1 15.Math-Review Monday 8/14/00

15.Math-Review2 General Mathematical Rules zAddition yBasics: ySummation Sign: yFamous Sum:

15.Math-Review3 General Mathematical Rules zMultiplication yBasics ySquares: yCubes:

15.Math-Review4 zMultiplication yGeneral Binomial Product: General Mathematical Rules yProduct Sign: yDistributive Property:

15.Math-Review5 zFractions yAddition: General Mathematical Rules yProduct:

15.Math-Review6 General Mathematical Rules zPowers yInterpretation: yGeneral rules: ySeries:

15.Math-Review7 General Mathematical Rules zLogarithms yInterpretation: xThe inverse of the power function. yGeneral rules and notation:

15.Math-Review8 zExercises: yWe know that project X will give an expected yearly return of $20 M for the next 10 years. What is the expected PV (Present Value) of project X if we use a discount factor of 5%? yHow long until an investment that has a 6% yearly return yields at least a 20% return? General Mathematical Rules

15.Math-Review9 zDefinition: zGraphical interpretation: The Linear Equation c 1 a -c/a y x

15.Math-Review10 zExample: Assume you have $300. If each unit of stock in Disney Corporation costs $20, write an expression for the amount of money you have as a function of the number of stocks you buy. Graph this function. zExample: In 1984, 20 monkeys lived in Village Kwame. There were 10 coconut trees in the village at that time. Today, the village supports a community of 45 monkeys and 20 coconut trees. Find an expression (assume this to be linear) for, and graph the relationship between the number of monkeys and coconut trees. The Linear Equation

15.Math-Review11 zSystem of linear equations 2x – 5y = 12(1) 3x + 4y = 20 (2) zThings you can do to these equalities: (a)add (1) to (2) to get: 5x – y = 32 (b)subtract (1) from (2) to get: x + 9y = 8 (c)multiply (1) by a factor, say, 4 8x – 20y = 48 zAll these operations generate relations that hold if (1) and (2) hold. The Linear Equation

15.Math-Review12 zExample: Find the pair (x,y) that satisfies the system of equations: 2x – 5y = 12(1) 3x + 4y = 20 (2) Now graph the above two equations. zExample: Solve, algebraically and graphically, 2x + 3y = 7 4x + 6y = 12 zExample: Solve, algebraically and graphically, 5x + 2y = 10 20x + 8y = 40 The Linear Equation

15.Math-Review13 zExercise: A furniture manufacturer has exactly 260 pounds of plastic and 240 pounds of wood available each week for the production of two products: X and Y. Each unit of X produced requires 20 pounds of plastic and 15 pounds of wood. Each unit of Y requires 10 pounds of plastic and 12 pounds of wood. How many of each product should be produced each week to use exactly the available amount of plastic and wood? The Linear Equation

15.Math-Review14 zDefinition: zGraphical interpretation: The Quadratic Equation y x When a<0 r2r2 r1r1 y x When a>0 r2r2 c r1r1 y x Can have only 1 or no root. r1r1

15.Math-Review15 zCompleting squares: The Quadratic Equation yAnother form of the quadratic equation: yThe point (h,k) is at the vertex of the parabola. In this case:

15.Math-Review16 zExample: Find the alternate form of the following quadratic equations, by completing squares, and their extreme point. The Quadratic Equation

15.Math-Review17 zSolving for the roots yWe want to find x such that ax 2 +bx+c=0. This can be done by: xFactoring. Finding r 1 and r 2 such that ax 2 +bx+c = (x- r 1 )(x- r 2 ) The Quadratic Equation Example: xFormula Example:

15.Math-Review18 zExercise: Knob C.O. makes door knobs. The company has estimated that their revenues as a function of the quantity produced follows the following expression: The Quadratic Equation zwhere q represents thousands of knobs, and f (q), represents thousand of dollars. yIf the operative costs for the company are 20M, what is the range in which the company has to operate? yWhat is the operative level that will give the best return?

15.Math-Review19 zDefinition: yFor 2 sets, the domain and the range, a function associates for every element of the domain exactly one element of the range. yExamples: xGiven a box of apples, if for every apple we obtain its weight we have a function. This maps the set of apples into the real numbers. xDomain=range=all real numbers. For every x, we get f(x)=5. For every x, we get f(x)=3x-2. For every x, we get f(x)=3 x +sin(3x) Functions

15.Math-Review20 zTypes of functions yLinear functions yQuadratic functions yExponential functions: f(x) = a x Example: Graph f(x) = 2 x, and f(x) = 1-2 -x. Example: I have put my life savings of $25 into a 10-year CD with a continuously compounded rate of 5% per year. Note that my wealth after t years is given by w = 25e 5t. Graph this expression to get an idea how my money grows. Functions

15.Math-Review21 zTypes of functions yLogarithmic functions f(x) = log(x) yLets finally see what this ‘log’ function looks like: Functions

15.Math-Review22 zGiven a function f(x), a line passing through f(a) and f(b) is given by: Convexity and Concavity zDefinition: yf(x) is convex in the interval [a,b] if yf(x) is concave in the interval [a,b] if Another definition is f(x) is concave if -f(x) is convex

15.Math-Review23 yThese ideas graphically: Convexity and Concavity y x b a f(a) f(b) x b a f(a) f(b)