Cengage Web Assign Mathskeller The Study J. Constable Ray Kremer Dr. Koester UK UK Math Department

UK MA162 Fall 2013

Schedule, Reading Assignments, and Course Materials

I expect you to read the assigned sections of the textbook. You should complete the relevant readings BEFORE the dates on the schedule. Don't worry if you don't immediately understand everything from the readings; the point of the lectures is to help clarify the subtle or difficult issues. However, the lectures will assume that you have at least attempted the readings.
Chapter 1: Linear Equations
  1. Overview of Chapter 1
    August 28 - September 4
    Most of the material in the first chapter should be review. However, you must take this chapter very seriously!
    • First, whereas chapter 1 reviews basic facts about lines in the plane, later chapters will be concerned with large numbers of lines and "line-like-objects" in higher dimensional spaces. If you do not have a solid foundation in the material from Chapter 1, you will struggle throughout these later chapters.
    • Many of the suggested practice problems from chapter 1 are word problems. Get used to this. Most problems throughout the course will be word problems.
  2. Chapter 1.1: Cartesian Coordinate System
    August 28, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    Do a few problems from each of the below block of problems. Answers to odd numbered problems are in back of book.
    1-12: Quadrants, intercepts, etc.
    13-20: Graphing points in plane
    Web Assign HW
    Due September 6, 2013
    Summary
    You won't be tested on the distance formula or the equation of a circle.
    Our variables will not always go by the names x and y, so I will often say horizontal axis instead of x-axis, vertical axis instead of y-axis, horizontal intercept instead of x-intercept, and vertical intercept instead of y-intercept.
  3. Chapter 1.2: Straight Lines
    August 28, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    11, 12: Interpreting the slope
    71, 73, 75: Application/word problems
    82
    Web Assign HW
    Due September 6, 2013
    Summary
    "Find slope, find equation of line." Seems like a repeat of every math class you've taken over the last few years.
    Focus on
    • General Form of a Line. Your previous math courses probably prefered point-slope form or slope-intercept form of a line. Start getting used to the general form, as this will be the prefered form for lines in this course. (Why? One reason is that the general form naturally generalizes to three or more variables. Another reason is that the general form does not require us to choose one of the variables to be the independent variable and the other to be the dependent variable.)
    • Word problems / Application problems / Interpret or explain problems. Many problems in this course will be in "word problem form." Spend extra time in this section practicing with the word problems.
    Hopefully the rest of this section is just a review (finding slope, finding equation of line, finding intercepts, etc.)
  4. Chapter 1.3: Linear Functions and Math Models
    September 4, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    15, 19, 21, 37, 39, 45: Application/Word problems
    Web Assign HW
    Due September 10, 2013
    Summary
    This section and the next introduce several important business applications. Familiarize yourself with these ideas now, as we will extend these applications over the next three chapters.
    The three super important business applications
    • Linear Depreciation
    • Cost, Revenue, Profit, Break-Even Analysis
    • Supply, Demand, Equilibrium Analysis
    Important terminology
    • Scrap value
    • Cost function
    • Revenue function
    • Profit function
    • Fixed cost
    • Variable cost
    • Break-even level, break-even quantity, break-even price
    • Demand equation, demand curve, demand function
    • Supply equation, supply curve, supply function
    • Equilibrium quantity, equilibrium price
    • Marginal cost, marginal revenue, marginal profit: these terms did not appear explicitly in the readings, but they are worth knowing. From an economic point of view, marginal cost measures the "cost of the next item," marginal revenue measures the "revenue generated by the next item," and marginal profit refers to the "profit earned on the next item." Mathematically, "marginals" are derivatives: \[ \mbox{Marginal cost } = C'(x) \] \[ \mbox{Marginal revenue } = R'(x) \] \[ \mbox{Marginal profit} = P'(x) \] (Calculus is not a prereq for this course, so don't worry if you don't know what a derivative is. In this course, our cost, revenue, and profit functions will always be linear, in which case "derivative" is just a fancy word for "slope".)
  5. Chapter 1.4: Intersections of Lines
    September 4, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    11, 13, 15, 17, 25, 27
    Web Assign HW
    Due September 10, 2013
    Summary
    The summary for Chapter 1.3 covers all I want to say about Chapters 1.3 and 1.4.
    The basic difference between the two sections is that Chapter 1.3 considers a single function at a time (e.g., do something with a cost function, do something with a revenue function, etc) whereas Chapter 1.4 is concerned with finding where two functions cross, e.g., break-even or equilibrium points.
Chapter 2: Systems of Linear Equations and Matrices
  1. Overview of Chapter 2
    September 9 - September 25
    The previous chapter discussed solving systems of two equations in two unknowns. The basic ideas can be extended to handle lots of equation in lots unknowns, but it is very easy to get lost as the number of equation and unknowns increases. The point of this chapter is to develop systematic methods for solving these larger systems of equations.
  2. Chapter 2.1: An Intro
    September 9, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    3, 5, 9, 13: 2 equations, 2 unknowns
    17, 18: Find \(k\) so that no solution or infinitely many solutions
    21, 23, 27: Word Problems: set up but do not solve
    Web Assign HW
    Due September 13, 2013
    Summary
    Solving many applied problems can be solved by determining the point at which two or more curves intersect. The next three sections will develop sytematic methods for finding the point (or points) or intersections of two or more lines (or "line-like objects"). I don't want to go into details of how to find the intersections here (that's what class is for!), so I'll just touch on a few qualitative aspects of Chapter 2.1 here.
    • Linear equations in \(n\) variables: \[a_1x_1 + a_2x_2 + \cdots + a_nx_n = c\]
      • If \(n = 2\), this is just the General form of a line from Chapter 1 \[a_1 x_1 + a_2 x_2 = c\]
      • For larger \(n\), the word linear is misleading at first. For example, \[a_1x_1 + a_2x_2 + a_3x_3 = c\] describes a plane in three dimensional space, NOT a line in three dimensional space.
    • Intersecting two lines: Three things can happen
      • Exactly one intersection point - this occurs for non-parallel lines
      • No intersection point - this occurs for parallel lines
      • Infinitely many intersection points - this occurs if the two lines are just copies of each other
    • Intersecting three planes: "Three" things can happen
      • Exactly one intersection point
      • No intersection point: picture on page 72 is a little misleading. Three parallel planes certainly won't intersect. There are other ways to have no intersection, for example, three planes in which each pair of planes intersect in a line, yet no common three-way intersection.
      • Infinitely many intersection points
  3. Chapter 2.2: Systems of Linear Equations; Unique Solutions
    September 11, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    9, 13, 17: Identifying matrices in row reduced form.
    19, 23, 27, 29: Gauss-Jordan, with guidance
    35, 39, 43, 47, 51: Gauss-Jordan, unguided
    57, 59, 61, 63: Solving application problems using Gauss-Jordan
    Web Assign HW
    Due September 20, 2013
    Summary
    The geometry behind Gauss-Jordan
    Gauss-Jordan: Infinitely many solutions and no solutions
    This section introduces the Gauss-Jordan method for solving systems of equations. I cannot overemphasize the importance of this technique. It will appear in nearly every problem we do throughout chapters 2, 3, and 4. Given the large number of arithmetic steps, this process can appear overwhelming at first. Just do a lot of practice with it!
  4. Chapter 2.3: Overdetermined and Underdetermined Systems
    September 16, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 5, 7, 9, 11: Identify if unique solution, no solution, infinitely many solutions, from row-reduced form
    13, 17, 19: Solve system of equations
    35, 39, 41: Word Problems
    Web Assign HW
    Due September 20, 2013
    Summary
    We continue to apply the Gauss-Jordan elimination method. In the previous section, the problems always worked out so that the left hand side of the augmented matrix would reduce to the identity matrix. In this section, we'll see what to do when the left hand side of the row-reduced augmented matrix is not the identity matrix.
  5. Addendum: A note on Row-reduced form
    September 16, 2013
    According to the definition of Row-Reduced Form of a Matrix, given on page 79 of the text:
    • Each row consisting entirely of zeros lies below all rows having nonzero entries
    • The first nonzero entry in each (nonzero) row is 1 (called a leading 1)
    • In any two successive (nonzero) rows, the leading 1 in the lower row lies to the right of the leading 1 in the upper row
    • If a column in the coefficient matrix contains a leading 1, then all other entries in that column are zero.
    On a recent Web Assign assignment, you were asked if the following matrix is in row-reduced form: \[ \left[\begin{array}{cc|c} 0 & 1 & 3 \\ 0 & 0 & 5 \\ \end{array}\right] \] This is not in row-reduced form, as it violates the second condition in the definition. However, the web homework system claims that this is in row-reduced form.
    Upon learning that the web homework was counting the correct answer as incorrect, I increased the number of allowed attempts on this problem to allow everybody to still get credit.
    Please look at example 3 on page 79 of the text for several (correctly) worked examples illustrating the difference between row-reduced and non-row-reduced matrices.
  6. Chapter 2.3: Overdetermined and Underdetermined Systems
    September 16, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 5, 7, 9, 11: Identify if unique solution, no solution, infinitely many solutions, from row-reduced form
    13, 17, 19: Solve system of equations
    35, 39, 41: Word Problems
    Web Assign HW
    Due September 20, 2013
    Summary
    We continue to apply the Gauss-Jordan elimination method. In the previous section, the problems always worked out so that the left hand side of the augmented matrix would reduce to the identity matrix. In this section, we'll see what to do when the left hand side of the row-reduced augmented matrix is not the identity matrix.
  7. Chapter 2.4: Matrices
    September 18, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 3, 7: basic matrix notation and terminology
    13, 15, 19: Addition, subtraction, scalar multiplication of matrices
    21: Matrix equality
    37, 39, 41: Word Problems
    Web Assign HW
    Due September 24, 2013
    Summary
    We introduced augmented matrices a few sections ago. In this section, we look at a more general notion of matrix. There is a lot of new terminology and notation in this section.
    • Size, or dimension, of a matrix
    • Entires, rows, columns
    • Row matrix, Column matrix, square matrix
    • \(a_{i,j}\) notation
    • Equality of matrices
    • Addition and subtraction of matrices
    • Commutativity and Associativity of matrices
    • The zero matrix
    • Transpose of a matrix
    • Scalar multiplication
    A few words of warning:
    • Equality of matrices is not the same as row equivalence of matrices. Saying that two matrices are row equivalent means that their associated systems of equations have the same solution sets. This is a much weaker condition than saying the two matrices a equal.
    • Addition and subtraction are only for matrices of the same size
  8. Chapter 2.5: Matrix Multiplication
    September 23, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 3: size restrictions on matrix products
    7, 13, 19, 23: Matrix products
    37, 39: Matrix multiplication and systems of equations
    43, 49, 53: Word Problems
    Web Assign HW
    Due September 27, 2013
    Summary
    The previous section introduced matrix addition and subtraction. Both of these were relatively easy, you just add or subtract the entries of the matrices term-wise. Matrix multiplication is a bit more involved.
    First, in order for \(AB\) to be defined, it is required that the number of columns of A is equal to the number of rows of B. In this case, the number of rows of \(AB\) will be equal to the number of rows of \(A\) and the number of columns of \(AB\) will be equal to the number of columns of \(B.\)
    As far as the actual muliplication computation is concerned, its best to look at the examples in the text and the slides and from class. Here, I just want to point out that matrix multiplication is NOT defined by simply multiplying corresponding entries together.
    Why do we multiply matrices in complex manner?
    • The example on page 115 of the text tries to motivate the definition of matrix multiplication.
    • I give a different motivating example in the slides.
    • As exercises 37-42 in the text show, the definition of matrix multiplication is related to systems of equations.
    • In more advanced courses, like MA 322, it is shown that matrix multiplication has interpretations in terms of "change of variables" as well as "composition of functions."
    My point is that even though matrix multiplication may seem unnatural at first, there are many reasons why this is the right way to define multiplication.
  9. Chapter 2.6: Matrix Inverses
    September 25, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    5, 9, 13: Computing some inverses
    25, 29: Using matrix inverses to solve systems of equations
    39, 43: Word Problems
    Web Assign HW
    Due September 27, 2013
    Summary
    In the previous section, we saw that a system of equations can be view as an equation involving matrix multiplication, \[ AX = B. \] Thus, to solve for X, we may try to "divide by the matrix A" \[ X = A^{-1}B \] Now, \(A^{-1}\) is called the inverse of the matrix \(A.\)
    Not every matrix has an inverse. First, in order for a matrix to have an inverse, the matrix must be square, meaning the matrix has the same number of rows as it has columns. However, not every square matrix even has an inverse.
    How can we tell which matrices even have an inverse? The best approach is to just try to compute the inverse.
    How do we compute the inverse? We can use a slight extension of the Gauss-Jordan process. Recall that the Gauss-Jordan process begins with an augmented matrix like \[ \left[\begin{array}{ccc|c} 1 & 2 & 3 & 4 \\ 3 & 2 & 1 & 0 \\ 5 & 10 & 5 & 15 \\ \end{array}\right] \] You perform row operations until the left hand side is reduced to a simple form. To find the inverse of \[ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 3 & 2 & 1 \\ 5 & 10 & 5 \\ \end{array}\right] \] we will perform the Gauss-Jordan process on a different kind of augmented matrix: \[ \left[\begin{array}{ccc|ccc} 1 & 2 & 3 & 1 & 0 & 0 \\ 3 & 2 & 1 & 0 & 1 & 0 \\ 5 & 10 & 5 & 0 & 0 & 1 \\ \end{array}\right] \] The original matrix has an inverse if the Gauss-Jordan process ends by placing the identity matrix on the left side of the vertical bar, in which case the inverse matrix is the stuff on the right side of the bar. Details are in the slides and the text.
Chapter 3: Linear Programming: A Geometric Approach
  1. Chapter 3: Linear Programming Overview
    October 2 through October 9
    Now that we've gotten plenty of practice with linear equalities, we will turn our attention of linear inequalities. In the previous sections, all of our application problems were selected because they had answers that made sense from a decision analysis point of view. However, in real applications we might not be so lucky. Consider a farmer, who has has \(100\) acres of land and wants to determine how much of the land to use to grow corn and how much land to use to grow wheat. Depending on how the numbers work out, the methods of the previous chapter might tell the farmer that she should use \(137\) acres for corn, and \(-37\) acres for wheat. -37 acres? The farmer can't use this information to make a decision!
    In this chapter we'll see that these unrealistic answers can be avoided, provided that we mix some inequalities in with the linear equations from the last chapter. In the case of this farmer, she would need to specify inequalities, like number of acres of corn must be nonnegative, and number of acres of wheat must be nonnegative.
    Handling these inequalities will require developing some new mathematical techniques. We'll develop a geometric approach in this chapter, and an algebraic approach in the next chapter. The geometric approach is rather natural, and works well provided that we only have two variables. The algebraic approach is less natural, but it can handle arbitrarily many variables and inequalities.
  2. Chapter 3.1: Graphing 2-Variable Inequalities
    October 2, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 5, 9: Graphing solution to single linear inequality
    11, 15, 17: Identifying system of inequalities given the graphical region.
    19, 25, 31, 35: Graphing solution to system of linear inequalities
    Web Assign HW
    Due October 8, 2013
    Summary
    There are two skills you must master in this section.
    • Graphing the solution set to a single linear inequality: If you can graph linear equations then you're 90% of the way there. For the inequality, its just a matter of deciding which side of the line to shade.
    • Graphing the solution set to a system of linear inequalities: In the homework, you'll approach this from two points of view:
      • Here is a graphical region, determine the inequalities that generated this region.
      • Here is a system of inequalities, determine the graphical region.
  3. Chapter 3.2: Linear Programming Problems
    October 7, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 5, 9, 13, 17, 21, 25: Setting up Linear Programming Problems (save your work, you'll be asked to revisit these problems in the next section)
    Web Assign HW
    Due October 11, 2013
    Key terms and defintions
    • Linear Programming Problems
    • Objective funtion
    • Constraints
    Summary
    Lots of word problems. As far as this section is concerned, you are not asked to solve problems, you are only asked to "set up" the problems. You may wish to review previous sections, like Chapter 2.1, which spent a bit of time developing "setting up" skills.
    Be sure to watch out for "implied constraints." For example, in the standard "farmer will plant corn and wheat" kind of problem, you will need to include "number of acres of wheat must be nonnegative" and "number of acres of corn must be nonnegative" as constraints, whether the statement of the problem explicitly states that the number of acres can't be negative. You will have to use some common sense.
  4. Chapter 3.3: Graphical Solution to LPs
    October 9, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1, 3, 5: Method of Corners, given the corners.
    7, 13, 15, 19, 23: Method of Corners, but you have to find the corners first.
    29, 33, 37, 41, 45: Applications using the method of corners.
    Web Assign HW
    Due October 15, 2013
    Key terms and defintions
    • Feasible set
    • Feasible solution
    • Optimal solution
    • Isoprofit line
    • Method of corners
    Summary
    The main idea in this section is the method of corners. You should look at Theorem 1 and Theorem 2 for the precise statement of this method, but basically it says that the solution to LP must be at one of the corners of the corresponding convex polygonal region. This means that LPs are very easy to solve, IF (and this is a very big if) you already know the corner points of the region.
    As far as the homework is concerned, there are three types of problems:
    • Find the solution to the LP, provided that the corner points are already given.
    • Find the solution to the LP, provided that the constraints are given. In this case, you'll need to use the techniques from Chapter 3.1 to turn the constraints into a geometric region, then determine the corners, then treat it as the above type of problem.
    • Find the solution to the LP, provided the problem is given in "application form." In this case, you'll need to use the techniques from Chapter 3.2 to determine the objective function and the constraints, then treat it as the above type of problem.
Chapter 4: Linear Programming: An Algebraic Approach
  1. Chapter 4.1: Simplex Method
    October 14 and 16, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    Web Assign HW
    Due October 18, 2013
    Due October 22, 2013
    Terminology
    • Standard Maximization Problem
    • Slack variables
    • Basic variables
    • Non-basic variables
    • Feasible solutions
    • Simplex Tableau
    • Simplex Method
    Summary
    This lengthy section introduces the Simplex Algorithm, which was developed by George Dantzig in 1939. The method is too long to give a simple description here, so see the text and the slides for worked out examples. Here I want to give some thoughts as to what the simplex algorithm does, and how it is related to, and how it is different from the method of corners from Chapter 3.
    Method of Corners: The method of corners tells us that in order to solve a linear programming problem, we need to draw the region, then find the corners of the region. The we plug each corner point into the objective function. The maximum value is then the largest value obtained by plugging in these corner points.
    What's wrong with this? Why do we need another method?
    • LPs with three variables will require finding corners or polyhedra in space, rather than corners of polygons in the plane. LPs with more than three variables require finding corners of polytopes in higher dimensional space. Since we can't see in these hihger dimensional spaces, we need to find a non-geometric approach to these problems.
    • Even in two dimensions, finding the corners can be hard and time consuming. Suppose an LP has \(20\) constraints. \(20\) constraints in two variables could give rise to a polygon with up to \(20\) corner points. Each corner of the feasible region will arise as an intersection between two of the contraints. However, many of the intersections are NOT corner points to the region. In fact, \(20\) constraints in two variables could give rise to as many as \(190\) intersection points. So, if you used the method of corners, you would begin by finding \(190\) intersection points. Then you would have to determine which of those points are corner points. At this point, you will throw out a large number of the intersection points. This is very wasteful and time consuming.
    • Even if you already know all of the corner points of the region, the method of corners can be slow and wasteful. If there are twenty corners, you must evaluate the objective function at each of these twenty corners. Even if you plugged in nineteen of the twenty corners, you still can't say much about the solution to the LP.
    The Simplex Algorithm presents a much smarter attack on the problem.
    • You begin with a single corner. The simplex algorithm will be able to tell if this corner gives rise to an optimal solution.
    • If the above corner does not give rise to an optimal solution, the simplex algorithm will tell us to move to a nearby corner. The new corner will be "better". For example, in a maximization problem, the objective function will be larger at the new corner than at the old corner.
    • This process repeats until we eventually arrive at the best corner.
    In other words, "are we currently the best? If not, find a better one, and repeat until we are the best."
    The simplex algorithm is able to make its decisions based purely on algebra (no geometry required) which means this method works regardless of the number of variables. The simplex algorithm does not require us to know all of the corners. It computes the corners as it goes along, and it only looks at the corners that it needs to look at.

    The above criticism of the method of corners are only valid when either the number of variables or the number of constraints get large. Is it important to be able to solve LP with large numbers of variables and constraints? YES! In standard real-life management decisions, there may be several variables attached to each employee, and likewise there may be several constraints attached to each employee. If the company has \(100\) employees, management may be faced with an LP with \(400\) or so variables and \(700\) or so constraints!
Chapter 6: Sets and Counting
  1. Chapter 6.1: Sets and Set Operations
    October 21, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    21-24
    33-36
    47-50
    Web Assign HW
    Due October 25, 2013
    Key terms and defintions
    • Sets
    • Union, \(\cup\)
    • Intersection, \(\cap\)
    • Complement, \({}^c\)
    • Difference, \( - \)
    Summary
    This section introduces the idea of a set. Sets are very tools for counting and probability theory. I recommend focusing more on the lecture slides than on the textbook for Chapter 6.1. Our ultimate goal in studying set theory is to develop some useful tools and notation to aid us in our discussions on counting (chapter 6) and probability theory (chapter 7); the text tends to lose sight of this goal develops some aspects of set theory that are not necessary for our purposes.
    Also notice that I talk about set differences in the notes, but this does not appear explicitly in the text. However, a difference of sets can be expressed in terms of intersections and complements, \[ A - B = A \cap B^c. \] I feel the difference notation is a lot clearer than the latter.
  2. Chapter 6.2: Number of elements in a Finite Set
    October 23, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    3,5,7,9,11,13,15,17: Two set inclusion-exclusion problems
    19,21,23,25: two set inclusion-exclusion word problems
    27,29,31: three set inclusion-exclusion problems
    33,35,37,39: three set inclusion-exclusion word problems
    Web Assign HW
    Due October 25, 2013
    This section introduces the Inclusion-Exclusion Principle, a fundamental principle of counting. In advanced counting problems, we need to be careful not to double count. For example, according to my class rolls I have 242 students in my MA 162 course and I have 363 students in my MA 123 course. Does this mean I am responsible for \(242 + 363 = 605\) students? Not quite. 7 students are enrolled in both my MA 123 and MA 162 courses. The \(605\) figure counted these students once for being in my MA 123 and once for being in my MA 162 course. After correcting for this double counting, we see that I am responsible for \[ 242 + 363 - 7 = 598 \] students. The inclusion-exclusion principle discusses the "correction factor for double counting" in more general counting problems.
  3. Chapter 6.3: Multiplication Principle
    October 30, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1 - 27
    Web Assign HW
    Due November 5, 2013
    Key terms and defintions
    • The Multiplication Principle
    Summary
    If a counting problem natuarally decomposes into multiple stages, in which one of m possibilities are selected in the first stage, then, independent of what choice was made in the first stage, one of n possibilities is selected in the second stage, then the total number of choices (stage one and stage two together) is given by the product mn.
    When using the multiplication principle, be careful to check that the number of choices at later stages in the problem are independent of the previous choices already made! Different techniques are required when the choices are dependent.
  4. Chapter 6.4: Permutations and Combinations
    November 4, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    23 - 77
    Web Assign HW
    Due November 8, 2013
    Key terms and defintions
    • Permutation of distinct items
    • Permutation of distinct items taken r at a time
    • Permutation of non-distinct items
    • Combinations
    Summary
    Many counting problems are one of the four above listed types. Each of these four types are well studied, and we have formulas to count the number of possibilities in each of these cases. Thus, if you can recognize your counting problem as one of these four types, then you should be able to immediately write down a solution. In more difficult counting problems, you will need to break the problem up into pieces, and hopefully each of the pieces will be one of these four standard types.
Chapter 7: Probability
  1. Chapter 7.1: Experiments, Samples Spaces, and Events
    November 6, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1-19 odd: Using the notation and terminology of Probability Theory
    21-43 odd: Determining Sample Spaces and Events
    Web Assign HW
    Due November 12, 2013
    Key terms and defintions
    • Experiment
    • Outcome
    • Sample Point
    • Sample Space
    • Event
    • Union, Intersection, Complement of Events
    • Mutual Exlusive
    Summary
    The foundations of Probability Theory have a lot in common with Counting. Two of the main things to focus on in this short section: (a) while many of the computations in probability theory are very similar to computations in counting, the terminology is not always the same (b) pay careful attention to the idea of a sample space.
  2. Chapter 7.2: Definition of Probability
    November 6, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1-7 odd: Identifying Simple Events
    9-45 odd: Computing Probabilities through relative frequency
    Web Assign HW
    Due November 12, 2013
    Key terms and defintions
    • Probability of Event
    • Simple Events
    • Probability Function or Probability Measure
    • Uniform Distribution
    Summary
    Apart from a few additional definitions, this section is basically focused on the idea that the probability of an event can be computed by, first, counting the number of outcomes in the event, then counting the total number of outcomes in the entire probability space, then dividing the former by the latter.
  3. Chapter 7.3: Rules of Probability
    November 11, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1-11 Odd
    25-47 Odd
    Web Assign HW
    Due November 15, 2013
    Key terms and defintions
    • Probability of two mutually exclusive events
    • Probability of two not-necessarily mutually exclusive events
    • Probability of the complement of an event
    Summary
    Many of the properties of probabilities have analogues in chapter 6. Two notions to pay particular attention to are mutually exclusive events and independent events.
    Mutually exclusive means the two events cannot occur simultaneously. For example, you may be a Freshman, you may be a Sophomore, but you cannot be both. Mutual exclusivity is very useful to exploit when finding the probability of a union of events, as you just need to add the probabilities of the events.
    Independent means the two events occur in sequence, and the outcome of the first event does not impact the outcome of the second event. For example, you may be an Accounting major, you may be a Junior. Whether you are an accounting major or not should not affect whether you are a junior or not. Independence is most useful to exploit when finding the probability of an intersection of events. In the case of independent events, the probability of the intersection is just the product of the individual probabilities.
  4. Chapter 7.4: Probability via Counting Techniques
    November 13, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    1-41 Odd
    Web Assign HW
    Due November 19, 2013
    Summary
    This section is basically focused on the idea that the probability of an event can be computed by, first, counting the number of outcomes in the event, then counting the total number of outcomes in the entire probability space, then dividing the former by the latter. This is similar in nature to Chapter 7.2, but the current section exploits some of the more advanced counting techniques from Chapter 6, like permutations of distinct items, permutations of non-distinct items, combinations, etc.
  5. Chapter 7.5: Conditional Probability
    November 18, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    5 - 7 Odd: Are events independent
    11 - 15 Odd: Are events independent
    19 - 49 Odd: Computations with Conditional Probabilities
    Web Assign HW
    Due November 22, 2013
    Key terms and defintions
    • Conditional Probability of B given A
    • \( P(B|A) = \frac{P(A \cap B)}{P(A)} \)
    • \( P(A \cap B) = P(A) P(B|A)\)
    Summary
    There has been much overlap between the ideas in Chapter 6 and Chapter 7 thus far. This section introduces Conditional Probability, a concept that does not have a direct analogue in Chapter 6. A conditional probability takes into account additional information about the underlying sample space. For example, a car insurance company wants to estimate the probability that a new customer will be involved in a serious accident. Not knowing anything about this driver, the insurance company may assign one probability to this driver. On the other hand, if the underwriting process reveals that this new customer has been involved in several accidents and has received several speeding tickets, then their probability of having another accident should be much greater than that of a randomly selected individual from the general population.
    Conditional probabilities provide us with the techniques needed to distinguish between the random driver and the driver with a bad record.
  6. Chapter 7.6: Baye's Theorem
    November 20, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    13 - 51 Odd
    Web Assign HW
    Due November 22, 2013
    Key terms and defintions
    • Baye's Theorem
    • A priori probability
    • A posteriori probability
    Summary
    Suppose a health insurance company insures some healthy people and some unhealthy people. Suppose they categorize \(85\%\) of their policy holders as healthy and \(15\%\) as unhealthy.
    Suppose the probability that a healthy individual will have an overnight hosptital stay in the next year is \(0.01\), and the probability that an unhealthy individual will have an overnight hospital stay in the next year is \(0.35\).
    A random policy holder has an overnight stay in the hospital. What is the probability they were catagorized as a healthy policy holder?
    Let \(H\) denote ``healthy policy holder,'' \(U\) denote ``unhealthy policy holder'' and let \(O\) denote ``overnight hospital stay.'' We want to find \( P(H | O)\). Notice, though, that we have ``the other conditional probability,'' \( P(O|H) = 0.01.\) The question, then, is whether we can somehow use \( P(O|H)\) to help determine \( P(H|O).\)
    From chapter 7.5, \[ P(H \cap O) = P(H|O) P(O) \] \[ P(O \cap H) = P(O|H) P(H) \] However, \(H \cap O = O \cap H\) and so \[ P(H|O) P(O) = P(O|H) P(H) \] which rearranges to \[ P(H|O) = \frac{P(O|H)P(H)}{P(O)} \] This rearranged formula goes by the name ``Baye's Theorem.''
    Furthermore, \[ P(O) = P(O|H)P(H) + P(O|U)P(U) = 0.01*0.85 + 0.35*0.15 = 0.061 \] and \[ P(O|H)P(H) = 0.01*0.85 = 0.0085 \] So \[ P(H|O) = \frac{0.0085}{0.061} \]
Chapter 5: Financial Mathematics
  1. Chapter 5.1: Compound Interest
    December 2, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    "Direct" Compound Interest Problems: 11 - 19, 25 - 29 Odd
    Compound Interest Word Problems: 39 - 51 Odd, 57, 61
    Web Assign HW
    Due December 6, 2013
    Key terms and defintions
    • Compound Interest
    • Nominal interest rate, effective interest rate, period interest rate
    • Conversion period, conversion frequency
    • Compound interest formulas, Accumulated Amount and Present Value

    Strategy for solving Compound Interest Word Problems
    1. First determine if you are dealing with a single cash flow or multiple cash flows.
    2. If your problem involves multiple cash flows, divide problem into several single cash flow problems
    3. For a single cash flow problem, determine if you need to find the initial value or the final value. Use the Present Value form of the Compound Interest Formula if you need to find the initial value, and use the Accumulated Value form of the Compound Interest Formula if you need to find the final value.
    4. Now determine the value of \(m\), the conversion frequency.
      • Annual: \(m = 1\)
      • Semi-annual: \(m = 2\)
      • Quarterly: \(m = 4\)
      • Monthly: \(m = 12\)
      • Weekly: \(m = 52\)
      • Daily: \(m = 365\)
    5. Now determine \(i\) and \(n\).
      • \(i = \frac{r}{m}\), where \(r\) is the nominal interest rate. When you plug into the formulas, be sure to convert the percentage into a decimal!
      • \(n = m\cdot t\), where \(t\) is the number of years of the investment
    6. Now plug \(i\) and \(n\) into the appropriate compound interest formula.

    As far as the exam is concerned, being able to identify which formula (or formulas) to use, and identifying the appropriate parameters for the formula is more important than getting the correct numerical value. For example, you may leave an answer in non-decimal form on the exam. In other words, your answer on the exam may involve unsimplified expressions like \( \frac{1200}{(1.0035)^{12}} - \frac{400}{(1.0035)^{6}}.\)

    As far as WebAssign is concerned, you will need to enter decimal approximations. Most calculators and spreadsheet programs have built in routines for evaluating interest functions. Check out the appendix to Chapter 5.1 (pages 281-284 in 10th edition of Tan) for instructions on using a computer or graphing calculator to assist in these calculations.
  2. Chapter 5.2: Annuities
    December 4, 2013
    Recommended Practice Problems, 10th edition of Tan's Finite Mathematics
    "Direct" Annuity Problems: 1 - 13 Odd
    Annuity Word Problems: 15 - 33 Odd
    Web Assign HW
    Due December 10, 2013
    Key terms and defintions
    • Annuity
    • Annuity Certain versus Contingent Annuity
    • Simple Annuity versus Complex Annuity
    • Ordinary Annuity versus Annuity Due

    Strategy for solving Finance Word Problems
    1. First determine if you are dealing with a single cash flow or multiple cash flows.
    2. If your problem involves a small number of cash flows and the cash flows are irregular, divide the problem into several single cash flow problems and use the techniques from Chapter 5.1 (see the flow chart for December 2.)
    3. If your problem involves a large number of cash flows and the cash flows are regular, then you are dealing with an annuity. In this case, now determine if you are trying to find the initial value of the annuity, the final value of the annuity, or the payment size.
    4. Now determine the value of \(m\), the conversion frequency. (In this course, we only consider simple annuities, meaning that the conversion frequency and the cash flow frequency will match.)
      • Annual: \(m = 1\)
      • Semi-annual: \(m = 2\)
      • Quarterly: \(m = 4\)
      • Monthly: \(m = 12\)
      • Weekly: \(m = 52\)
      • Daily: \(m = 365\)
    5. Now determine \(i\) and \(n\).
      • \(i = \frac{r}{m}\), where \(r\) is the nominal interest rate. When you plug into the formulas, be sure to convert the percentage into a decimal!
      • \(n = m\cdot t\), where \(t\) is the number of years of the investment
    6. Now determine the
    7. Now plug \(i\) and \(n\) into the appropriate annuity formula.
      • If payment size R is given and you want to find final value S, use \[S = R \cdot \require{enclose}s_{\enclose{actuarial}{n}i} \]
      • If payment size R is given and you want to find present value P, use \[P = R \cdot \require{enclose}a_{\enclose{actuarial}{n}i} \]
      • If final value S is given and you want to find payment size R, use \[R = \frac{S}{\require{enclose}s_{\enclose{actuarial}{n}i}} \]
      • If initial value P is given and you want to find payment size R, use \[R = \frac{P}{\require{enclose}a_{\enclose{actuarial}{n}i}} \]
    8. If you wish to convert the annuity symbols \(\require{enclose}s_{\enclose{actuarial}{n}i} \) or \(\cdot \require{enclose}s_{\enclose{actuarial}{n}i}\) into decimals, use
      • \[\require{enclose}s_{\enclose{actuarial}{n}i} = \frac{(1 + i)^{n} - 1}{i}\]
      • \[\require{enclose}a_{\enclose{actuarial}{n}i} = \frac{1 - (1 + i)^{-n}}{i}\]
      • Or use this excel spreadsheet
      • If the above hyperlink does not work, copy and paste http://www.ms.uky.edu/~pkoester/teaching/Fall13/MA162/Annuity.xlsx into the URL line of your web browser

    As far as the exam is concerned, being able to identify which formula (or formulas) to use, and identifying the appropriate parameters for the formula is more important than getting the correct numerical value. For example, you may leave an answer in non-decimal form on the exam. In other words, your answer on the exam may involve unsimplified expressions like \( 1200 \cdot \require{enclose}s_{\enclose{actuarial}{24}0.02} \)

    As far as WebAssign is concerned, you will need to enter decimal approximations. Most calculators and spreadsheet programs have built in routines for evaluating annuity functions. Check out the appendix to Chapter 5.2 (pages 293-296 in 10th edition of Tan) for instructions on using a computer or graphing calculator to assist in these calculations.
  3. Chapter 5.3: Amortization and Sinking Funds
    December 9, 2013

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