Fundamentals of Math Analysis Course Outline 1-11 (1).pdf

FUNDAMENTALS OF MATH ANALYSIS (MATHA05S-M) COURSE OUTLINE Prepared by: ARTHUR GLENN A. GUILLEN Faculty, Mathematics Department...

FUNDAMENTALS OF MATH ANALYSIS (MATHA05S-M) COURSE OUTLINE Prepared by: ARTHUR GLENN A. GUILLEN Faculty, Mathematics Department Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Contents Fundamentals of Math Analysis Course Outline - Week 1.......................................4 Real Numbers..........................................................................................................4 Elementary Properties of Real Numbers.................................................................4 Order of Operations...............................................................................................5 Inequalities............................................................................................................5 Absolute Value......................................................................................................6 Intervals and Bounded Sets...................................................................................6 Fundamentals of Math Analysis Course Outline - Week 2.......................................7 Algebraic Expressions.............................................................................................7 Polynomials...........................................................................................................7 Operations on Polynomials....................................................................................7 Special Products....................................................................................................7 Factoring Polynomials...........................................................................................7 Fundamentals of Math Analysis Course Outline - Week 3.......................................8 Rational Expressions..............................................................................................8 Rational Expressions.............................................................................................8 Simplification of Rational Expressions...................................................................8 Negative Integral Exponents..................................................................................8 Operations on Rational Expressions......................................................................8 Complex Fractions.................................................................................................8 Fundamentals of Math Analysis Course Outline - Week 4.......................................9 Radical Expressions................................................................................................9 Radical Expressions..............................................................................................9 Fractional or Rational Exponents..........................................................................9 Rules Governing Fractional Exponents..................................................................9 Simplification of Radicals......................................................................................9 Operations on Radicals..........................................................................................9 Fundamentals of Math Analysis Course Outline - Week 5.....................................10 Equations.............................................................................................................. 10 First-Degree Equations in One Variable............................................................... 10 Fractional Equations........................................................................................... 10 Quadratic Equations........................................................................................... 10 Radical Equations............................................................................................... 10 Polynomial Equations.......................................................................................... 10 Fundamentals of Math Analysis Course Outline - Week 6.....................................11 Inequalities.............................................................................................................11 Linear Inequalities............................................................................................... 11 Quadratic Inequalities......................................................................................... 11 Absolute Value Inequalities................................................................................. 11 Rational Inequalities............................................................................................ 11 Fundamentals of Math Analysis Course Outline - Week 7.....................................12 Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner.  Graphs and Functions............................................................................................ 12 Circles................................................................................................................. 12 Cartesian Coordinate System.............................................................................. 12 Symmetry............................................................................................................12 Distances Between Two Points............................................................................12 Standard Equation of a Circle.............................................................................12 Fundamentals of Math Analysis Course Outline - Week 8.....................................13 Graphs and Functions............................................................................................ 13 Straight Lines......................................................................................................13 Fundamentals of Math Analysis Course Outline - Week 9.....................................14 Graphs and Functions............................................................................................ 14 Functions............................................................................................................14 Graphing Functions............................................................................................ 14 Linear and Quadratic Functions..........................................................................14 Piecewise-Defined Function................................................................................. 14 Absolute Value and Greatest Integer Function..................................................... 14 Combining Functions.......................................................................................... 14 Elementary Functions......................................................................................... 15 Transformations of Functions.............................................................................. 15 Inverse Functions................................................................................................ 15 Fundamentals of Math Analysis Course Outline - Week 10...................................16 Polynomial and Rational Functions........................................................................16 Polynomial Functions.......................................................................................... 16 Graphing Polynomial Functions..........................................................................16 Rational Zeros of Polynomials.............................................................................. 16 Rational Functions.............................................................................................. 16 Partial Fractions..................................................................................................16 Fundamentals of Math Analysis Course Outline - Week 11...................................17 Exponential and Logarithmic Functions.................................................................17 MAJOR EXAMINATION GUIDE................................................................................ 18 Examination Types............................................................................................... 18 Multiple Response............................................................................................. 18 Error Analysis....................................................................................................18 Statement Analysis........................................................................................... 18 Table Completion.............................................................................................. 19 Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 1 Real Numbers Elementary Properties of Real Numbers 1. Definition of Real Numbers: Real numbers include (ℝ) all the numbers on the number line, encompassing rational and irrational numbers. o Rational Numbers: Numbers that can be expressed as the quotient of 1 two integers (e.g., , 3, −4, −0.75). 2 o Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., 𝜋, √2, 𝑒). 2. Specific Examples of Real Numbers: o 2 (an integer (ℤ), rational number (ℚ)) o −5 (an integer (ℤ), rational number (ℚ)) o 0 (an integer (ℤ), rational number (ℚ)) 3 o 4 (a rational number (ℚ)) o 0.5 (a rational number (ℚ)) o √2 (an irrational number (ℝ\ℚ)) o 𝜋 (an irrational number (ℝ\ℚ)) o 𝑒 (an irrational number (ℝ\ℚ)) 3. Complex Numbers: Complex numbers (ℂ) include all numbers of the form 𝑎 + 𝑏𝑖, where 𝑎 and 𝑏 are real numbers, and 𝑖 is the imaginary unit with the property that 𝑖 2 = −1. Examples of Complex Numbers: o 2 + 3𝑖 o −1 − 𝑖 o 0 + 5𝑖 (which is purely imaginary) 4. Properties of Real Numbers: o Commutative Property: The order in which two numbers are added or multiplied does not change their sum or product. ▪ Addition: 𝑎 + 𝑏 = 𝑏 + 𝑎 ▪ Multiplication: 𝑎 ⋅ 𝑏 = 𝑏 ⋅ 𝑎 o Associative Property: The way in which numbers are grouped in addition or multiplication does not change their sum or product. ▪ Addition: (𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) ▪ Multiplication: (𝑎 ⋅ 𝑏) ⋅ 𝑐 = 𝑎 ⋅ (𝑏 ⋅ 𝑐) o Distributive Property: The sum of two numbers times a third number is equal to the sum of each addend times the third number. Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner.  ▪ 𝑎 ⋅ (𝑏 + 𝑐) = (𝑎 ⋅ 𝑏) + (𝑎 ⋅ 𝑐) o Identity Property: Adding 0 or multiplying by 1 leaves a number unchanged. ▪ Addition: 𝑎 + 0 = 𝑎 ▪ Multiplication: 𝑎 ⋅ 1 = 𝑎 o Inverse Property: Every number has an additive inverse (opposite) and a multiplicative inverse (reciprocal) that result in the identity for the operation. ▪ Addition: 𝑎 + (−𝑎) = 0 1 ▪ Multiplication: 𝑎 ⋅ 𝑎 = 1 for 𝑎 ≠ 0 o Closure Property: Performing an operation (addition or multiplication) on any two real numbers results in another real number. ▪ Addition: The sum of any two real numbers is a real number. ▪ Multiplication: The product of any two real numbers is a real number. Order of Operations 1. Definition: Order of operations is the sequence in which operations should be performed in a mathematical expression to ensure consistent and correct results. 2. PEMDAS Rule: An acronym to remember the order of operations: Parentheses, Exponents, Multiplication, and Division (left-to-right), Addition and Subtraction (left-to-right). o Parentheses o Exponents (including roots, such as square roots) o Multiplication and Division (left-to-right) o Addition and Subtraction (left-to-right) Inequalities 1. Definition: Inequalities are mathematical statements that compare two values or expressions using inequality symbols. 2. Symbols: Inequality symbols are used to show the relative size of two values. o Greater than (>) o Less than ( 𝑏 and 𝑏 > 𝑐, then 𝑎 > 𝑐. o Addition Property: If 𝑎 > 𝑏, then 𝑎 + 𝑐 > 𝑏 + 𝑐. o Multiplication Property: If 𝑎 > 𝑏 and 𝑐 > 0, then 𝑎 ⋅ 𝑐 > 𝑏 ⋅ 𝑐. Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Absolute Value 1. Definition: The absolute value of a number is its distance from zero on the number line, always a non-negative value. 2. Notation: |𝑥| 3. Properties: o |𝑥| = 𝑥 if 𝑥 ≥ 0 o |𝑥| = −𝑥 if 𝑥 < 0 o |𝑎𝑏| = |𝑎| ⋅ |𝑏| 𝑎 |𝑎| o |𝑏 | = |𝑏| for 𝑏 ≠ 0 Intervals and Bounded Sets 1. Definition of Intervals: An interval is a set of real numbers between two endpoints. 2. Types of Intervals: o Closed Interval: [𝑎, 𝑏] includes all numbers between 𝑎 and 𝑏, including the endpoints. o Open Interval: (𝑎, 𝑏) includes all numbers between 𝑎 and 𝑏, excluding the endpoints. o Half-Open Interval: [𝑎, 𝑏) or (𝑎, 𝑏] includes all numbers between 𝑎 and 𝑏, including only one endpoint. 3. Bounded Sets: o A set is bounded if it has both upper and lower bounds. o Upper Bound: A number 𝑀 is an upper bound of a set 𝑆 if 𝑥 ≤ 𝑀 for all 𝑥 ∈ 𝑆. o Lower Bound: A number 𝑚 is a lower bound of a set 𝑆 if 𝑚 ≤ 𝑥 for all 𝑥 ∈ 𝑆. Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 2 Algebraic Expressions Polynomials 1. Definition: A polynomial is an algebraic expression composed of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. 2. Form: 𝑃(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0 3. Degree: The highest power of the variable in the polynomial. 4. Types of Polynomials: o Monomial: A polynomial with one term. o Binomial: A polynomial with two terms. o Trinomial: A polynomial with three terms. Operations on Polynomials 1. Addition: Combine like terms. o Example: (2𝑥 2 + 3𝑥 + 4) + (𝑥 2 + 2𝑥 + 1) = 3𝑥 2 + 5𝑥 + 5 2. Subtraction: Distribute the negative sign and combine like terms. o Example: (2𝑥 2 + 3𝑥 + 4) − (𝑥 2 + 2𝑥 + 1) = 𝑥 2 + 𝑥 + 3 3. Multiplication: Use distributive property. o Example: (2𝑥 + 3)(𝑥 + 1) = 2𝑥 2 + 5𝑥 + 3 4. Division: Polynomial long division or synthetic division. Special Products 1. Square of a Binomial: (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 2. Difference of Squares: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) 3. Sum and Difference of Cubes: o 𝑎3 + 𝑏 3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏 2 ) o 𝑎3 − 𝑏 3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏 2 ) Factoring Polynomials 1. Factoring by Grouping: Group terms to factor out common factors. o Example: 𝑎𝑥 + 𝑎𝑦 + 𝑏𝑥 + 𝑏𝑦 = (𝑎 + 𝑏)(𝑥 + 𝑦) 2. Factoring Quadratic Polynomials: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 o Find two numbers that multiply to 𝑎𝑐 and add to 𝑏. o Example: 𝑥 2 + 5𝑥 + 6 = (𝑥 + 2)(𝑥 + 3) 3. Factoring Perfect Square Trinomials: 𝑎2 + 2𝑎𝑏 + 𝑏 2 = (𝑎 + 𝑏)2 4. Factoring Difference of Squares: 𝑎2 − 𝑏 2 = (𝑎 + 𝑏)(𝑎 − 𝑏) Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 3 Rational Expressions Rational Expressions 1. Definition: A rational expression is a ratio of two polynomials. 𝑃(𝑥) o Example: where 𝑃(𝑥) and 𝑄(𝑥) are polynomials and 𝑄(𝑥) ≠ 0. 𝑄(𝑥) Simplification of Rational Expressions 1. Simplifying Rational Expressions: Factor both the numerator and the denominator and then cancel out any common factors. 𝑥2−1 (𝑥−1)(𝑥+1) 𝑥−1 o Example: 𝑥2+2𝑥+1 = (𝑥+1)2 = 𝑥+1 for 𝑥 ≠ −1. Negative Integral Exponents 1. Definition: A negative exponent indicates the reciprocal of the base raised to the absolute value of the exponent. 1 o Example: 𝑎 −𝑛 = 𝑎𝑛 for 𝑎 ≠ 0. Operations on Rational Expressions 1. Addition and Subtraction: Find a common denominator, combine the numerators, and then simplify. 1 2 (𝑥+1)+2𝑥 3𝑥+1 o Example: 𝑥 + 𝑥+1 = 𝑥(𝑥+1) = 𝑥(𝑥+1). 2. Multiplication and Division: Multiply or divide the numerators and denominators, then simplify. 𝑎 𝑐 𝑎𝑐 o Multiplication Example: 𝑏 ⋅ 𝑑 = 𝑏𝑑. 𝑎 𝑐 𝑎 𝑑 𝑎𝑑 o Division Example: 𝑏 ÷𝑑 = 𝑏⋅𝑐 = 𝑏𝑐 Complex Fractions 1. Definition: A complex fraction is a fraction where the numerator, the denominator, or both, contain fractions. 2. Simplification: Simplify by finding a common denominator for the fractions within the numerator and the denominator, then perform the division. 𝑎 𝑎 𝑑 𝑎𝑑 o Example: 𝑏 𝑐 =𝑏⋅𝑐 = 𝑏𝑐 𝑑 Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 4 Radical Expressions Radical Expressions 1. Definition: Expressions containing roots, such as square or cube roots. 3 o Example: √𝑥 𝑜𝑟 √𝑥. Fractional or Rational Exponents 1. Definition: Exponents that are fractions, where the numerator is the power, and the denominator is the root. 𝑚 𝑛 o Example: 𝑎 𝑛 = √𝑎𝑚. Rules Governing Fractional Exponents 1. Properties: 𝑚 𝑛 o 𝑎 𝑛 = √𝑎 𝑚 𝑝 𝑚 𝑞 𝑚𝑝 o (𝑎 ) = 𝑎 𝑛𝑞 𝑛 𝑚 𝑝 𝑚𝑞+𝑛𝑝 o 𝑎 𝑛 ∙ 𝑎𝑞 = 𝑎 𝑛𝑞 Simplification of Radicals 1. Simplifying Radicals: Express the radical in its simplest form by factoring out squares. o Example: √18 = √9 ∙ 2 = 3√2. Operations on Radicals 1. Addition and Subtraction: Combine like radicals. o Example: 5√3 + 3√2 + 4√2 − 3√3 = 7√2 + 2√3. 2. Multiplication: If possible, use the distributive property and combine under the same radical. o Example: √𝑎 ∙ √𝑏 = √𝑎𝑏. 3. Division: Rationalize the denominator if necessary. √𝑎 𝑎 o Example: = √𝑏. √𝑏 Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 5 Equations First-Degree Equations in One Variable 1. Definition: Linear equations of the form 𝑎𝑥 + 𝑏 = 0, where 𝑎 and 𝑏 are constants. 4 o Example: 3𝑥 + 4 = 0 ⇒ 𝑥 = − 3. Fractional Equations 1. Definition: Equations that involve fractions. 2. Solution: Clear the fractions by multiplying through by the least common denominator (LCD). 𝑥 3 5 𝑥 3 1 o Example: 2 + 4 = 6 ⇒ 12 (2 + 4) = 12(56) ⇒ 6𝑥 + 9 = 10 ⇒ 𝑥 = 6 Quadratic Equations 1. Definition: Equations of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, where 𝑎 ≠ 0. 2. Solution Methods: o Factoring: 𝑥 2 − 5𝑥 + 6 = 0 ⇒ (𝑥 − 2)(𝑥 − 3) = 0 ⇒ 𝑥 = 2 or 𝑥 = 3. −𝑏±√𝑏2 −4𝑎𝑐 o Quadratic Formula: 𝑥 = 2𝑎. o Completing the Square: 𝑥 2 + 4𝑥 + 4 = (𝑥 + 2)2 = 0 ⇒ 𝑥 = −2. Radical Equations 1. Definition: Equations involving radicals. 2. Solution: Isolate the radical on one side and then square both sides to eliminate the radical. o Example: √𝑥 + 2 = 3 ⇒ 𝑥 + 2 = 9 ⇒ 𝑥 = 7. Polynomial Equations 1. Definition: Equations involving polynomials of degree greater than two. 2. Solution Methods: o Factoring: Express the polynomial as a product of its factors. o Synthetic Division: Used to divide polynomials and find roots. o Remainder Theorem: If a polynomial 𝑃(𝑥) is divided by 𝑥 − 𝑐, the remainder is 𝑃(𝑐). o Factor Theorem: 𝑥 − 𝑐 is a factor of 𝑃(𝑥) if and only if 𝑃(𝑐) = 0. Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 6 Inequalities Linear Inequalities 1. Definition: Inequalities of the form 𝑎𝑥 + 𝑏 > 𝑐, where 𝑎, 𝑏, and 𝑐 are constants. 2. Solution: Solve the inequality similarly to equations, then graph the solution on a number line. o Example: 2𝑥 + 3 > 7 ⇒ 2𝑥 > 4 ⇒ 𝑥 > 2 Quadratic Inequalities 1. Definition: Inequalities of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 > 0, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 < 0, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ≥ 0, or 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 ≤ 0. 2. Solution: Find the roots of the corresponding equation 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, and then test intervals between the roots to determine where the inequality holds. o Example: 𝑥 2 − 3𝑥 + 2 > 0 ⇒ (𝑥 − 1)(𝑥 − 2) > 0. Absolute Value Inequalities 1. Definition: Inequalities that involve absolute values. 2. Solution: Express as two separate inequalities without the absolute value. o Example: |𝑥 − 3| < 5 ⇒ −5 < 𝑥 − 3 < 5 ⇒ −2 < 𝑥 < 8 Rational Inequalities 1. Definition: Inequalities involving rational expressions. 2. Solution: Find the critical points by setting the numerator and denominator equal to zero, then test intervals. 𝑥−1 o Example: 𝑥+2 ≥ 0 ⇒ 𝑥 − 1 = 0 ⇒ 𝑥 = 1 𝑎𝑛𝑑 𝑥 + 2 = 0 ⇒ 𝑥 = −2 Other Example: 𝒙𝟐 + 𝟐𝒙 0, left if 𝑐 < 0) 2. Reflections: o Across the x-axis: −𝑓(𝑥) o Across the y-axis: 𝑓(−𝑥) 3. Expansions and Contractions: o Vertical Expansion: 𝑐𝑓(𝑥) if 𝑐 > 1 o Vertical Contraction: 𝑐𝑓(𝑥) if 0 < 𝑐 < 1 Inverse Functions 1. Definition: The inverse function 𝑓 −1 (𝑥) of a function 𝑓(𝑥) reverses the roles of inputs and outputs. 2. Properties: o One-to-One Function: A function 𝑓(𝑥) has an inverse if it is one-to-one (passes the horizontal line test). o Finding the Inverse: Solve 𝑦 = 𝑓(𝑥) for 𝑥, then interchange 𝑥 and 𝑦. o Symmetry Property: The graph of 𝑓(𝑥) and 𝑓 −1 (𝑥) are symmetric with respect to the line 𝑦 = 𝑥. Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 10 Polynomial and Rational Functions Polynomial Functions 1. Definition: A polynomial function is a function that can be represented by a polynomial. 2. Division Algorithm: For any polynomials 𝑓(𝑥) and 𝑑(𝑥), there are unique polynomials 𝑞(𝑥) and 𝑟(𝑥) such that 𝑓(𝑥) = 𝑑(𝑥)𝑞(𝑥) + 𝑟(𝑥) where the degree of 𝑟(𝑥) is less than the degree of 𝑑(𝑥). Graphing Polynomial Functions 1. Graph Characteristics: o Degree: The highest power of the variable. o Leading Coefficient Test: Determines the end behavior of the graph. o Roots/Zeros: Points where the graph crosses the x-axis. Rational Zeros of Polynomials 1. Fundamental Theorem of Algebra: Every non-zero single-variable polynomial with complex coefficients has at least one complex root. 𝑝 2. Rational Root Theorem: If a polynomial has a rational root , then 𝑝 is a 𝑞 factor of the constant term and 𝑞 is a factor of the leading coefficient. Rational Functions 1. Definition: A rational function is the ratio of two polynomials. 2. Vertical and Horizontal Asymptotes: o Vertical Asymptotes: Occur where the denominator is zero. o Horizontal Asymptotes: Determined by comparing the degrees of the numerator and the denominator. Partial Fractions 1. Partial Fraction Decomposition: Expressing a rational function as a sum of simpler fractions. 𝑃(𝑥) 𝐴 𝐵 o Example: = (𝑥−𝑟 ) + (𝑥−𝑟 ) + ⋯ 𝑄(𝑥) 1 2 Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Fundamentals of Math Analysis Course Outline - Week 11 Exponential and Logarithmic Functions Exponential Functions 1. Definition: An exponential function is of the form 𝑓(𝑥) = 𝑎 ⋅ 𝑏 𝑥 , where 𝑏 > 0 and 𝑏 ≠ 1. o Example: 𝑓(𝑥) = 2 ⋅ 3𝑥 Exponential Function with Base 𝒆 1. Definition: The natural exponential function is 𝑓(𝑥) = 𝑒 𝑥 , where 𝑒 ≈ 2.71828. o Example: 𝑓(𝑥) = 𝑒 2𝑥 Logarithmic Functions 1. Definition: The inverse of the exponential function, 𝑦 = log 𝑏 (𝑥) if and only if 𝑏 𝑦 = 𝑥, where 𝑏 > 0 and 𝑏 ≠ 1. o Example: 𝑦 = log 2(𝑥) means 2𝑦 = 𝑥 Common and Natural Logarithm 1. Common Logarithm: The logarithm with base 10, denoted as log(𝑥). o Example: 𝑙𝑜𝑔(100) = 2 because 102 = 100 2. Natural Logarithm: The logarithm with base 𝑒, denoted as ln(𝑥). Note: ln(𝑥) = log 𝑒 (𝑥) o Example: 𝑙𝑛(𝑒 2 ) = 2 because 𝑒 2 = 𝑒 2 Exponential and Logarithmic Equations 1. Solving Exponential Equations: Rewrite the equation so that the bases are the same, then set the exponents equal to each other. o Example: 2𝑥+1 = 8⇒2𝑥+1 = 23 ⇒ 𝑥 + 1 = 3 ⇒ 𝑥 = 2 2. Solving Logarithmic Equations: Use the properties of logarithms to combine or separate logs, then exponentiate both sides to eliminate the logarithm. o Example: log(𝑥)+log(2) = 1 ⇒ log(2𝑥) = 1 ⇒ 2𝑥 = 10 ⇒ 𝑥 = 5 Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. MAJOR EXAMINATION GUIDE Examination Types Multiple Response Instruction: Choose the letter of the correct answer. Sample item: Sir Glenn provided BSIT 1A with the opportunity to articulate their insights following the lesson on the Elementary Properties of Real Numbers. The students' observations are outlined below: I. 𝜋 is not an integer. 2 II. 5 is not an integer. √2 III. 2 is a rational number. IV. 89 is not a prime number. V. All whole numbers are integers. VI. All integers are rational numbers. VII. All irrational numbers are rational numbers. A. All the students’ observations are correct. B. Only six students’ observations are correct. C. Only five students’ observations are correct. D. Only four students’ observations are correct. Error Analysis Instruction: Shade the letter or letters of the incorrect part of the solution/answer. If there is NO ERROR in the given solution/answer, shade x. Sample item: 𝟑𝒙 𝟓𝒙 1. 𝟐 − 𝟑 + 𝟑𝒙 − 𝟒 = 𝟏𝟑 3𝑥 5𝑥  6( 2 − 3 + 3𝑥 − 4) = (13)6  9𝑥 − 10𝑥 + 18𝑥 − 24 = 78  17𝑥 = 78 + 24 17𝑥 102  17 = 17  𝑥 = 16  NO ERROR Statement Analysis Instruction: Given the two statements, shade  if both statements are TRUE, shade  if the first statement is TRUE and the second statement is FALSE, shade  if the first statement is FALSE and the second statement is TRUE, and shade  if both statements are FALSE. Sample item: Statement 1: All whole numbers are integers. Statement 2: All negative integers are irrational numbers.  Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner. Table Completion Instruction: Fill in the table completely. There is a corresponding point in every column and row. 𝒙𝟐 − 𝒙 − 𝟏𝟐 ≤ 𝟎 Critical Points: 𝑥 =  Interval Test Value 𝑥 2 − 𝑥 − 12 ≤ 0            Solution Set:   Fundamentals of Math Analysis Course Outline, developed by Arthur Glenn A. Guillen, MAT., LPT. this August 6, 2024. No part of this material may be reproduced, stored in a retrieval system, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without prior or written permission from the owner.