling the footballs is \$4000 (for licensing and delivery) plus \$30 per football Where do we break even? 18 9

10 Putting it All Together Graphically 19 Putting it All Together Algebraically 20 10

11 Solving Simultaneous Equations Many business problems involve solving two or more equations at the same time Examples: Breakeven: When is revenue = cost where both revenue and cost are functions of quantity Parity: When will the unit cost of two machines be equal when each machine has a different cost function Tracking Portfolio: How many of two different assets do I need to buy to get the same payoffs as another asset 21 Solving Simultaneous Linear Equations When the equations are linear, solving them simultaneously is equivalent to finding their graphical intersections Example: x + 2y = 5 3x + 5y = 11 What are x and y? Note that we have two equations and two unknowns 22 11

12 Substitution Method Express one variable in terms of the other and substitute x + 2y = 5 Find x as a f(y): x = 5 2y 3x + 5y = 11 3*(5 2y) + 5y = y + 5y = y = 11 y = 4 x + 2*4 = 5 x = 3 Solution ( 3, 4) Substitute f(y) for x in other equation and solve Substitute this solved value back into the first equation 23 Solving Graphically 24 12

13 Solving Systems of Equations In general, you will solve systems of two equations with two unknowns Remember: In your MBA, you can only solve systems of n equations and n unknowns 25 Example Solve for x and y: 2x 3y 4 x 2y

14 Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 27 Rates of Change Business is frequently concerned with the rates at which quantities grow or shrink Sales, profits, number of customers, costs, etc. Rate of change of a linear function is easy: the slope! It is equal to the y/ x It is equal to the rise/run It is equal to the m in y = mx + b How can we identify rates of change when the relationships are non linear? 28 14

15 Rates of Return With a curve, the slope is always changing The slope at any point is equal to the tangent The tangent is the y/ x for infinitely small x 29 Differentiation How do we determine the slope of the tangent? We could draw line segments on our graph that became increasingly smaller and measure the slope each time to see if the value converges upon some number And do this for numerous points on the curve to see if there is a relationship Or we could use a shortcut: differentiation 30 15

16 General Differentiation Rule The slope of the curve y = x 2 is 2x More generally: The slope of a curve y = x n for any value x is nx n 1 Terminology: Finding the slope = differentiation Find the slope f(y) = x 5 = differentiate x 5 with respect to x Terminology: The slope of y = f(x) = the rate of change of y with respect to x = the derivative of y with respect to x = dy/dx = δy/δx = ( dee y by dee x ) 31 Some Rules for Differentiation For the purpose of the MBA, you should only be concerned with polynomials: Basics Straight line: Powers n n 1 2 then f ' x 2x 3 2 If f x k then f ' x 0 If f x x then f ' x 1 If f x k g x then f ' x k g' x If f x g x + h x then f ' x g' x h' x If f x ax b then f ' x a If f x x then f ' x nx If f x x If f x x then f ' x 3x 32 16

17 Examples Calculate the derivative with respect to x of: f x 4x f x 3x x 1 33 Differentiating More Than Once Sometimes we are not just interested in the rate of change but in how the rate of change is changing! When driving: You have velocity: d(distance,s)/d(time,t) = ds/dt But you also have acceleration: d(velocity)/d(time,t) = d(ds/dt)/dt = d 2 s/dt 2 This is referred to as the second derivative Example: If distance travelled is: s = 8t 2 +7t (quadratic) then velocity is: ds/dt = 16t+7 (linear) and acceleration is: d(16t+7)/dt = 16 (constant) 34 17

18 A Turning Point 35 Finding Local Maxima and Minima 36 18

19 Zero Slope When a function has a zero slope it is either: A maximum Point A on previous figure A minimum Point B on previous figure An inflection point Point C on previous figure In all cases, dy/dx = 0, but the d 2 y/dx 2 differ At a maximum d 2 y/dx 2 < 0 At a minimum d 2 y/dx 2 > 0 At an inflection point d 2 y/dx 2 = 0 37 Procedure for Finding Turning Points To find the turning points of function y = f(x) Find dy/dx Find the value(s) of x for which dy/dx = 0 For each of these values, find the value of d 2 y/dx 2 If d 2 y/dx 2 is positive, the point is a minimum If d 2 y/dx 2 is negative, it is a maximum If d 2 y/dx 2 is zero, it is probably an inflection point 38 19

20 Back to the Football Example Where do we profit maximize? 39 Profit Maximization Profit = Revenue Cost π = (130n 0.25n 2 ) ( n) π = 100n 0.25n

21 The Football Example Graphically At what price should we sell the football in order to maximize profits? 41 Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 42 21

22 Counting Example You are having dinner at a restaurant that offers 3 appetizers, 8 entrees, and 6 desserts. How many different meals can you have? 43 Fundamental Counting Principle If there are n(a) ways in which an event A can occur, and if there are n(b) ways in which a second event B can occur after the first event has occurred, then the two events can occur in n(a) n(b) ways. The Principle can be extended to any number of events as long as they are independent 44 22

23 Another Counting Example You are planning your wedding and it is time to decide how to seat your 6 closest friends at the head table. In how many arrangements can they be seated in the side by side 6 chairs? 45 Factorials Special products such as are frequent in counting theory, so we have a special notation to denote them: 6! = 6 factorial = 720 For any positive integer n, we define n factorial as: n! = n (n 1) (n 2) Calculator example: What is 25 factorial? 46 23

24 More Counting Examples There are 200 guests at your wedding dinner. How many seating combinations are there? If a state permits either a letter or a nonzero digit to be used in each of six places on its license plates, how many different plates can it issue? 47 Permutations Example: Suppose that you have 8 best friends, but only 6 can be seated at the head table. How many arrangements can be made? This is called the number of permutations of 8 things taken 6 at a time: 8P 6 = 48 24

25 Permutations The number of possible distinct arrangements of r objects chosen from a set of n objects is called the number of permutations of n objects taken r at a time, and it equals: n P r n! n r! 49 Permutations Example In how many ways can a president, a vice president, a secretary, and a treasurer be selected from an organizations with 20 members? 50 25

26 Combinations Permutations are used when order is a factor in the selection. What do we do if that is not the case? Example: You are the president of a company and you want to pick 2 secretaries to work for you. If 5 people are qualified, how many different pairs of people can you select? 51 Combinations The number of ways in which r objects can be chosen from a set of n objects, without regard to the order of selection, is called the number of combinations of n objects taken r at a time, and it equals: n npr n! nc r r r! r! n r! 52 26

27 Combination Example Ten men have volunteered to serve on a committee. How many committees can be formed containing three of the volunteers? 53 Final Combination Example Ten men and eight women have volunteered to serve on a committee. How many committees can be formed containing three men and three women? 54 27

28 Conclusion This was a (fast!) review of basic algebraic tools you will encounter during your MBA If you are uncomfortable with this materials, please take a couple of hours this week to work on the attached problem set! See you tomorrow for more 55 28

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