# Review of Fundamental Mathematics

Size: px
Start display at page:

Transcription

1 Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools that you will learn can provide numerical answers to questions such as: What price should Harley-Davidson charge for its 008 model Sportster? At Doctors Walk-In Clinic, what combination of doctors and nurses provides the expected level of medical services at the lowest possible total cost? What is the forecasted price of West Texas crude oil six months from now? How many PCs should Dell manufacture and sell each month in order to maximize profit? In order to apply the decision-making rules developed in managerial economics, you need to understand some fundamental mathematics, most of which you learned in high school algebra. No calculus is used in the body of the textbook, and none is required to work the Technical Problems and Applied Problems at the end of each chapter in the textbook. Exercises are provided throughout this Review. Answers to the exercises can be found at the end of this Review. Mathematical Functions CONCEPT OF A FUNCTION The relation between decision-making variables such as output, labor and capital employment, price, cost, and profit can be expressed using mathematical functions. A function shows mathematically the relation between a dependent variable and one or more independent variables. The dependent variable is often denoted by y, and the independent variable is denoted by x. The idea that y is related to x or that y is a function of x is expressed symbolically as y = f( x) This mathematical notation expresses the relation between y and x in the most general functional form. In contrast to this general form of expression, a specific functional form uses an equation to show the exact mathematical relation between y and x using an equation. For example, y = 100 x gives a specific function relating y to values of x. 1

2 The equation showing the specific functional relation between y and x provides a formula for calculating the value of y for any specific value of x. In the specific function given above, when x equals 0, y equals 60 ( = ). In economic analysis, it is frequently convenient to denote the dependent and independent variable(s) using notation that reminds us of the economic variables involved in the relation. Suppose, for example, that the quantity of golfing lessons (q) that a professional golf instructor gives each week depends on the price charged by the golf pro ( p ). The specific functional relation which is called the golf pro s demand for lessons can be expressed as q= f( p) = 100 p Instead of using the generic mathematical names y and x, the dependent variable is denoted by q to suggest quantity, and the independent variable is denoted by p to suggest price. If the golf pro charges \$40 per lesson, then she will give 0 ( = ) lessons per week. INVERSE FUNCTIONS Sometimes it is useful to reverse a functional relation so that x becomes the dependent variable and y becomes the independent variable: 1 x = f 1 ( y) This function, known as an inverse function, gives the value of x for given values of y. The inverse of y = f( x) can be derived algebraically by expressing x as a function of y. Consider again the demand for golfing lessons. The inverse function gives the price that a golf pro can charge for a given quantity of lessons each week: p = f( q) = q If the golf pro wishes to sell 0 lessons per week, she charges a price of \$40 ( = ) per lesson. Notice that all of the values of q and p satisfying the function q= 100 palso satisfy the inverse function p = q. FUNCTIONS OF TWO OR MORE INDEPENDENT VARIABLES In many economic relations, the dependent variable is a function of, or depends on, the values of more than one independent variable. If y depends on both w and z, for example, the general functional relation is expressed as y = f( w, z) Suppose a firm employs two inputs, labor and capital, to produce its product. The price of labor is \$30 per hour, and the price of using capital is \$60 per hour. The total cost of production (C) can be expressed as a function of the amount of labor employed per hour (L) and the amount of capital used per hour (K): 1 For the function y = f ( x), mathematicians denote its inverse function as x = f 1 ( y). The 1" is not an exponent here, but rather it denotes this function to be the inverse function of y = f ( x). Inverse functions do not exist for all functions. In this review, we will not investigate the conditions that ensure the existence of an inverse. In cases where inverse functions are required for managerial decision making, the required inverse functions generally do exist and can be rather easily derived.

3 C = f( L, K) = 30L+ 60K From this cost function, the manager calculates the cost of employing 10 workers and 5 units of capital to be \$600 ( = ). Exercises (Answers to Exercises follow this Review on pages ) 1. In the following functions, find the value of the dependent variable when x = 10. a. y = 300 0x b. y = x+ x c. w= 0 + 3x d. s= 40x+. Find the inverse functions. a. y = x b. q= 1, p 3. In the functions below, find the value of the dependent variable for the values of the independent variables given. a. y = f( x, w) = x + w, for x = 5 and w = 5 b. z = f( w, r) = 50w+ 150r, for w = 00 and r = 3 Linear Functions Functions can be either linear or nonlinear in form. This section of the Review discusses the properties of linear functions, and then the next section examines nonlinear functions. The primary distinction between linear and nonlinear functions is the nature of their slopes: linear functions have constant slopes while curvilinear (i.e., nonlinear) functions have varying slopes. DEFINITION OF LINEAR FUNCTIONS A function, y = f( x), is a linear function if a graph of all the combinations of x and y that satisfy the equation y = f( x) lie on a straight line. Any linear relation between y and x can be expressed in the algebraic form y = a+ bx where y is the dependent variable, x is an independent variable, a is the intercept parameter, and b is the slope parameter. The terms a and b are called parameters, rather than variables, because their values do not change along the graph of a specific linear function. The intercept parameter, a, gives the value of y when x = 0. For the line representing y = f( x), when x = 0 the line crosses the y-axis. Hence the name intercept parameter. 3 Every line is characterized by a unique pair of values for a and b. 3 Sometimes a is called the y-intercept. When the dependent variable is not named y, the term y-intercept can be confusing. It is best to think of the intercept parameter a as giving the point where the line for this function crosses the axis of the variable on the other side of the equal sign. If, for example, c = 30 5q, the line passes through the c axis at the value c = 30 ( = ). Also note that c need not be graphed on the vertical axis; 30 is where the line crosses the c-axis whether c is plotted on the vertical or on the horizontal axis. 3

4 Figure A.1 shows a graph of the golf pro s demand function which relates the number of lessons given each week (q) to the price charged for a lesson ( p). The values of the parameters of this linear function are a = 100 and b =. Notice that the line passes through the q axis at 100 lessons per week. The slope of the line is. The slope of lines and curves are so important that we will now discuss in detail the meaning of the slope of a line and the slope of a curve. FIGURE A.1: Demand function for golfing lessons SLOPE OF A LINE The slope of a line representing the function y = f( x) is defined as the change in the dependent variable y divided by the change in the independent variable x: y slope of a line = b = x where the symbol indicates the change in a variable. 4 When y is plotted on the vertical axis and x is plotted on the horizontal axis, the change is y can be called rise, the change in x can be called run, so the slope of the line is frequently referred to as rise over run. In Figure A.1, moving from the top of the line where it intersects the q axis down to the bottom of the line where it intersects the p axis, the rise is 100 ( = q ). The run is +50 ( p ), and so the slope is. The slope of a line can be either positive, negative, or zero. If x and y move in the same direction, x and y have the same algebraic sign. When an increase (decrease) in x causes an increase (decrease) in y, x and y are said to be directly or positively related. When x and y are directly or positively related, the slope of y = f( x) is positive (i.e., b > 0). The graph of a line is upward sloping when x and y are directly or positively related. See Panel A in Figure A. for an example of a direct or positive relation. Alternatively, x and y may move in opposite directions: an increase (decrease) in x causes a decrease (increase) in y. In this situation, x and y are said to be inversely or negatively related, and the slope of y = f(x) is negative (i.e., b < 0). The graph of a line is downward sloping when variables are inversely or negatively related. See Panel B in Figure A. for an example of an inverse or negative relation. 4 The change in a variable is calculated by taking the final value of the variable ( x 1 ) and subtracting the initial value ( x 0 ): x = x x. When a variable increases (decreases) in value, the final value is greater (less) than the 1 0 initial value of the variable, and the change in the variable is positive (negative). 4

5 In some situations, changes in x do not cause y to increase or to decrease. If changes in x have no effect on y, x and y are unrelated or independent. Suppose, for example, y = 50 for all values of x. The parameter values for this situation are a = 50 and b = 0. The graph of this linear function is a horizontal line at y = 50 (see Panel C of Figure A.). In general, when x and y are independent, the slope of the line is zero. FIGURE A.: The slope of a line CALCULATING THE SLOPE OF A LINE Calculating the slope of a line is straightforward. First, locate any two points on the line. Then calculate y and x between the two points. Finally, divide y by x. Consider, for example, points R and S in Panel A of Figure A.. Moving from R to S, y = 0 10 =+ 10 and x = 4 0=+ 4. Now divide: y x= =+.5. Since the slope is positive, the line is upward sloping and the variables y and x are directly (or positively) related. As discussed above, when variables are inversely (negatively) related, the slope of the line is negative. In Panel B of Figure A., consider moving from R to S. Note that y = =+ 80 because the movement from R to S is upward (by 80 units). Since moving from R to S is a leftward movement, the change in x is negative: x = 0 0= 0. Now divide the changes in y and x: y x= = 4. Moving along the line in Panel C, y is always zero for any change in x. Thus, the slope of the horizontal line is zero. INTERPRETING THE SLOPE OF A LINE AS A RATE OF CHANGE Students usually learn to calculate the slope of lines rather quickly. It is surprising, however, that many students who can correctly calculate the slope of a line cannot explain the meaning of slope or interpret the numerical value of slope. To understand the meaning and usefulness of slope in decision making, you must learn to think of slope as the rate of change in the dependent variable as the independent variable changes. Since the slope parameter is a ratio of two changes, y and x, slope can be interpreted as the rate of change in y per unit change in x. 5

6 In Panel A of Figure A., the slope of the line is.5 ( b= y x=+.5 ). This means that y changes.5 units for every one-unit change in x, and y and x move in the same direction (since b > 0). Thus, if x increases by units, y increases by 5 units. If x decreases by 6 units, then y decreases by 15 units. Y changes.5 times as much as x changes and in the same direction as x changes. In Panel B of Figure A., the slope of the line is 4 ( b= y x= 4 ). This means that y changes 4 units for every one unit x changes, and y and x move in opposite directions (since b < 0). Thus, if x increases by units, y decreases by 8 units. If x decreases by 6 units, then y increases by 4 units. Y changes 4 times as much as x changes but in the opposite direction of x. When a linear function has more than one independent variable, each independent variable has a slope parameter that gives the rate of change in y per unit change in that particular independent variable by itself, or holding other independent variables constant. For example, consider a linear function with three independent variables x, y, and z: w = f ( x, y, z) = a + bx + cy + dz The slope parameters b, c, and d give the impacts of one-unit changes in x, y, and z, respectively, on w. Each slope parameter measures the rate of change in w attributable to a change in a specific independent variable. For example, if w= 30 + x 3y+ 4z then each one unit increase in x causes w to increase by units, each one-unit increase in y causes w to decrease by 3 units, and each one-unit increase in z causes w to increase by 4 units. Exercises 4. Consider again the golf pro s demand for lessons, which is graphed in Figure A.1. Let the golf pro increase the price of her lessons from \$0 to \$5. a. The quantity of lessons given at a price of \$5 is lessons per week. b. The change in p is p =. c. The change in q is q =. d. The slope of the demand line is. e. Since a \$1 increase in price causes the quantity of lessons each week to by lessons per week, a \$5 increase in price causes the quantity of lessons per week to by lessons per week. 5. In Panel A of Figure A., change the coordinates of point S to x = 30 and y = 100. Let point R continue to be at x = 0 and y = 10. a. Moving from point R to the new point S, y = =. b. Moving from point R to the new point S, x = =. c. The slope of the line =. d. The equation of the new line is. e. Along the new line, y changes times as much as x changes and in the direction. f. If x increases by 4 units, y by units. g. The parameter values for the new line are a = and b =. 6

7 6. In Panel B of Figure A., change the coordinates of point R to x = 40 and y = 40. Let point S continue to be at x = 0 and y = 180. a. Moving from the new point R to point S, y = =. b. Moving from the new point R to point S, x = =. c. The slope of the line =. d. The equation of the new line is. e. Along the new line, y changes times as much as x changes and in the direction. f. If x increases by 5 units, y by units. g. The parameter values for the new line are a = and b =. 7. Let w= 30 + x 3y+ 4 z. a. Evaluate w at x = 10, y = 0, and z = 30. b. If x decreases by 1 unit, w by units. c. If y increases by 1 unit, w by units. d. If z increases by 3 units, w by units. e. Evaluate w at x = 9, y = 1, and z = 33. Compare this value of w to the value of w in part a: the change in w equals. f. Add the individual impacts on w in parts b, c, and d. Does the sum of the individual impacts equal the total change in w found in part e? Curvilinear Functions Linear functions are easy to use because the rate of change in the dependent variable as the independent variable changes is constant. For many relations, however, the rate of change in y as x changes is not constant. Functions for which the rate of change, or slope, varies are called curvilinear functions. As the name suggests, the graph of a curvilinear function is a curve rather than a straight line. While some curvilinear functions can be difficult to use, the curvilinear functions used in managerial economics are generally quite easy to use and interpret. Any relation between y and x that cannot be expressed algebraically in the form y = a+ bx is a curvilinear function. Examples of curvilinear functions include: x y = a+ bx+ cx, y = sin x, y = ln x, and y = e. We begin this section with a discussion of how to measure and interpret the slope of a curvilinear function. Then we examine the properties of polynomial functions and logarithmic functions. MEASURING SLOPE AT A POINT ON A CURVE The slope of a curve varies continuously with movements along the curve. It is useful in studying decision making to be able to measure the slope of a curve at any one point of interest along the curve. You may recall from high school algebra or a pre-calculus class in college that the slope at a point on a curve can be measured by constructing the tangent line to the curve at the point. 5 The slope of a curve at the point of tangency is equal to the slope of the tangent line itself. To illustrate how to measure the slope of a curve at a point, consider the curvilinear function y = f( x) = 800x 0.5x, which is graphed in Figure A.3. As x increases, the value of y increases (the curve is positively sloped), reaches its peak value at x = When a line is tangent to a curve, it touches the curve at only one point. For smooth, continuous curves, there is one and only one line that is tangent to a curve at any point on the curve. Consequently, the slope of a curve at a point is unique and equal to the slope of the tangent line. 7

8 (point M), and then decreases until y = 0 at x = 1,600. Let s find the slope of this curve at point A, which is the point x = 1,000 and y = 300,000. First, a tangent line labeled TT in the figure, is constructed at point A. 6 Then, the slope of TT is calculated to be 00 ( = 500,000/,500 ). The slope of the curve at point A indicates that y changes 00 times as much as x changes, and in the opposite direction because the slope is negative. FIGURE A.3: Measuring slope at a point on a curve y T Blow up at A 500, ,000 T 300,000 99, A + T 300,000 75,000 M A T 1,000 1,00 00,000 y = 800 x 0.5x 100, ,000 1,600,500 x It is important to remember that the slope of a curve at a point measures the rate of change in y at precisely that point. If x changes by more than an infinitesimal amount, which it usually does in practical applications, the slope of the tangent measures only approximately the rate of change in y relative to the change in x. However, the smaller the change in x, the more precisely the change in y can be approximated by the slope of the tangent line. To illustrate this rather subtle point, let x increase by units from 1,000 (at point A) to 1,00. Since the slope of the curve at point A is 00, the increase in x of units causes y to decrease by approximately 400 units. We say y decreases by approximately 400 units, because a -unit change in x, which is a rather small change when x is equal to 1,000, is still large enough to create a tiny amount of error. Since the change in x is quite small relative to the point of measure, the actual change in y will be quite close to 400 though not exactly 400. The blow up at point A in Figure A.3 confirms that the change in y is not exactly 400 but rather is 40. When the change in x is small, we must emphasize that the tiny error can be ignored for practical purposes. 6 A tangent line that you draw will only approximate an exact tangent line because your eyesight is probably not so sharp that you can draw precisely a line tangent to a curve. Fortunately, for the purposes of managerial economics, you only need to be able to sketch a line that is approximately tangent to a curve. If you have taken a course in calculus, you know that taking the derivative of a function and evaluating this derivative at a point is equivalent to constructing a tangent line and measuring its slope. 8

9 Exercises 8. Calculate the slope at point A on the following curves: y y y 70 3 T T A 15 T A A T 0 90 x x x 0 0 (a) ( b) (c) 9. In Figure A.3, find the slope of the curve at x = 500. [Hint: Use a ruler and a sharp pencil to draw the appropriate tangent line.] POLYNOMIAL FUNCTIONS One of the simplest kinds of curvilinear functions is a polynomial function. A polynomial function takes the form y= f x = a + ax+ ax + + ax ( ) where n is an integer and a 0, a 1, a,, a n are parameters. The highest power to which the independent variable x is raised is called the degree of the polynomial. Many of the curvilinear relations in managerial economics involve polynomial functions of degree or degree 3. A polynomial of degree 1 is a linear function, which we have discussed. When the relation between y and x is a polynomial of degree, the function is known as a quadratic function, which can be expressed as follows: y = a+ bx+ cx The graph of a quadratic function can be either U -shaped or I -shaped, depending on the algebraic signs of the parameters. Two of the most common quadratic functions in managerial economics have the properties shown in Panels A and B of Figure A.4. We now discuss the restrictions on the algebraic signs of the parameters associated with the U -shaped or I -shaped curves shown in Figure A.4. As shown in Figure A.4, when b is negative (positive) and c is positive (negative), the quadratic function is U - shaped ( I -shaped).7 Point M, which denotes either the minimum point for a U -shaped curve in Panel A or the maximum point for a I -shaped curve in Panel B, occurs at x = b/cin either case. As just mentioned, b and c are of opposite algebraic signs, thus x is positive when these curves reach either a minimum or a maximum. 8 n n 7 In Panel b, the constant term is zero (a = 0) because the curve passes through the origin. 8 Since economic variables generally take only positive values, an additional mathematical property, b < 4ac, ensures that point M occurs at a positive, rather than negative, value of y. We do not emphasize this property here or in the textbook because, as it turns out, the condition is always met in practice when the parameters of the quadratic equation are estimated using economic data for which x and y take only positive values. 9

10 FIGURE A.4: Two common quadratic functions A polynomial function of degree 3 is often referred to as a cubic function. The specific form of a cubic function used in managerial economics takes the form: 3 y = ax+ bx + cx Figure A.5 summarizes the shapes of two cubic functions that you will see later in this course. The graph of a cubic function is either S-shaped or reverse S-shaped, depending on the values of the parameters. In Panel A, the reverse S-shaped cubic function is characterized by parameters with algebraic signs a > 0, b < 0, and c > 0. In Panel B, the S- shaped cubic function has parameter restrictions b > 0 and c < 0. (Note that a = 0 in Panel B.) 9 9 For cubic equations, the condition b < 4ac ensures that the slope of the curve at the point of inflexion (I ) is upward (positive). As explained in footnote 8, this condition is not usually of much concern in practice. 10

11 FIGURE A.5: Two common cubic functions The inflexion point on an S-shaped or a reverse S-shaped curve is the point at which the slope of the curve reaches either its minimum value (Panel A) or its maximum value (Panel B). Consider Panel A. Beginning at zero and moving toward the point of inflexion (labeled I in both panels), the slope of the curve is decreasing. Then, beyond the inflexion point, the slope of the curve begins increasing. 10 Note that y is always increasing as x increases, but at first (over the range 0 to I ) y increases at a decreasing rate, then beyond I, y increases at an increasing rate. It follows that the slope is smallest at point I in Panel A. In Panel B, y first increases at an increasing rate, and then beyond the inflexion point, y increases at a decreasing rate. Thus, the rate of change in y is greatest at inflexion point I in Panel B. The point of inflexion I occurs at x = b/ 3cfor both S-shaped and reverse S-shaped curves. Exercises 10. Consider the function y = x x. a. Is this function U -shaped or I -shaped? How do you know? b. The dependent variable y reaches a (minimum, maximum) value when x =. c. The value of y at the minimum/maximum point found in part b is. 10 To see that slope is decreasing over the segment of the curve from 0 to I, visualize a series of lines tangent to the curve at points along the curve between 0 and I. Since these visualized tangent lines are getting flatter between 0 and I, the slope of the curve is getting smaller. Moving beyond I, the visualized tangent lines are getting steeper, so the slope of the curve is rising. 11

12 11. Consider the function y = 0.05x x. a. Is this function U -shaped or I -shaped? How do you know? b. The dependent variable y reaches a (minimum, maximum) value when x =. c. The value of y at the minimum/maximum point found in part b is. 1. In Panel A of Figure A.5, choose 5 points along the curve and sketch the tangent lines at the points. a. Visually verify that the tangent lines get flatter over the portion of the curve from 0 to I. b. Visually verify that the tangent lines get steeper over the portion of the curve beyond I. c. At point I, is the slope of the curve positive, negative, or zero? How can you tell? Consider the function y = 10x 0.03x x. a. Is this function S-shaped or reverse S-shaped? How do you know? b. What is the value of the y-intercept? c. The inflexion point occurs at x =. At the inflexion point, the slope of the curve reaches its (maximum, minimum) value. d. What is the value of y when x = 1,000? EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS We now discuss two more curvilinear functions, exponential functions and natural logarithmic functions, which have mathematical properties that can be quite useful in many applications arising in managerial economics and finance. Exponential Functions An exponential function takes the form y = f( x) = a where a is any positive constant (called the base of the exponent), and the independent variable x is the power to which the base is raised. Exponential functions differ from 3 polynomial functions, such as y = ax+ bx + cx, because x is an exponent in exponential functions but is a base in polynomial functions. Exponential functions have numerous algebraic properties that are quite helpful in applications you will see later in this course. x n 1. a = a a... a ( ntimes) 4. i j i j aa = a + 0. a = 1 5. ( a i ) j = a ij i n 1 a i j 3. a = 6. = a n j a a A commonly used base in economics and finance applications is the constant e, 1 which is the limiting value of the expression (1 + ) n, as n gets very large (i.e., approaches 1 n

13 infinity): 1 e = lim n n Nearly all hand calculators have a key labeled e or e x that enters the value into the calculator s display for exponential computations. Verify that you can use this 4 key on your calculator by making the calculation e = Natural Logarithms Natural logarithms are logarithms for which e is the base x y= e where x is the power to which e must be raised to get y. For example, e must be raised to the power of 4 to obtain the number Thus, the natural logarithm of is 4. In general, the natural logarithm of any positive number y can be expressed as x = ln y The symbol ln is used to distinguish this base e logarithm from the base 10 logarithm that is used in some scientific applications. 11 Notice that the natural logarithmic function x = ln y is the inverse of the exponential function y = e x. Figure A.6 illustrates the inverse nature of the two functions. Natural logarithms have the following convenient algebraic properties that can be quite useful: 1. ln rs= ln r+ ln s. s ln r = sln r 3. ln r = ln r ln s s FIGURE A.6: The inverse relation between exponential and natural logarithmic functions n 11 The natural logarithm function key on your hand calculator is labeled ln or ln x, whereas the key for base 10 logarithms is usually labeled log or log10. You will always use the ln or ln x key in this Student Workbook (and Textbook). 13

14 Exercises 14. Evaluate the following: a. b. 0 5 e. 3 f. a b c. ( K ) g. 1 e = lim 1 + n n n i. ln1,000 ax 4 e b j. ln ( ) 3 e k. ln ( K L ) d. a+ b b L L h. ln8 l. 0 e Finding Points of Intersection You may recall from your algebra courses that the point at which two lines intersect can be found mathematically by solving two equations containing the two variables. Similarly, the two points where a straight line crosses a quadratic function can be found using the quadratic formula. While you no doubt hoped to avoid seeing these techniques again, you will see that finding points of intersection plays a crucial role in solving many business decision-making problems. We now review this important algebraic skill. FINDING THE INTERSECTION OF TWO LINES Consider two lines represented by the two linear equations y= a+ bx y= c+ dx The point at which these two lines intersect assuming they are not parallel (b = d ) can be found algebraically by recalling that the values of y and x at the point of intersection solve both equations. Setting the two equations equal to each other and solving for x provides the value of x at the point of intersection ( x ) a c x = d b To find the value of y at the point of intersection, x is substituted into either linear equation to find y ad bc y = d b Figure A.8 illustrates how to find the point of intersection of the two linear equations: y = 10 xand y = + x. In this example, a = 10, b = 1, c =, and d = 1. Substituting these parameter values into the above formulas, the solution is found to be x = 4 and y = 6. Verify that the point x = 4, y = 6 solves both equations. 14

15 FIGURE A.8: Finding the intersection of two lines FINDING THE INTERSECTION OF A LINE AND A QUADRATIC CURVE A line crosses a quadratic curve either a U -shaped or a I -shaped curve at two points. Figure A.9 shows the linear function y = x, as well as the quadratic function y = x x. As in the case of intersecting lines, the values of y and x at the points of intersection between a line and a quadratic curve solve both equations. In general, to find the points of intersection, the equations for the line and the quadratic curve are set equal to each other and then solved for the two values of x at which the line and the curve cross. To illustrate this technique, set the equation for the line, y = x, equal to the equation for the curve, y = x x x = x x Now there is one variable in one equation, and the equation is a quadratic equation. To solve a quadratic equation, the quadratic equation must be expressed in the form 0 = A+ Bx+ Cx where A is the constant term, B is the coefficient on the linear term, and C is the coefficient on the quadratic term. After setting expressions equal, terms are rearranged to get zero on one side of the equality: 0 = x x It is important to note that A and B are not equal to the parameter values a and b of the quadratic curve. In this problem, for example, a = 0 and b = 0.14, but A = 36and B = 0.1. The solution to the quadratic equation 0 = A+ Bx+ Cx is the familiar quadratic formula: B ± B 4AC x1, x = C Substituting the values of A, B, and C into the quadratic formula yields the two solutions for x: x 1 = 00 and x =

16 FIGURE A.9: Finding the intersection of a line and a quadratic curve Exercises 15. Find the point of intersection of the following two lines: y = 1,000 0x and y = 5 + 5x 16. Consider the following linear function and quadratic function: y = 3 0.0x y = 50 0.x x The line intersects the quadratic curve at two points: x 1 =, y 1 = and x =, y =. 16

17 ANSWERS TO EXERCISES 1. a. y = 100 ( = ) b. y = 140 ( = ) c. w = 10 ( = ) d. s = 40 ( = ). a. x = f( y) = y b. p = f( q) = q 3. a. y = 650 ( = ) b. z = 14,800 ( = 10, ,800) 4. a. 50 ( = 100 5) b. + 5( = 5 0) c. 10 ( = 50 60) d. ( = q p = ) e. decrease; ; decrease; a. y = =+ 90 b. x = 30 0 =+ 30 c. + 3 ( = y x = ) d. y = x e. 3; same f. increases; 1 g. a = 10; b = 3 6. a. y = = b. x = 0 40= 40 c. 3.5 ( = y x = ) d. y = x e. 3.5; opposite f. decreases; 17.5 g. a = 180; b= a. 110 ( = ) b. decreases; c. decreases; 3 d. increases; 1 e. +7; because 30+( 9)+( 3 1)+(4 33)=117, which is 7 units greater than the value of w in part a. f. ( x )+( y 3)+( z 4) = = +7; yes, w = a. slope at point A = slope of TT = 70 / 90 = 8 b. slope at point A = slope of TT = 15/ 3 = + 5 c. slope at point A = slope of tangent = 0 a the minimum point 9. When x = 500, y = 75,000 (confirm this with the equation). Construct a tangent line at the dot on the curve at x = 500 and y = 75,000, and extend the tangent line to cross the y-axis. If precisely drawn, the tangent line crosses the y-axis at 15,000. If your tangent line is precisely constructed, the slope of the tangent line is [ = (75, , 000) / 500]. Be satisfied if you came close to +300; calculus is required to get a precisely accurate measure of slope. 10. a. U -shaped; The polynomial is a quadratic and the parameters have the sign pattern associated with a U -shaped curve: a > 0, b < 0, c > 0. b. minimum; 300 ( = b/ c = 0.03/ ) c. 5.5 [ = (300) (300) ] 17

18 11. a. I -shaped; The polynomial is a quadratic and the parameters have the sign pattern associated with a I -shaped curve:, b > 0, c < 0. b. maximum; 9 ( = b/ c = 1.45/ 0.05) c [ = 0.05(9) (9)] 1. The figure shows the sketched tangent lines at five additional points along the curve in Panel A of Figure A.5. a. You can verify visually that the three tangent lines between 0 and I get flatter as x increases. b. You can verify visually that the two tangent lines to the right of point I get steeper as x increases. c. At point I, the slope is positive because the sketched tangent line at point I slopes upward. 13. a. Reverse S-shaped, as in Panel A of Figure A.5. The parameters have the signs a > 0, b < 0, and c > 0. b. 0 c. 00; minimum d. 3 30,000 [ = 10(1,000) 0.03(1,000) (1,000) ] 14. a. 1 b = = + c. ab K d. L a = L a + b b e = e f g h i j. ln a+ bln x k. ln K ln L l. 0 1, because a = x = (1,000 5) / (5 ( 0)) = 39 and y = 1,000 (0 39) = 0 = 5 + (5 39) 16. Solve the quadratic equation, 0 = x x, to get two solutions for x: x 1 = 600, y 1 = 0 x = 10, y =

### Elements of a graph. Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on

### Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

### 3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

### Week 1: Functions and Equations

Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.1-2.2, and Chapter

### Section 1.1 Linear Equations: Slope and Equations of Lines

Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

### Understanding Basic Calculus

Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other

### Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

### Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS

DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding

### Elasticity. I. What is Elasticity?

Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### x 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1

Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only

Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials

### JUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson

JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials

9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### 2013 MBA Jump Start Program

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 Equation of

### The degree of a polynomial function is equal to the highest exponent found on the independent variables.

DETAILED SOLUTIONS AND CONCEPTS - POLYNOMIAL FUNCTIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE NOTE

### Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

### EL-9650/9600c/9450/9400 Handbook Vol. 1

Graphing Calculator EL-9650/9600c/9450/9400 Handbook Vol. Algebra EL-9650 EL-9450 Contents. Linear Equations - Slope and Intercept of Linear Equations -2 Parallel and Perpendicular Lines 2. Quadratic Equations

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Polynomial and Rational Functions

Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving

### Some Lecture Notes and In-Class Examples for Pre-Calculus:

Some Lecture Notes and In-Class Examples for Pre-Calculus: Section.7 Definition of a Quadratic Inequality A quadratic inequality is any inequality that can be put in one of the forms ax + bx + c < 0 ax

### Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.

Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

### 100. In general, we can define this as if b x = a then x = log b

Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,

### 6.4 Normal Distribution

Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

### Chapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs

Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s

### What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### Graphing Linear Equations

Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

### Chapter 4 Online Appendix: The Mathematics of Utility Functions

Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

### Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

### www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

### The Point-Slope Form

7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

### Graphical Integration Exercises Part Four: Reverse Graphical Integration

D-4603 1 Graphical Integration Exercises Part Four: Reverse Graphical Integration Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Laughton Stanley

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

Introduction to Quadratic Functions The St. Louis Gateway Arch was constructed from 1963 to 1965. It cost 13 million dollars to build..1 Up and Down or Down and Up Exploring Quadratic Functions...617.2

### List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

### EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

### Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0

College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

### MAT12X Intermediate Algebra

MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### PLOTTING DATA AND INTERPRETING GRAPHS

PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### 10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED

CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### https://williamshartunionca.springboardonline.org/ebook/book/27e8f1b87a1c4555a1212b...

of 19 9/2/2014 12:09 PM Answers Teacher Copy Plan Pacing: 1 class period Chunking the Lesson Example A #1 Example B Example C #2 Check Your Understanding Lesson Practice Teach Bell-Ringer Activity Students

### ALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals

ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an

### Algebra and Geometry Review (61 topics, no due date)

Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties

### PARABOLAS AND THEIR FEATURES

STANDARD FORM PARABOLAS AND THEIR FEATURES If a! 0, the equation y = ax 2 + bx + c is the standard form of a quadratic function and its graph is a parabola. If a > 0, the parabola opens upward and the

### 1.7 Graphs of Functions

64 Relations and Functions 1.7 Graphs of Functions In Section 1.4 we defined a function as a special type of relation; one in which each x-coordinate was matched with only one y-coordinate. We spent most

### TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

### 1 Mathematical Models of Cost, Revenue and Profit

Section 1.: Mathematical Modeling Math 14 Business Mathematics II Minh Kha Goals: to understand what a mathematical model is, and some of its examples in business. Definition 0.1. Mathematical Modeling

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

### CURVE FITTING LEAST SQUARES APPROXIMATION

CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

### http://www.aleks.com Access Code: RVAE4-EGKVN Financial Aid Code: 6A9DB-DEE3B-74F51-57304

MATH 1340.04 College Algebra Location: MAGC 2.202 Meeting day(s): TR 7:45a 9:00a, Instructor Information Name: Virgil Pierce Email: piercevu@utpa.edu Phone: 665.3535 Teaching Assistant Name: Indalecio

### 1 Solving LPs: The Simplex Algorithm of George Dantzig

Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### Linear Programming. Solving LP Models Using MS Excel, 18

SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

### 7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

### Law of Demand: Other things equal, price and the quantity demanded are inversely related.

SUPPLY AND DEMAND Law of Demand: Other things equal, price and the quantity demanded are inversely related. Every term is important -- 1. Other things equal means that other factors that affect demand

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

### IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

### Objectives. Materials

Activity 4 Objectives Understand what a slope field represents in terms of Create a slope field for a given differential equation Materials TI-84 Plus / TI-83 Plus Graph paper Introduction One of the ways

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### REVIEW OF ANALYTIC GEOMETRY

REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.

### Part 1: Background - Graphing

Department of Physics and Geology Graphing Astronomy 1401 Equipment Needed Qty Computer with Data Studio Software 1 1.1 Graphing Part 1: Background - Graphing In science it is very important to find and

### BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES.

BEST METHODS FOR SOLVING QUADRATIC INEQUALITIES. I. GENERALITIES There are 3 common methods to solve quadratic inequalities. Therefore, students sometimes are confused to select the fastest and the best

### Graphing calculators Transparencies (optional)

What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used

### SPECIAL PRODUCTS AND FACTORS

CHAPTER 442 11 CHAPTER TABLE OF CONTENTS 11-1 Factors and Factoring 11-2 Common Monomial Factors 11-3 The Square of a Monomial 11-4 Multiplying the Sum and the Difference of Two Terms 11-5 Factoring the

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### 3.3 Real Zeros of Polynomials

3.3 Real Zeros of Polynomials 69 3.3 Real Zeros of Polynomials In Section 3., we found that we can use synthetic division to determine if a given real number is a zero of a polynomial function. This section

### 1) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-1) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = 7) (5)(-4) = 8) (-3)(-6) = 9) (-1)(2) =

Extra Practice for Lesson Add or subtract. ) (-3) + (-6) = 2) (2) + (-5) = 3) (-7) + (-) = 4) (-3) - (-6) = 5) (+2) - (+5) = 6) (-7) - (-4) = Multiply. 7) (5)(-4) = 8) (-3)(-6) = 9) (-)(2) = Division is

### CHAPTER 1 Linear Equations

CHAPTER 1 Linear Equations 1.1. Lines The rectangular coordinate system is also called the Cartesian plane. It is formed by two real number lines, the horizontal axis or x-axis, and the vertical axis or

### M 1310 4.1 Polynomial Functions 1

M 1310 4.1 Polynomial Functions 1 Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a, a,..., a, a, a n n1 2 1 0, be real numbers, with a

### 5 Systems of Equations

Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

### chapter Behind the Supply Curve: >> Inputs and Costs Section 2: Two Key Concepts: Marginal Cost and Average Cost

chapter 8 Behind the Supply Curve: >> Inputs and Costs Section 2: Two Key Concepts: Marginal Cost and Average Cost We ve just seen how to derive a firm s total cost curve from its production function.

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method