Preparatory Math Courses

Transcription

1 Preparatory Math Courses For 1 st Semester ISP Students, Faculty of Economics and Management at Otto-von-Guericke University Magdeburg Lecturer: Mr. Alexandr Polujan Tutors: Ms. Juliane Selle and Ms. Mozhdeh Mousazadehkamdar Duration of the Course: Contents Brief Description of The Course 2 1 Real Numbers Structure of Real Numbers Properties of Real Numbers Computations over Reals Absolute Value Computations with Power and Roots Computations with Logarithms Equations Linear Equations Quadratic Equations Root Equations Logarithmic and Exponential Equations Inequalities Set Theory Terminology Linear Inequalities Quadratic Inequalities General Approach. The Method of Intervals Rational Inequalities with Absolute Values Other Inequalities Text Problems 7

2 Brief Description of The Course The main goal of the course is to remember how to solve certain classes of equations and inequalities. The structure of the course is the following: (C 1 ) The rst chapter is dedicated to atomic objects real numbers (or briey reals): here we will explain the structure of reals and describe its basic properties; (C 2 ) In the second chapter will talk about mathematical terms, which include real numbers. We recollect, how to deal with absolute values, power and roots, and, nally, logarithms. Moreover, we also learn, how to simplify certain terms; (C 3 ) The third chapter is devoted to equations in one variable: we present a classication of equations and learn how to solve each of the type; (C 4 ) In the fourth chapter we do the same things, as in the previous one, but for inequalities. (C 5 ) In the last chapter, we consider some examples, which show, how one can use mathematics in order to solve various problems. The main literature is the book [1], chapters 2-4 and some elements of chapters 1 and 8. 2

3 1 Real Numbers Real numbers (R, +,, ); + and are binary operations over R; is the relation of order, which is called inequality. Further, we will consequently learn the structure of the set of real numbers and its properties, answering by the way the following questions: (Q 1 ) What are the main subsets of the set of real numbers R? Are they nice enough and big enough? If no, how can we extend them? (Q 2 ) What are the properties of binary operations + and and the relation of the order? (Q 3 ) How +, and are related? 1.1 Structure of Real Numbers N N N Z Figure 1: Integers Z on the number line N positive integers, N negative integers, N 0 non-negative integers N the set of natural numbers; N 0 the set of non-negative integers; Z the set of integers; Division over integers, divisor; The set of prime numbers; Prime factor representation of n Z; Least common multiple of a, b Z. Q the set of rational numbers Fraction, numerator and denominator of the fraction; Equivalent transformations of fractions; Proper, improper and mixed fractions; Expand and reduce the fraction: 3

4 Rules for working with fractions: 1. Addition and subtraction of fractions, having the same denominators; 2. Addition and subtraction of fractions, having dierent denominators; Least common denominator; Expansion factors. 3. Division and multiplication of fractions. Double fraction; Characterization of rational numbers via decimal expansions; Can one solve x 2 = 2 over Q? I the set of irrational numbers: How many irrational numbers? Examples of irrational numbers; Characterization of irrational numbers via decimal expansions. R = Q I the set of real numbers. 1.2 Properties of Real Numbers Addition; Multiplication; Distributive law and: Generalizations; Binomial formulas. The relation of order ; Connection between +, and ; One-to-one correspondence between real numbers and points on the number line. Figure 2: gure: Reals on the Number Line 4

5 2 Computations over Reals Mathematical term. 2.1 Absolute Value Absolute value: Denition; Geometrical interpretation; Properties; The distance of the real number x from the real number a. 2.2 Computations with Power and Roots Exponentiation; The n-th root of a number x R 0, surd and radical; Rules for working with powers and roots; Connections between powers and roots. 2.3 Computations with Logarithms Logarithm; Rules for working with logarithms; Natural and decimal logarithm; Change-of-base formula for logarithms. 5

6 3 Equations Function of a real variable; Domain, range and graph of a function; Polynomial, degree of polynomial; Equation and equivalent transformations of equations; Solution of the equation. Geometrical interpretation. 3.1 Linear Equations Which equation is called linear? How one can get the solution of the linear equation; Geometrical interpretation of the solution. Slope-intercept form of the line equation; Proportion, factor of proportionality. System of two linear equations with two variables. How one can solve such a system? 3.2 Quadratic Equations Which equation is called quadratic? Normal form of the quadratic equation; Solution of the quadratic equation. Discriminant. Geometrical interpretation of the solution, depending on the sign of discriminant. Parabola; Factorization of the quadratic equation; Generalization of the quadratic equations biquadratic equation. How one can solve it? 3.3 Root Equations Denition, examples and elementary techniques, which could be used to solve simple root equations; 3.4 Logarithmic and Exponential Equations Logarithmic Equation: Denition, examples and elementary techniques, which could be used to solve simple root equations; Exponential Equation: Denition, examples and elementary techniques, which could be used to solve simple root equations. 6

7 4 Inequalities Denition of inequality; Solution of inequality. 4.1 Set Theory Terminology Intervals: bounded, unbounded, graphical denotion. Empty set; Union and intersection of two sets. 4.2 Linear Inequalities Denition, examples and elementary techniques, which could be used to solve simple linear inequalities; Graphical interpretation of the solution. 4.3 Quadratic Inequalities Denition, examples and elementary techniques, which could be used to solve quadratic inequalities. Complete square; Graphical interpretation of the solution. 4.4 General Approach. The Method of Intervals Rational function; Rational inequality; The method of intervals. Description. Examples, which show, how one can use this approach in order to solve rational inequalities. 4.5 Rational Inequalities with Absolute Values Examples, which show, how one can use this approach in order to solve rational inequalities with absolute values. 4.6 Other Inequalities Inequalities with absolute values, roots and logarithms. 5 Text Problems Examples of real life problems, which could be solved with a help of the tools, considered in previous chapters. 7

8 References [1] Werner F.: A Refresher Course in Mathematics, 1 Edition, ISBN: , Peer reviewed by Dr. Larysa Burtseva, The Engineering Institute of the Autonomous University of Baja California, Mexicali, Mexico 8