...A polynomial is a mathematical expression consisting of a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. The simplest polynomials have one variable but they can have two three or more variables. Knowing the formula and how to use and brake down polynomial can help us in our everyday lives. It can help you figure out how much interest you can accrue from a deposit or investment that you made with a given interest rate. Let’s take a closer look at an equation that deals with polynomials. “P dollars is invested at annual interest rate r for 1 year. If the interest is compounded annually then the polynomial P(1 + r)2 represents the value of the investment after 1year” (Dugopolski, 2012). P(1 + r/2)2 Original expression P(1 + r/2)(1 + r/2) Simplify the expression by using foil this means multiply first outer, Inner, last P(1+ r/2 + r/2 + r2/4) Combine the like terms P(1 + 2(r/2) + r2/4) Distribute P across the trinomial P + Pr + Pr2/4 Put all variables in descending order Now let’s try our formula with a given set of numerical information P=200 r =10% interest rate .10 as a decimal. P + Pr + Pr2/4 expanded formula 200 + 2/4(200)(.10) + 200(.10)2 substitute values into formula 200 + 10 + 200(.01) multiply 200 + 10 + 2 ...
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...COURSE: MATHEMATICS I (MTH 101) SYLLABUS I. Basic algebra: Linear and quadratic equations, Solving linear and quadratic equations, Application of equations: profit, pricing, savings, revenue, sales tax, investment, bond redemption, linear inequalities, applications of inequalities: profit, renting verses purchasing, leasing versus purchasing, revenue, current ratio, investment, Maple session on solving linear, quadratic and higher degree equations, solving inequalities II. Functions and Graphs: Introduction to functions, domain and range of a function, Applications: demand, supply and profit functions, demand and supply schedule, value of business, depreciation, Special functions: polynomial, rational, piecewise defined functions, Absolute value function, and evaluation of such functions. Combination of functions. Applications: cost, investment, sales, profit, business, Graphs of functions: linear, quadratic, piecewise defined functions, graphing of quadratic functions by finding vertex, Applications on graphs: inventory, debt payment, pricing, revenue and profit, demand and supply curves, Maple session on functions and graphs III. Lines and Systems: Equation of a straight line, slope and intercept of a line, parallel and perpendicular lines, Applications: price-quantity relationship, production levels, cost, revenue, demand and supply equations, isocost line, isoprofit line, depreciation, appreciation, systems of linear equations, solution of system...
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...Complex Analysis Complex Numbers: A complex number z is an ordered pair (x,y) of real numbers x and y. z = (x,y) = x + iy The Real Part of z ie.Re(z) = x and the Imaginery part of z ie. Im(z) = y. Moreover,i2 = -1 which is an imaginery unit. a. The two imaginery numbers x + iy and a + ib are equal iff x = a and y =b, b. For z = x + iy, if x = 0,then z = iy (A pure imaginery number) and if y = 0 then z = x ( Pure real number). If z1 = x1 + iy1 and z2 = x2 + iy2, then the Addition, Multiplication, Subtraction and Division of two complex numbers respectively is defined as follows: z1 + z2 = (x1 + x2) + i(y1 + y2) z1 z2 = (x1 x2 - y1 y2) + i(x1 y2 + x2 y1) z1 − z2 = (x1 - x2) + i(y1 - y2) z = x/y = x + iy,where x = , y = ,z2 ≠ 0. Complex Conjugate Number The complex conjugate of the number z = x +iy is = x-iy Re(z) = x = (z + ) and Im(z) = (z - ) When z is real, z = x then z = Polar Form of Complex Numbers Let (x,y) be the Cartesian coordinates and (r,Ө) be the polar coordinates,then x = r cos Ө , y = r sin Ө Therefore, z = x+iy = r (cos Ө+ isin Ө) r = which is the absolute value or the modulus of z. Ө = arg z = tan which is the argument of z. Important Properties Generalized Triangle Inequality : Let Then, De Moivre’s formula : Nth Root of z : Limit, Continuity and Derivatives of Function of Complex variable: Limit :...
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...Systems of Linear Equations in Three Variables Answer the following questions to complete this lab. Show all of your work for each question to get full credit. 1. Solve the following system of equations: a. x –2y+z=6 b. 2x+y –3z= –3 c. x –3y+3z=10 Add equation b and c: 2X+Y-3Z=-3 +X +3Y+3Z=10= 3X-2Y=7 Add equation a and c, attempting to cancel out variable c. (multiply equation a by -3. -3(x-2Y+Z=6): -3X+6Y-3Z=-18 + X-3Y+3Z=10 = -2X+3Y=-8 Add the new equations together to isolate one variable. Multiple each equation by the necessary coefficient to cancel out a variable. 3(3X-2Y=7) =9X-6Y=21 and 2(-2X+3Y=-8)=-4X+6Y=-16 add together 5X=5. X=1 Back substitute X=1 into one of the two variable equations. -2(1)+3Y=-8, 3Y=-6. Y=-2 Back substitute Y=-2 and X=1 into one of the original three variable equations. 2X+Y-3Z=-3 , 2(1)+(-2)-3Z=-3, -3Z=-3 Z=1 Now check you variables by plugging into one of the original equations and making sure statement is true. X-2Y+Z=6. (1)-2(-2)+(1)=6 True. 2. The opening night of a theater sold a total of 1,000 tickets. The front orchestra area cost $80 a seat, the back orchestra area cost $60 a seat, and the balcony area cost $50 a seat. Total revenue from ticket sales for the night was $62,800. The combined number of tickets sold for the front and back orchestra seats was equal to the number of balcony...
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...The Production Function for Wilson Company By using the EViews software, we get the result below by using Least Square method: Dependent Variable: Y | | | Method: Least Squares | | | Date: 06/18/12 Time: 03:24 | | Sample: 1 15 | | | Included observations: 15 | | | | | | | | | | | | Variable | Coefficient | Std. Error | t-Statistic | Prob. | | | | | | | | | | | C | -130.0086 | 129.8538 | -1.001192 | 0.3365 | L | 0.359450 | 0.245593 | 1.463601 | 0.1690 | K | 0.027607 | 0.006051 | 4.562114 | 0.0007 | | | | | | | | | | | R-squared | 0.838938 | Mean dependent var | 640.3800 | Adjusted R-squared | 0.812094 | S.D. dependent var | 227.9139 | S.E. of regression | 98.79645 | Akaike info criterion | 12.20086 | Sum squared resid | 117128.9 | Schwarz criterion | 12.34247 | Log likelihood | -88.50643 | F-statistic | 31.25263 | Durbin-Watson stat | 1.458880 | Prob(F-statistic) | 0.000017 | | | | | | | | | | | 1. In standard form the estimated Cobb-Douglas equation is written as: Q= α Lβ1 Kβ2 The multiplicative exponential Cobb-Douglass Function can be estimated as a linear regression relation by taking logarithm: Log Q = log α + β1 log L + β2 log K Therefore: log(y) = -130.0086 + 0.359450*log(L) + 0.027607*log(K) The output elasticity of capital is 0.027607 and the output elasticity of labor is 0.359450. 2...
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...border) - indicator terminal (thin border) Data Flow Programming - block diagram does NOT execute left to right. - nodes execute when data is available to all inputs Block Diagram Colors 1. Number – Orange 2. Boolean – Green 3. Integer – Blue 4. String – Pink Wires 1. Scalar – Thin 2. 1D Array – Thick 3. 2D Array – Double Chapter 3, 4 , 5 – MatLab Basics Matrices Array - A= [ 2 , 4 , 5 ; 3 , 16 , 7] ; Array operations (element by element) 1. Scalar – Array Addition A+b 2. Scalar – Array Subtraction A-b 3. Scalar – Array Multiplication A.*B 4. Scalar – Array Division A./B Polynomial Functions > roots(x) roots of a polynomial >poly(x) coefficient of a polynomial whose roots are specified by the array x. > polyval(a,x) evaluates a polynomial at specified values of its independent variable x Format Command > format short – 4 digits after decimal > format long – 14 digits after decimal > format short e > 5 digits plus exponent > format long e > 16 digits plus exponent > format bank > 2 decimal digits > format rat > approximate ratio X-Y Plotting > plot(x,y) – plots any pair of vectors x and y versus eachother (vectors must be same length) Multiple X-Y plots >...
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...SOME PROBLEMS IN CONFORMAL MAPPING D. C. SPENCER 1. Introduction. Attention will be confined to a group of problems centering around so-called schlicht functions—that is, functions regular in a given domain and assuming no value there more than once. The type of problem we consider involves determination of precise bounds for certain quantities depending on the function/, as ƒ ranges over the schlicht functions in question. Since, for suitable normalization of the functions at some fixed point of the domain, the resulting family of functions is compact or normal, the extremal schlicht functions always exist and the problem is to characterize them. Interest was focused on this category of questions by the work of Koebe in the years 1907-1909, who established for the family of funct i o n s / o f the form ƒ(z) = z+a2Z2+aszz+ • • • , schlicht and regular in \z\ < 1 , a series of properties, among them the theorem of distortion bearing Koebe's name. This theorem asserts the existence of bounds for the absolute value of the derivative ƒ'(s), these bounds depending only on \z\. Further efforts were directed toward finding the precise values of the bounds asserted by Koebe's theorem, but success was not attained until 1916 when Bieberbach, Faber, Pick and others gave a final form to the theorem of distortion. At the same time the precise bound for | a2\ was given, namely 2, and the now famous conjecture was made that \an\ ^n for every n. Since 1916 this group of problems has attracted...
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...Kierra O’Neal Dr.Fletcher P12 11:00-12:20 October 15, 2015 Solve the equation. * Example 1: 5x+4=3x-8 Step one: Move the smallest variable to one side and subtract * 2x+4= -8 Step two: Move the constant term to the other side and subtract * 2x= -12 Step three: Divide by 2 to get what “X” is * X= -6 * Example 2: 6(x-1)+4= 3(7x+1) Step one: Distribute the 6 and the 3 * 6x-6+4= 21x+3 Step two: Combine like terms * 6x-2= 21x+3 Step three: Move smaller variable to one side and subtract * -2= 15x+3 Step four: Move the three to the other side * -5= 15x Step five: Divide by 15 * -5/15= x Step six: ALWAYS simplify * -1/3= x * Example 3: 7x/9-12= x Step one: Multiply each side by 9 * 7x-108=9x Step two: Add 108 to both sides * 7x= 9x+108 Step three: Subtract 9x from both sides * -2x=108 Step four: Divide by negative 2 to find what “X” is * X= -54 Solve and graph the inequality. Write the solution set in interval notation. * Example 1: 7x-9 > 6x-12 Step one: Move the smallest variable to one side and subtract * X-9 > -12 Step two: Move the constant term to other side and add * X > -3 Step three: Write in interval notation * [-3, infinity) Step four: Graph * Example 2: 2x+8 > 10x-14 Step one: Subtract 8 from both sides * 2x > 10x-22 Step two: Subtract 10x from both sides * -8x > -22 Step three: Multiply both sides by -1 (reverse...
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...上海大学 20 4 ～ 2015 学年 秋季学期研究生课程考试 课程名称： Econometrics 论文题目（Title）: 研究生姓名(Name): Hoang Thi Lan Huong 学号(ID): 14760006 研究生班级： 留学生班 成 绩: 任课教师: 叶明确 评阅日期: 1. Introduction (Background and Purpose) The data use for this paper was constructed in the following way. The data are drawn from the OPE Campus Safety and Security Statistics website database to which crime statistics (as of the 2010 data collection) are submitted annually, via a web-based data collection, by all postsecondary institutions that receive Title IV funding (i.e., those that participate in federal student aid programs). This data collection is required by the Jeanne Clery Disclosure of Campus Security Policy and Campus Crime Statistics Act and the Higher Education Opportunity Act. The outcome of this process was that the enrolments is not influenced significantly by the price or living conditions of the campus. When reading some articles I was personally very surprised about this since normally the living conditions should be influent more than the other factors in the enrolments. Based on this, I was very interested which other factors influence to students’ decisions. That is why I chose this dataset to analyze for the final term paper in the course of econometrics. 2. Multiple regressions 2.1 Data source The numerous data used in this paper (reference see below), consists of observations on six variables. The variables are: • enroll ...
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...Algebra I Quarter 3 Exam Name/Student Number:__________________________ Score:_______/________ Directions: For each question show all work that is required to arrive at the solution. Save this document with your answers and submit as an attachment to be graded. Simplify each expression. Use positive exponents. 1. m3n–6p0 2. a 4 b 3 ab 2 3. (x–2y–4x3) –2 4. Write the explicit formula that represents the geometric sequence -2, 8, -32, 128 5. Evaluate the function f (x) 4 • 7x for x 1 and x = 2. Show your work. 6. Simplify the quotient 4.5 x 103 9 x 107 . Write your answer in scientific notation. Show your work. Simplify the expressions. Show your work. 7. 3x(4x4 – 5x) 8. (5x4 – 3x3 + 6x) – ( 3x3 + 11x2 – 8x) 9. (x – 2) (3x-4) 10. (x + 6)2 Factor each expression. Show your work. 11. r2 + 12r + 27 12. g2 – 9 13. 2p3 + 6p2 + 3p + 9 Solve each quadratic equation. Show your work. 14. (2x – 1)(x + 7) = 0 15. x2 + 3x = 10 16. 4x2 = 100 17. Find the roots of the quadratic equation x2 – 8x = 9 by completing the square. Show your work. 18. Use the discriminant to find the number of real solutions of the equation 3x2 – 5x + 4 = 0. Show your work. A water balloon is tossed into the air with an upward velocity of 25 ft/s. Its height h(t) in ft after t seconds is given by the function h(t) = − 16t2 + 25t + 3. Show your work. 19. After how many seconds will the balloon hit the ground? (hint: Use the...
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...Week 7 DQ The best way to determine the number of solutions a quadratic equation has is to graph the equation. The number of times the equation crosses the x-axis is the number of real solutions the equation will have. If you are given the solutions (roots) p and q of a quadratic equation then you can find the equation by plugging p and q into the formula (x-p)(x-q) = 0. It is possible for two quadratic equations to have the same solutions. An example of that would be -x^2 + 4 and x^2 - 4. Each of these equations have roots at x = -2 and 2 Example: x^6-3 Example: -x^6+3 Week 7 DQ Quadratic formula: In my opinion, this is likely the best overall. It will always work, and if you memorize the formula, there is no guessing about how to apply it. The formula allows you to find real and complex solutions. Graphing: graphing the equation will only give valid results if the equation has real solutions. The solutions are located where the graph crosses the x axis. If the solutions are irrational or fractions with large denominators, this method will only be able to approximate the solutions. If you have a graphing calculator, this method is the quickest. If you don't have a calculator, it can be difficult to graph the equation. Completing the square: This is probably the most difficult method. I find it hardest to remember how to apply this method. Since the quadratic formula was derived from this method, I don't think there is a good reason to use completing the square when...
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...expression. Give the exact answer. 4. Solve the following logarithmic equation: log ( x + 16) = log x + log 16 Reject any value of x that is not in the domain of the original logarithmic expression. Give the exact answer. 5. The population of the world has grown rapidly during the past century. As a result, heavy demands have been made on the world's resources. Exponential functions and equations are often used to model this rapid growth, and logarithms are used to model slower growth. The formula 0.0547 16.6 t Ae models the population of a US state, A , in millions, t years after 2000. a. What was the population in 2000? b. When will the population of the state reach 23.3 million? 6. The goal of our financial security depends on understanding how money in savings accounts grows in remarkable ways as a result of compound interest. Compound interest is computed on your original investment as well as on any accumulated interest. Complete the table for a savings account subject to four compounding periods yearly. Use the following formula to solve this problem: 1 nt r AP n MA131 0 : Module 2 Exponential a nd Logarithmic Functions...
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... Feb. 11, 2015 I. Objectives a. Define polynomials b. Classify polynomials according to number c. Perform operation involving polynomials, addition and subtraction. II. Subject Matter Topic: Polynomials Materials: visuals, chalk and board Reference/s: College Algebra, pages 1-4 III. Presentation A. Preliminaries 1. Opening prayer 2. Checking of Attendance 3. Checking of Assignments 4. Motivation (Video clip) B. Lesson Proper Teacher’s Activity In the video that you’ve watched, what will be our lesson for today? Yes, George? Yes it has. Anyone can tell what it is? Yes, Fred. Very good. Can someone tell what is a polynomial is? Yes Claire. Yes, very well said, any additional information? Ok, so let’s proceed. So terms that different only in their constant coefficients are called “like terms”. Polynomials and algebraic expressions can be classified(according to the number of term) as; Monomial – having one term Binomial – having two terms Trinomials – having three termsMultinomials – having more than three terms. The degree of a polyomial is determined by the hiegst exponent of its variable. Someone give me an examples of polynomials. (called several students). Thank you for your answers. Is there any question? Oko next is adding and subtracting polynomials: Rule:Add/subtract the constant coefficients of like termsEx. Add 5x3 - 7x2 + 10x – 4 and ...
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...MAT 116 COMPLETE CLASS To purchase this visit here: http://www.activitymode.com/product/mat-116-complete-class/ Contact us at: SUPPORT@ACTIVITYMODE.COM MAT 116 COMPLETE CLASS MAT 116 FINAL EXAM Week 2: Exercise: Week Two Concept Check Post your 50-word response to the following: How do you know when an equation has infinitely many solutions? How do you know when an equation has no solution? Assignment: Expressions and Equations Complete Appendix C to apply the skills learned in Ch. 1 and sections 2.1–2.6 of Ch. 2 to a real-life situation. Use Equation Editor® to write mathematical equations and expressions in Appendix C. Week 4: Exercise: Week Four Concept Check Post your 50-word response to the following: Explain in your own words why the line x = 4 is a vertical line. Assignment: Solving Inequalities and Graphing Equations Complete Appendix D to apply skills learned in Ch. 2 & 3 to a real-life situation. Use Equation Editor® to write mathematical equations and expression in Appendix D. Week 6: Exercise: Week Six Concept Check Post your 50-word response to the following: How can you determine if two lines are perpendicular? Assignment: Functions and Their Graphs Complete Appendix E to apply the skills learned in Ch. 7 to a real-life situation. Use Equation Editor® to write mathematical expressions and equations in Appendix E. MAT 116 COMPLETE CLASS To purchase this visit here: http://www.activitymode.com/product/mat-116-complete-class/ ...
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...university of phoenix | Correlation Paper | | | Amber Kluever | 2/29/2016 | | Correlation is a measure of association that tests whether a relationship exists between two variables. It indicates both the strength of the association and its direction, direct or inverse. I am trying to find out if there is a relationship between A. PTSD and B. AODA, whether there is a relationship between the two depends on the strengths between them. Each method views variables not in isolation, but instead as systematically and meaningfully associated with, or related to, other variables. For example, using correlation coefficient which indicate the strength of association between two variables the (X,Y) it also describes correlation that reflects mutual relations of r of 1.0 (positive or negative) indicates an perfect linear relation, while 0 indicates that neither X or Y can be predicted by a linear equation. In these types of cases when the r is positive then there is an increase in both X and Y. Now if the r is negative it’s an increase in just the X and a decrease in the Y. Another method that is commonly used is the dichotomous variable also knows as the discrete variable which has two separate parts. An example of this would be measuring sound waves but having to measure in two different parts one high and the other low. The advantages of correlation are that it is used for research to be carried out, either by using experiments or taking surveys. The major advantage...
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