Mathematics Foundations for Engineering and Science – Examples, Exercises & Solutions 📚 By Ronen Kolton Yehuda (MKR: Messiah King RKY), with the assistance of ChatGPT AI

*Note: need to update with more explanations

📚 Mathematics Foundations for Egineering and Science – Examples, Exercises & Solutions

By Ronen Kolton Yehuda (MKR:Messiah King RKY), with the assistance of ChatGPT AI

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📘 Book I – Foundations & Pre-Calculus

Preface

• About the Series

• How to Use This Book

• The Importance of Mathematical Foundations

Chapter 0 – The Very Basics of Mathematics

• What is Mathematics? (Counting, Measuring, Patterns)

• Mathematical Symbols and Notation (+, –, ×, ÷, =, >, <)

• Counting Numbers and Zero

• Place Value (Units, Tens, Hundreds)

• Basic Arithmetic (Addition, Subtraction, Multiplication, Division)

• Order of Operations (PEMDAS/BODMAS)

• Fractions (Simplification, Comparison)

• Decimals (Conversion, Place Value)

• Percentages (Conversion, Applications)

• Negative Numbers (Number Line, Operations)

• Introduction to Variables and Simple Expressions

Chapter 1 – Numbers, Sets, and Logic

• Number Systems (ℕ, ℤ, ℚ, ℝ, Irrationals)

• Sets and Operations (Union, Intersection, Complement)

• Logic Basics (Truth Values, Connectives, Truth Tables)

• Proof Methods (Direct, Contradiction, Induction)

Chapter 2 – Algebra Basics

• Algebraic Expressions & Simplification

• Linear Equations & Inequalities

• Quadratic Equations (Factoring, Formula, Completing the Square)

• Absolute Value Equations and Inequalities

Chapter 3 – Functions

• Definition, Domain & Range

• Linear and Quadratic Functions

• Piecewise and Step Functions

• Exponential and Logarithmic Functions (Intro)

Chapter 4 – Trigonometry Basics

• Angles, Radians, and the Unit Circle

• Trigonometric Ratios

• Pythagorean Identities

• Basic Trigonometric Equations

• Applications (Heights, Distances)

Chapter 5 – Vectors in 2D

• Magnitude and Direction

• Vector Addition and Scalar Multiplication

• Dot Product (Intro Applications)

• Applications in Physics (Forces, Displacement)

Chapter 6 – Review & Challenge Problems

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📘 Book II – Advanced Algebra & Trigonometry

Chapter 1 – Complex Numbers

• Definition and Operations

• Complex Plane

• De Moivre’s Theorem

Chapter 2 – Advanced Polynomials

• Division of Polynomials

• Factor & Remainder Theorems

• Roots & Factorization

• Graphing Polynomials

Chapter 3 – Sequences and Series

• Arithmetic Progressions

• Geometric Progressions

• Sigma Notation

• Proof by Induction

Chapter 4 – Advanced Trigonometry

• Double & Half Angle Identities

• Sum-to-Product & Product-to-Sum

• Advanced Trigonometric Equations

• Trig Graphs and Applications

Chapter 5 – Analytic Geometry (Conics)

• Circles

• Parabolas

• Ellipses and Hyperbolas

• Applications

Chapter 6 – Vectors in 3D

• Representation of 3D Vectors

• Scalar and Vector Products

• Applications to Mechanics

Chapter 7 – Review & Challenge Problems

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📘 Book III – Calculus I

Chapter 1 – Limits and Continuity

• Concept of a Limit

• Techniques and Laws

• One-Sided & Infinite Limits

• Continuity and Types of Discontinuity

Chapter 2 – Differentiation Rules

• Derivative as Rate of Change and Slope

• Rules of Differentiation (Product, Quotient, Chain)

• Higher-Order Derivatives

• Implicit Differentiation

Chapter 3 – Applications of Derivatives

• Tangents and Normals

• Maxima, Minima, Optimization Problems

• Curve Sketching

• Newton’s Method (Intro)

Chapter 4 – Related Rates and Kinematics

• Motion Along a Line

• Related Rates Word Problems

• Growth and Decay Models

Chapter 5 – Introduction to Integration

• Antiderivatives and Indefinite Integrals

• Basic Integration Rules

• Substitution Method

Chapter 6 – Definite Integrals & Area Problems

• Definite Integral Definition

• Fundamental Theorem of Calculus

• Area Under Curves and Between Curves

Chapter 7 – Review & Challenge Problems

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📘 Book IV – Calculus II & Linear Algebra

Chapter 1 – Advanced Integration Techniques

• Integration by Parts

• Trigonometric Integrals

• Partial Fractions

• Improper Integrals

Chapter 2 – Differential Equations (First-Order Models)

• Separable Equations

• Linear First-Order Equations

• Applications: Population Growth, Cooling, Circuits

Chapter 3 – Multivariable Functions

• Functions of Two Variables

• Partial Derivatives

• Gradient and Directional Derivatives

Chapter 4 – Multiple Integrals

• Double Integrals over Rectangles

• Double Integrals in Polar Coordinates

• Triple Integrals

Chapter 5 – Introduction to Linear Algebra

• Matrices and Determinants

• Systems of Linear Equations (Gaussian Elimination)

• Applications to Engineering

Chapter 6 – Eigenvalues and Eigenvectors

• Definition and Computation

• Simple Applications in Physics & Engineering

Chapter 7 – Review & Challenge Problems

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📘 Book V – Probability, Statistics & Applied Mathematics

Chapter 1 – Combinatorics & the Binomial Theorem

• Factorials and Counting Principles

• Permutations and Combinations

• Binomial Expansion

Chapter 2 – Probability Basics

• Classical Probability

• Conditional Probability

• Bayes’ Theorem

Chapter 3 – Random Variables and Distributions

• Discrete Random Variables

• Continuous Random Variables

• Expectation and Variance

Chapter 4 – Statistics

• Measures of Central Tendency (Mean, Median, Mode)

• Measures of Dispersion (Variance, Standard Deviation)

• Normal Distribution and z-Scores

• Confidence Intervals (Intro)

Chapter 5 – Mathematical Modeling Applications

• Trusses and Force Calculations

• Electrical Circuits using Differential Equations

• Exponential and Logistic Growth Models

Chapter 6 – Capstone Challenge Problems

• Integrated Problems Connecting Algebra, Calculus, Probability

• 

📘 Book I – Foundations & Pre-Calculus

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 0 – The Very Basics of Mathematics

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🔹 0.1 What is Mathematics?

Mathematics is the universal language of patterns, quantities, and relationships. It allows us to:

• Count objects and events.

• Measure distances, time, and weight.

• Compare sizes and quantities.

• Predict how things behave (like motion, growth, or decay).

From ancient civilizations (Babylonians, Egyptians, Greeks) to modern engineering, mathematics has been the foundation of science and technology.

📘 Historical Note

• Babylonians used base-60 numbers (which is why a circle has 360°).

• Indians introduced zero as a number.

• Greeks developed geometry (Euclid, Pythagoras).

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🔹 0.2 Mathematical Symbols and Notation

Mathematics is built on a symbolic language:

• Addition (+): combine numbers.

• Subtraction (–): remove or find difference.

• Multiplication (× or ⋅): repeated addition.

• Division (÷ or /): sharing into equal groups.

• Equality (=): both sides are the same.

• Inequalities: greater than (>), less than (<), greater-or-equal (≥), less-or-equal (≤).

📘 Example 0.1

$$3+5=8,10\text{–}6=4,4\times 3=12,12÷4=3$$

📊 (Graph: number line showing +, – moves)

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🔹 0.3 Counting Numbers and Zero

• Counting (Natural) Numbers: 
  $1,2,3,4,\dots$

• Zero (0): represents “nothing” but is crucial for place value and algebra.

📘 Example 0.2

A basket has 5 apples. If you remove all 5, you are left with 0 apples.

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🔹 0.4 Place Value System

Every digit in a number has a place value depending on its position:

• In 347 →

  • 3 = hundreds = 300

  • 4 = tens = 40

  • 7 = ones = 7

$$347=3\times 100+4\times 10+7\times 1$$

📘 Example 0.3

Write 5,062 in words → Five thousand and sixty-two.

📊 (Graph: place value table — Units, Tens, Hundreds, Thousands)

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🔹 0.5 Basic Arithmetic

Addition: combining.

Subtraction: finding difference.

Multiplication: repeated addition.

Division: equal sharing.

📘 Example 0.4

$$27+16=43;50\text{–}28=22;6\times 7=42;45÷5=9$$

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🔹 0.6 Order of Operations (PEMDAS/BODMAS)

Order matters in math. Rules:

1. Parentheses / Brackets

2. Exponents / Orders

3. Multiplication & Division (left to right)

4. Addition & Subtraction (left to right)

📘 Example 0.5

$$3+2\times (5\text{–}2)=3+2\times 3=3+6=9$$

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🔹 0.7 Fractions

A fraction represents part of a whole:

$$\frac{𝑎}{𝑏},𝑏\mathrm{\ne }0$$

• Simplify by dividing numerator & denominator by their greatest common factor.

• Equivalent fractions: 
  $\frac{1}{2}=\frac{2}{4}=\frac{50}{100}$
  .

📘 Example 0.6

$$\frac{12}{18}=\frac{2}{3}$$

📊 (Graph: pizza cut into slices to illustrate ½, ¼, etc.)

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🔹 0.8 Decimals

Decimals are fractions with denominator powers of 10.

• 0.5 = ½

• 0.25 = ¼

• Place value after the decimal: tenths, hundredths, thousandths.

📘 Example 0.7

Convert: 

$\frac{3}{4}=0.75$

.

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🔹 0.9 Percentages

“Percent” means “per 100.”

• 50% = 
  $\frac{50}{100}=0.5$
  .

• To find a percent of a number: multiply by fraction form.

📘 Example 0.8

Find 20% of 200.

$$200\times 0.20=40$$

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🔹 0.10 Negative Numbers

• Numbers less than zero.

• Represented on a number line left of 0.

• Used in temperatures, bank balance, altitude.

📘 Example 0.9

$$(\text{–}3)+(\text{–}5)=\text{–}\mathrm{8 };(\text{–}7)\text{–}(\text{–}2)=\text{–}5$$

📊 (Graph: number line showing negative and positive integers)

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🔹 0.11 Introduction to Variables

• A variable is a symbol (like 
  $𝑥$
  or 
  $𝑦$
  ) that represents an unknown.

• Basis of algebra.

📘 Example 0.10

Solve: 

$𝑥+3=7$

.

$$𝑥=4$$

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📝 Exercises

Exercise 0.1 – Basic Practice

1. Fill in the missing numbers:
   a) 
   $8+\mathrm{\_}\mathrm{\_}=13$
   
   
   b) 
   $\mathrm{\_}\mathrm{\_}\text{–}7=12$
   
   
   c) 
   $9\times \mathrm{\_}\mathrm{\_}=36$
   
   
   d) 
   $25÷\mathrm{\_}\mathrm{\_}=5$

Exercise 0.2 – Fractions & Decimals

1. Simplify: 
   $\frac{20}{28}$
   .

2. Convert: 
   $\frac{2}{5}$
   into decimal and percent.

Exercise 0.3 – Percentages

A jacket costs $80. A store gives 25% discount. What is the new price?

Exercise 0.4 – Word Problem

The temperature is –2°C in the morning and rises by 7°C. What is the new temperature?

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✅ Solutions

Solution 0.1

a) 5 , b) 19 , c) 4 , d) 5

Solution 0.2

1. $\frac{20}{28}=\frac{5}{7}$

2. $\frac{2}{5}=0.4=40\mathrm{\%}$

Solution 0.3

25% of $80 = 

$80\times 0.25=20$

. New price = $60.

Solution 0.4

–2 + 7 = 5°C.

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🔥 Challenge Problems

1. Write 1,234,567 in words.

2. Simplify 
   $\frac{84}{126}$
   .

3. A smartphone costs $500. A holiday sale gives 15% discount. What is the final price?

4. A submarine is at –300 m below sea level. It ascends 120 m. What is its new depth?

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✅ Summary of Chapter 0

In this chapter, you learned:

• The symbols and language of mathematics.

• Place value, arithmetic, order of operations.

• Fractions, decimals, percentages, and negative numbers.

• Introduction to variables.

This is the alphabet of mathematics — every higher concept in algebra, calculus, and engineering is built upon these fundamentals.

📊 (Diagrams suggested: number line, fraction circle, percent bar, function box illustration.)

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📘 Book I – Foundations & Pre-Calculus

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 1 – Numbers, Sets, and Logic

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🔹 1.1 Number Systems

Numbers are organized into sets that build upon each other. Each expansion allows us to solve new kinds of problems.

1. Natural Numbers (ℕ):
   $1,2,3,4,\dots$
    – counting numbers.

2. Whole Numbers:
   $0,1,2,3,\dots$
    – includes zero.

3. Integers (ℤ):
   $\dots ,\text{–}3,\text{–}2,\text{–}1,0,1,2,3,\dots$
   .

4. Rational Numbers (ℚ): fractions of integers, like 
   $\frac{2}{3},\text{–}\frac{7}{4},\frac{5}{1}$
   .

5. Irrational Numbers: cannot be written as fractions. Examples: 
   $𝜋,\sqrt{2},𝑒$
   .

6. Real Numbers (ℝ): all rational + irrational numbers.

📘 Example 1.1

Classify the following numbers:

a) 

$7$

→ Natural, Whole, Integer, Rational, Real.

b) 

$\text{–}12$

→ Integer, Rational, Real.

c) 

$\frac{49}{7}=7$

→ Rational, Integer, Natural, Real.

d) 

$𝜋$

→ Irrational, Real.

📊 (Graph: number line showing nested sets of numbers)

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🔹 1.2 Sets and Operations

A set is a well-defined collection of objects, written in braces { }.

• Union (A ∪ B): all elements in A or B.

• Intersection (A ∩ B): elements common to both A and B.

• Difference (A – B): elements in A but not in B.

• Complement (A′): all elements not in A (relative to universal set U).

📘 Example 1.2

Let 

$𝐴=\{1,2,3,4\},𝐵=\{3,4,5,6\},𝑈=\{1,2,3,4,5,6,7\}$

.

• $𝐴\cup 𝐵=\{1,2,3,4,5,6\}$

• $𝐴\cap 𝐵=\{3,4\}$

• $𝐴\text{–}𝐵=\{1,2\}$

• $𝐵\text{′}=\{1,2,7\}$

📊 (Graph: Venn diagram showing union, intersection, difference)

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🔹 1.3 Logic Basics

Logic is the language of reasoning in mathematics. A statement is something that is either true (T) or false (F).

Logical operators:

• NOT (¬p): the opposite of p.

• AND (p ∧ q): true only if both p and q are true.

• OR (p ∨ q): true if at least one is true.

• IMPLIES (p → q): “if p, then q.”

📘 Example 1.3

Let p = “It is raining,” q = “I take an umbrella.”

Truth table for p → q:

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

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🔹 1.4 Proof Methods (Introduction)

Mathematics requires rigorous justification. Three common methods:

1. Direct Proof: Start from known facts, use logic, reach conclusion.

2. Proof by Contradiction: Assume the opposite of what you want to prove, show a contradiction.

3. Mathematical Induction:

   • Step 1: Base case (true for n=1).

   • Step 2: Assume true for n=k.

   • Step 3: Prove true for n=k+1.

📘 Example 1.4 – Induction

Prove that

$$1+2+3+\dots +𝑛=\frac{𝑛(𝑛+1)}{2}$$

✅ Solution:

• Base case: n=1 → LHS=1, RHS=1 → true.

• Assume true for n=k: 
  $1+\dots +𝑘=\frac{𝑘(𝑘+1)}{2}$
  .

• For n=k+1:

$$1+\dots +𝑘+(𝑘+1)=\frac{𝑘(𝑘+1)}{2}+(𝑘+1)=\frac{(𝑘+1)(𝑘+2)}{2}$$

So formula holds for k+1. ✅

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📝 Exercises

Exercise 1.1 – Classify Numbers

State which sets the number belongs to:

a) –25

b) 0

c) √16

d) 2.718… (e)

Exercise 1.2 – Sets

Let A = {2,4,6,8}, B = {1,2,3,4,5}, U = {1,2,3,4,5,6,7,8}. Find:

a) A ∪ B

b) A ∩ B

c) A – B

d) B′

Exercise 1.3 – Logic

Construct a truth table for (p ∧ q) ∨ ¬p.

Exercise 1.4 – Proof by Contradiction

Prove that √3 is irrational.

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✅ Solutions

Solution 1.1

a) Integer, Rational, Real

b) Whole, Integer, Rational, Real

c) 4 → Natural, Whole, Integer, Rational, Real

d) Irrational, Real

Solution 1.2

a) {1,2,3,4,5,6,8}

b) {2,4}

c) {6,8}

d) {6,7,8}

Solution 1.3

p	q	(p ∧ q) ∨  ¬p
T	T	T
T	F	F
F	T	T
F	F	T

Solution 1.4

Assume √3 = p/q in lowest terms. Then p²=3q² → p divisible by 3. Let p=3k → q²=3k² → q divisible by 3. Contradiction: both divisible by 3. Therefore √3 irrational. ✅

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🔥 Challenge Problems

1. 1. Show that the sum of any two even integers is even. 

   2. Show that the product of a rational and an irrational number is irrational. 

   3. Construct a truth table for (p→q)∧(q→p)(p → q) ∧ (q → p). 

   4. Prove there are infinitely many prime numbers (hint: Euclid’s proof). 

    

   ✅ Summary of Chapter 1 

   In this chapter, you learned: 

   • The hierarchy of numbers: natural, whole, integers, rationals, irrationals, and reals. 

   • How to describe and operate with sets. 

   • The basics of logic and truth tables. 

   • First steps into proof methods, including induction. 

These are the building blocks of rigorous mathematics. You will use them constantly in algebra, calculus, and applied science. 

1. By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 2 – Algebra Basics

1. ---

🔹 2.1 Algebraic Expressions & Simplification

1. Definition:
   An algebraic expression is a combination of numbers, variables, and operations.
   Examples:

   • $3𝑥+2$

   • $4{𝑎}^2\text{–}5𝑏+7$

   Simplification rules:

   1. Combine like terms (same variable and exponent).

   2. Apply distributive law: 
      $𝑎(𝑏+𝑐)=𝑎𝑏+𝑎𝑐$
      .

   📘 Example 2.1
   Simplify 
   $3𝑥+5𝑥\text{–}7+2$
   .
   ✅ Solution: Combine like terms → 
   $8𝑥\text{–}5$
   .
   📘 Example 2.2
   Expand 
   $2(𝑥+4)\text{–}(3𝑥\text{–}1)$
   .
   ✅ Solution: 
   $2𝑥+8\text{–}3𝑥+1=\text{–}𝑥+9$
   .

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🔹 2.2 Linear Equations & Inequalities

1. Linear equation: an equation of the form 
   $𝑎𝑥+𝑏=0$
   .

   • Solution is where the expression equals zero.

   📘 Example 2.3
   Solve 
   $2𝑥+5=11$
   .
   ✅ Solution: 
   $2𝑥=6\to 𝑥=3$
   .

   Inequalities: Similar rules, but note:

   • Multiplying/dividing by a negative reverses inequality sign.

   📘 Example 2.4
   Solve 
   $\text{–}3𝑥>9$
   .
   ✅ Solution: Divide by –3 → 
   $𝑥<\text{–}3$
   .

   📊 (Graph: number line with inequality solution region)

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🔹 2.3 Quadratic Equations

1. Quadratic = equation of form 
   $𝑎{𝑥}^2+𝑏𝑥+𝑐=0$
   .

   Methods of solving:

   1. Factoring (if possible).

   2. Quadratic Formula:

      $$𝑥=\frac{\text{–}𝑏\pm \sqrt{{𝑏}^2\text{–}4𝑎𝑐}}{2𝑎}$$

   3. Completing the square.

   📘 Example 2.5 (Factoring)
   Solve 
   ${𝑥}^2\text{–}5𝑥+6=0$
   .
   ✅ Solution: 
   $(𝑥\text{–}2)(𝑥\text{–}3)=0\to 𝑥=2,3$
   .
   📘 Example 2.6 (Quadratic Formula)
   Solve 
   $2{𝑥}^2+3𝑥\text{–}2=0$
   .
   ✅ Solution:
   $$𝑥=\frac{\text{–}3\pm \sqrt{9+16}}{4}=\frac{\text{–}3\pm 5}{4}$$
   → 
   $𝑥=\frac{1}{2},\text{–}2$
   .
   📘 Example 2.7 (Completing the square)
   Solve 
   ${𝑥}^2+6𝑥+5=0$
   .
   ✅ Solution:
   ${𝑥}^2+6𝑥+9\text{–}9+5=0\to (𝑥+3{)}^2\text{–}4=0$
   .
   So 
   $(𝑥+3{)}^2=4\to 𝑥=\text{–}3\pm 2\to 𝑥=\text{–}1,\text{–}5$
   .

   ---

🔹 2.4 Absolute Value Equations & Inequalities

1. Definition:

   $$\mathrm{\mid }𝑎\mathrm{\mid }=\{\begin{array}{ll}𝑎& 𝑎\ge 0\\ \text{–}𝑎& 𝑎<0\end{array}$$

   Equations: Split into two cases.

   📘 Example 2.8
   Solve 
   $\mathrm{\mid }𝑥\text{–}3\mathrm{\mid }=5$
   .
   ✅ Solution: 
   $𝑥\text{–}3=5\to 𝑥=8$
   or 
   $𝑥\text{–}3=\text{–}5\to 𝑥=\text{–}2$
   .

   Inequalities:

   • $\mathrm{\mid }𝑥\mathrm{\mid }<𝑎\to \text{–}𝑎<𝑥<𝑎$

     .

   • $\mathrm{\mid }𝑥\mathrm{\mid }>𝑎\to 𝑥<\text{–}𝑎$

     or 
     $𝑥>𝑎$
     .

   📘 Example 2.9
   Solve 
   $\mathrm{\mid }2𝑥\text{–}1\mathrm{\mid }\le 3$
   .
   ✅ Solution: –3 ≤ 2x–1 ≤ 3 → –2 ≤ 2x ≤ 4 → –1 ≤ x ≤ 2.

   📊 (Graph: V-shape graph of absolute value)

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📝 Exercises

1. Exercise 2.1 – Expressions
   Simplify:
   a) 
   $4𝑥+7𝑥\text{–}3$
   
   
   b) 
   $2(𝑎+5)\text{–}3(𝑎\text{–}2)$
   Exercise 2.2 – Linear Equations/Inequalities
   a) Solve 
   $7𝑥\text{–}4=17$
   .
   b) Solve 
   $\text{–}5𝑦\le 15$
   .
   Exercise 2.3 – Quadratics
   a) Solve 
   ${𝑥}^2\text{–}7𝑥+12=0$
   .
   b) Solve 
   $3{𝑥}^2\text{–}2𝑥\text{–}1=0$
   .
   Exercise 2.4 – Absolute Value
   a) Solve 
   $\mathrm{\mid }𝑥+4\mathrm{\mid }=7$
   .
   b) Solve 
   $\mathrm{\mid }𝑥\text{–}2\mathrm{\mid }>3$
   .

   ---

✅ Solutions

1. Solution 2.1
   a) 
   $11𝑥\text{–}3$
   .
   b) 
   $2𝑎+10\text{–}3𝑎+6=\text{–}𝑎+16$
   .
   Solution 2.2
   a) 
   $7𝑥=21\to 𝑥=3$
   .
   b) Divide by –5 (reverse inequality): 
   $𝑦\ge \text{–}3$
   .
   Solution 2.3
   a) 
   ${𝑥}^2\text{–}7𝑥+12=(𝑥\text{–}3)(𝑥\text{–}4)\to 𝑥=3,4$
   .
   b) Formula: 
   $𝑥=\frac{2\pm \sqrt{4+12}}{6}=\frac{2\pm 4}{6}$
   .
   So 
   $𝑥=1,\text{–}\frac{1}{3}$
   .
   Solution 2.4
   a) 
   $𝑥+4=7\to 𝑥=3$
   or 
   $𝑥+4=\text{–}7\to 𝑥=\text{–}11$
   .
   b) 
   $𝑥\text{–}2>3\to 𝑥>5$
   OR 
   $𝑥\text{–}2<\text{–}3\to 𝑥<\text{–}1$
   .

   ---

🔥 Challenge Problems

1. 1. Solve 
      $2𝑥+1=\mathrm{\mid }𝑥\text{–}4\mathrm{\mid }$
      .

   2. Show that if 
      $𝑎{𝑥}^2+𝑏𝑥+𝑐=0$
      has real solutions, then the discriminant 
      ${𝑏}^2\text{–}4𝑎𝑐\ge 0$
      .

   3. Prove that 
      $\mathrm{\mid }𝑎+𝑏\mathrm{\mid }\le \mathrm{\mid }𝑎\mathrm{\mid }+\mathrm{\mid }𝑏\mathrm{\mid }$
      (triangle inequality).

   4. A rectangle has length 
      $𝑥+3$
      and width 
      $𝑥\text{–}2$
      . If its area is 40, find x.

   ---

✅ Summary of Chapter 2

1. In this chapter, you learned:

   • How to simplify algebraic expressions.

   • Solving linear equations and inequalities.

   • Different methods for solving quadratic equations: factoring, quadratic formula, completing the square.

   • How to solve absolute value equations and inequalities.

   Algebra is the language of mathematics — it turns real-world problems into equations that we can solve.

   📊 (Suggested figures: graphs of linear equations, parabola for quadratics, absolute value V-shape).

   ---

📘 Book I – Foundations & Pre-Calculus

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 3 – Functions

---

🔹 3.1 Definition of a Function

A function is a rule that assigns each input (x) to exactly one output (y).

We often write 

$𝑦=𝑓(𝑥)$

.

• Domain: all possible inputs (x-values).

• Range: all possible outputs (y-values).

⚠️ Important: A function cannot give two different outputs for the same input.

📘 Example 3.1

Is 

${𝑦}^2=𝑥$

a function of x?

✅ Solution: For 

$𝑥=4$

, we get 

$𝑦=2$

or 

$𝑦=\text{–}2$

. Two outputs → Not a function.

---

🔹 3.2 Linear Functions

Form: 

$𝑦=𝑚𝑥+𝑏$

.

• $𝑚$

  = slope (rise over run).

• $𝑏$

  = y-intercept (where line crosses y-axis).

📘 Example 3.2

Graph 

$𝑦=2𝑥+1$

.

• Slope = 2 → rise 2 for every run 1.

• Intercept = 1 → line crosses (0,1).

---

🔹 3.3 Quadratic Functions

Form: 

$𝑦=𝑎{𝑥}^2+𝑏𝑥+𝑐$

.

• Graph is a parabola.

• Opens upward if 
  $𝑎>0$
  , downward if 
  $𝑎<0$
  .

• Vertex: 
  $𝑥=\text{–}\frac{𝑏}{2𝑎}$
  .

📘 Example 3.3

Find vertex of 

$𝑦={𝑥}^2\text{–}4𝑥+3$

.

• $𝑎=1,𝑏=\text{–}4$

  .

• Vertex at 
  $𝑥=\text{–}(\text{–}4)\mathrm{/}(2(1))=2$
  .

• $𝑦(2)=4\text{–}8+3=\text{–}1$

  .
  So vertex = (2, –1).

---

🔹 3.4 Piecewise and Step Functions

Piecewise functions are defined by different rules for different intervals.

📘 Example 3.4

$$𝑓(𝑥)=\{\begin{array}{ll}𝑥+2& 𝑥<0\\ {𝑥}^2& 𝑥\ge 0\end{array}$$

Evaluate 

$𝑓(\text{–}3)$

and 

$𝑓(2)$

.

✅ Solution:

• $𝑓(\text{–}3)=\text{–}3+2=\text{–}1$

  .

• $𝑓(2)=2^2=4$

  .

Step functions: constant on intervals, e.g. greatest integer function 

$𝑓(𝑥)=\lfloor 𝑥\rfloor$

.

---

🔹 3.5 Exponential and Logarithmic Functions (Intro)

• Exponential function: 
  $𝑓(𝑥)={𝑎}^{𝑥}$
  , where 
  $𝑎>0,𝑎\mathrm{\ne }1$
  .

  • Always positive.

  • Growth if 
    $𝑎>1$
    , decay if 
    $0<𝑎<1$
    .

• Logarithmic function: inverse of exponential.

  • $𝑦={\log }_{𝑎}(𝑥)$

    means 
    ${𝑎}^{𝑦}=𝑥$
    .

  • Defined only for 
    $𝑥>0$
    .

📘 Example 3.5

Solve 

$2^{𝑥}=16$

.

✅ Solution: 

$16=2^4$

, so 

$𝑥=4$

.

📘 Example 3.6

Solve 

${\log }_{10}(1000)$

.

✅ Solution: 

${10}^3=1000$

, so 

${\log }_{10}(1000)=3$

.

---

📝 Exercises

Exercise 3.1

State the domain and range:

a) 

$𝑓(𝑥)=2𝑥+5$

b) 

$𝑓(𝑥)={𝑥}^2$

c) 

$𝑓(𝑥)=\sqrt{𝑥}$

Exercise 3.2

Find slope and intercept of:

a) 

$𝑦=5𝑥\text{–}3$

b) 

$𝑦=\text{–}\frac{1}{2}𝑥+4$

Exercise 3.3

Find the vertex of:

a) 

$𝑦={𝑥}^2+6𝑥+5$

b) 

$𝑦=\text{–}2{𝑥}^2+8𝑥\text{–}3$

Exercise 3.4

Evaluate the piecewise function:

$$𝑓(𝑥)=\{\begin{array}{ll}2𝑥\text{–}1& 𝑥<1\\ {𝑥}^2& 𝑥\ge 1\end{array}$$

a) 

$𝑓(0)$

b) 

$𝑓(3)$

Exercise 3.5

Solve:

a) 

$3^{𝑥}=81$

b) 

${\log }_2(32)$

---

✅ Solutions

Solution 3.1

a) Domain: all real, Range: all real.

b) Domain: all real, Range: 

$𝑦\ge 0$

.

c) Domain: 

$𝑥\ge 0$

, Range: 

$𝑦\ge 0$

.

Solution 3.2

a) slope = 5, intercept = –3.

b) slope = –1/2, intercept = 4.

Solution 3.3

a) Vertex at 

$𝑥=\text{–}6\mathrm{/}2=\text{–}3$

, y = 

$(\text{–}3{)}^2+6(\text{–}3)+5=\text{–}4$

. Vertex = (–3, –4).

b) Vertex at 

$𝑥=\text{–}8\mathrm{/}(2\cdot \text{–}2)=2$

. y = –2(4) + 16 – 3 = 5. Vertex = (2, 5).

Solution 3.4

a) For 

$𝑥=0<1$

, 

$𝑓(0)=2(0)\text{–}1=\text{–}1$

.

b) For 

$𝑥=3\ge 1$

, 

$𝑓(3)=9$

.

Solution 3.5

a) 

$81=3^4$

, so 

$𝑥=4$

.

b) 

${\log }_2(32)=5$

because 

$2^5=32$

.

---

🔥 Challenge Problems

1. Find the intersection point(s) of 
   $𝑦=2𝑥+1$
   and 
   $𝑦={𝑥}^2$
   .

2. Solve: 
   ${\log }_3(𝑥)=4$
   .

3. A bacteria population doubles every 3 hours. If the initial population is 500, write the exponential function and find the population after 12 hours.

4. Draw the graph of the piecewise function:

$$𝑓(𝑥)=\{\begin{array}{ll}\text{–}𝑥& 𝑥<0\\ 𝑥+1& 𝑥\ge 0\end{array}$$

---

✅ Summary of Chapter 3

In this chapter, you learned:

• What a function is (input-output rule, domain, range).

• Linear and quadratic functions, with slope, intercepts, and vertex.

• Piecewise and step functions.

• Introduction to exponential and logarithmic functions.

Functions are the core language of mathematics. They will be used extensively in calculus, physics, and engineering.

---

📘 Book I – Foundations & Pre-Calculus

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 4 – Trigonometry Basics

---

🔹 4.1 Angles and Radians

• Angle: the measure of rotation between two rays.

• Units:

  • Degrees (°): full circle = 360°.

  • Radians: full circle = 
    $2𝜋$
    radians.

    • $180\mathrm{°}=𝜋$

      radians.

    • Conversion: 
      $\text{radians}=\frac{𝜋}{180\mathrm{°}}\times \text{degrees}$
      .

📘 Example 4.1

Convert 120° to radians.

✅ Solution: 

$120\mathrm{°}\times \frac{𝜋}{180}=\frac{2𝜋}{3}$

.

---

🔹 4.2 The Unit Circle

• Circle of radius 1 centered at (0,0).

• Every point has coordinates 
  $(\cos 𝜃,\sin 𝜃)$
  .

• Useful for defining trig functions beyond acute angles.

📘 Example 4.2

On the unit circle, find coordinates at 

$𝜃=90\mathrm{°}=\frac{𝜋}{2}$

.

✅ Solution: (0,1).

---

🔹 4.3 Trigonometric Ratios

For a right triangle with angle 

$𝜃$

:

• $\sin 𝜃=\frac{\text{opposite}}{\text{hypotenuse}}$

• $\cos 𝜃=\frac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan 𝜃=\frac{\text{opposite}}{\text{adjacent}}=\frac{\sin 𝜃}{\cos 𝜃}$

• Reciprocal functions:

  • $\csc 𝜃=\frac{1}{\sin 𝜃}$

    , 
    $\sec 𝜃=\frac{1}{\cos 𝜃}$
    , 
    $\cot 𝜃=\frac{1}{\tan 𝜃}$
    .

📘 Example 4.3

In a right triangle, angle 

$𝜃=30\mathrm{°}$

. Find 

$\sin 𝜃$

and 

$\cos 𝜃$

.

✅ Solution: 

$\sin 30\mathrm{°}=\frac{1}{2},\cos 30\mathrm{°}=\frac{\sqrt{3}}{2}$

.

---

🔹 4.4 Pythagorean Identities

Fundamental identity:

$${\sin }^2𝜃+{\cos }^2𝜃=1$$

From this:

• $1+{\tan }^2𝜃={\sec }^2𝜃$

• $1+{\cot }^2𝜃={\csc }^2𝜃$

📘 Example 4.4

If 

$\sin 𝜃=\frac{3}{5}$

, find 

$\cos 𝜃$

.

✅ Solution: 

${\sin }^2𝜃+{\cos }^2𝜃=1$

.

${\left(\frac{3}{5}\right)}^2+{\cos }^2𝜃=1$

.

$\frac{9}{25}+{\cos }^2𝜃=1$

.

${\cos }^2𝜃=\frac{16}{25}\to \cos 𝜃=\frac{4}{5}$

or 

$\text{–}\frac{4}{5}$

.

---

🔹 4.5 Basic Trigonometric Equations

• To solve: use identities and inverse functions.

📘 Example 4.5

Solve 

$\sin 𝑥=\frac{1}{2}$

, 

$0\le 𝑥<2𝜋$

.

✅ Solution: 

$𝑥=\frac{𝜋}{6},\frac{5𝜋}{6}$

.

---

🔹 4.6 Applications – Heights and Distances

Trigonometry is used to measure heights and distances indirectly.

📘 Example 4.6

A tree casts a shadow of 20 m. The angle of elevation of the sun is 30°. Find the height of the tree.

✅ Solution:

$\tan 30\mathrm{°}=\frac{ℎ}{20}$

.

$\frac{1}{\sqrt{3}}=\frac{ℎ}{20}\to ℎ=\frac{20}{\sqrt{3}}\approx 11.55𝑚$

.

---

📝 Exercises

Exercise 4.1 Convert:

a) 45° to radians.

b) 

$\frac{3𝜋}{4}$

radians to degrees.

Exercise 4.2 Find the coordinates on the unit circle for:

a) 

$𝜃=0$

b) 

$𝜃=180\mathrm{°}$

Exercise 4.3 Solve:

a) 

$\sin 60\mathrm{°}$

, 

$\cos 60\mathrm{°}$

, 

$\tan 60\mathrm{°}$

.

b) 

$\sin 45\mathrm{°}$

, 

$\cos 45\mathrm{°}$

.

Exercise 4.4 If 

$\cos 𝜃=\frac{5}{13}$

, find 

$\sin 𝜃$

(assuming 

$𝜃$

is acute).

Exercise 4.5 Solve:

a) 

$\cos 𝑥=0$

, 

$0\le 𝑥<2𝜋$

.

b) 

$\tan 𝑥=1$

, 

$0\le 𝑥<2𝜋$

.

Exercise 4.6 A building is 50 m tall. From the top, the angle of depression to a car is 30°. If the car is on flat ground, how far is it from the base of the building?

---

✅ Solutions

Solution 4.1

a) 

$45\mathrm{°}\times \frac{𝜋}{180}=\frac{𝜋}{4}$

.

b) 

$\frac{3𝜋}{4}\times \frac{180}{𝜋}=135\mathrm{°}$

.

Solution 4.2

a) (1,0)

b) (–1,0)

Solution 4.3

a) 

$\sin 60\mathrm{°}=\frac{\sqrt{3}}{2},\cos 60\mathrm{°}=\frac{1}{2},\tan 60\mathrm{°}=\sqrt{3}$

.

b) 

$\sin 45\mathrm{°}=\cos 45\mathrm{°}=\frac{\sqrt{2}}{2}$

.

Solution 4.4

${\cos }^2𝜃+{\sin }^2𝜃=1$

.

$(\frac{5}{13}{)}^2+{\sin }^2𝜃=1\to {\sin }^2𝜃=\frac{144}{169}$

.

$\sin 𝜃=\frac{12}{13}$

.

Solution 4.5

a) 

$\cos 𝑥=0\to 𝑥=\frac{𝜋}{2},\frac{3𝜋}{2}$

.

b) 

$\tan 𝑥=1\to 𝑥=\frac{𝜋}{4},\frac{5𝜋}{4}$

.

Solution 4.6

$\tan 30\mathrm{°}=\frac{50}{𝑑}$

.

$\frac{1}{\sqrt{3}}=\frac{50}{𝑑}\to 𝑑=50\sqrt{3}\approx 86.6𝑚$

.

---

🔥 Challenge Problems

1. Convert 225° into radians, then find its sine and cosine using the unit circle.

2. Solve: 
   $\sin 𝑥=\cos 𝑥$
   , 
   $0\le 𝑥<2𝜋$
   .

3. A ladder 10 m long leans against a wall at an angle of 60°. How high does it reach on the wall?

4. Prove that 
   $\sin (90\mathrm{°}\text{–}𝜃)=\cos 𝜃$
   using a right triangle.

---

✅ Summary of Chapter 4

In this chapter, you learned:

• How to work with angles and radians.

• The unit circle and its role in defining trig functions.

• Trigonometric ratios (sin, cos, tan, and reciprocals).

• Pythagorean identities and how to use them.

• Solving basic trig equations.

• Practical applications to heights and distances.

Trigonometry connects algebra and geometry, and it is essential for calculus, physics, and engineering.

---

📘 Book I – Foundations & Pre-Calculus

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 5 – Vectors in 2D

---

🔹 5.1 Introduction to Vectors

• A vector is a quantity that has both magnitude (length) and direction.

• Represented as an arrow in the plane.

• Written as:

  • $\overrightarrow{𝑣}=⟨{𝑣}_{𝑥},{𝑣}_{𝑦}⟩$

    (component form), or

  • $\overrightarrow{𝑣}={𝑣}_{𝑥}\mathbf{𝑖}+{𝑣}_{𝑦}\mathbf{𝑗}$

    (unit vector form).

📘 Example 5.1

Vector from point A(2,1) to B(5,4):

$$\overrightarrow{𝐴𝐵}=⟨5\text{–}2,4\text{–}1⟩=⟨3,3⟩$$

---

🔹 5.2 Magnitude of a Vector

The length (magnitude) of 

$\overrightarrow{𝑣}=⟨{𝑣}_{𝑥},{𝑣}_{𝑦}⟩$

:

$$\mathrm{\mid }\overrightarrow{𝑣}\mathrm{\mid }=\sqrt{{𝑣}_{𝑥}^2+{𝑣}_{𝑦}^2}$$

📘 Example 5.2

Find the magnitude of 

$\overrightarrow{𝑣}=⟨6,8⟩$

.

✅ Solution: 

$\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$

.

---

🔹 5.3 Direction of a Vector

The direction angle

$𝜃$

 is measured from the positive x-axis:

$$𝜃={\tan }^{-1}\text{\hspace{0.17em}⁣}\left(\frac{{𝑣}_{𝑦}}{{𝑣}_{𝑥}}\right)$$

📘 Example 5.3

Find the direction of 

$\overrightarrow{𝑣}=⟨3,4⟩$

.

✅ Solution: 

$𝜃={\tan }^{-1}(4\mathrm{/}3)\approx 53.13\mathrm{°}$

.

---

🔹 5.4 Vector Addition and Subtraction

If 

$\overrightarrow{𝑢}=⟨{𝑢}_{𝑥},{𝑢}_{𝑦}⟩$

, 

$\overrightarrow{𝑣}=⟨{𝑣}_{𝑥},{𝑣}_{𝑦}⟩$

:

• Addition: 
  $\overrightarrow{𝑢}+\overrightarrow{𝑣}=⟨{𝑢}_{𝑥}+{𝑣}_{𝑥},{𝑢}_{𝑦}+{𝑣}_{𝑦}⟩$

• Subtraction: 
  $\overrightarrow{𝑢}\text{–}\overrightarrow{𝑣}=⟨{𝑢}_{𝑥}\text{–}{𝑣}_{𝑥},{𝑢}_{𝑦}\text{–}{𝑣}_{𝑦}⟩$

📘 Example 5.4

$\overrightarrow{𝑢}=⟨2,5⟩,\overrightarrow{𝑣}=⟨\text{–}1,3⟩$

.

$$\overrightarrow{𝑢}+\overrightarrow{𝑣}=⟨2+(\text{–}1),5+3⟩=⟨1,8⟩$$

---

🔹 5.5 Scalar Multiplication

If 

$𝑘$

is a number (scalar):

$$𝑘\cdot \overrightarrow{𝑣}=⟨𝑘{𝑣}_{𝑥},𝑘{𝑣}_{𝑦}⟩$$

📘 Example 5.5

$3\cdot ⟨2,\text{–}4⟩=⟨6,\text{–}12⟩$

.

---

🔹 5.6 Dot Product

The dot product of 

$\overrightarrow{𝑢}=⟨{𝑢}_{𝑥},{𝑢}_{𝑦}⟩,\overrightarrow{𝑣}=⟨{𝑣}_{𝑥},{𝑣}_{𝑦}⟩$

:

$$\overrightarrow{𝑢}\cdot \overrightarrow{𝑣}={𝑢}_{𝑥}{𝑣}_{𝑥}+{𝑢}_{𝑦}{𝑣}_{𝑦}$$

Also:

$$\overrightarrow{𝑢}\cdot \overrightarrow{𝑣}=\mathrm{\mid }\overrightarrow{𝑢}\mathrm{\mid }\mathrm{\mid }\overrightarrow{𝑣}\mathrm{\mid }\cos 𝜃$$

where θ = angle between vectors.

📘 Example 5.6

$\overrightarrow{𝑢}=⟨3,4⟩,\overrightarrow{𝑣}=⟨2,\text{–}1⟩$

.

$$\overrightarrow{𝑢}\cdot \overrightarrow{𝑣}=(3)(2)+(4)(\text{–}1)=6\text{–}4=2$$

---

🔹 5.7 Applications in Physics

• Force: Vectors represent forces with magnitude and direction.

• Displacement: Movement in 2D space is a vector.

• Work:
  $𝑊=\overrightarrow{𝐹}\cdot \overrightarrow{𝑑}$
  .

📘 Example 5.7

A force 

$\overrightarrow{𝐹}=⟨10,0⟩$

N moves an object by 

$\overrightarrow{𝑑}=⟨3,4⟩$

m. Find the work.

✅ Solution:

$𝑊=\overrightarrow{𝐹}\cdot \overrightarrow{𝑑}=(10)(3)+(0)(4)=30𝐽$

.

---

📝 Exercises

Exercise 5.1

Find the vector from A(–2,1) to B(4,7).

Exercise 5.2

Find the magnitude of:

a) 

$⟨5,12⟩$

b) 

$⟨\text{–}8,6⟩$

Exercise 5.3

Find the direction angle of 

$⟨\text{–}4,4⟩$

.

Exercise 5.4

Compute:

a) 

$⟨3,2⟩+⟨\text{–}1,5⟩$

b) 

$2\cdot ⟨\text{–}3,7⟩$

Exercise 5.5

Find the dot product:

a) 

$⟨1,2⟩\cdot ⟨3,4⟩$

b) 

$⟨\text{–}2,5⟩\cdot ⟨4,\text{–}1⟩$

Exercise 5.6

A force 

$\overrightarrow{𝐹}=⟨6,8⟩$

N moves an object 5 m in the direction 

$⟨3,4⟩$

. Find the work done.

---

✅ Solutions

Solution 5.1

$\overrightarrow{𝐴𝐵}=⟨4\text{–}(\text{–}2),7\text{–}1⟩=⟨6,6⟩$

.

Solution 5.2

a) 

$\sqrt{5^2+{12}^2}=\sqrt{25+144}=13$

.

b) 

$\sqrt{(\text{–}8{)}^2+6^2}=\sqrt{64+36}=10$

.

Solution 5.3

$𝜃={\tan }^{-1}(4\mathrm{/}(\text{–}4))={\tan }^{-1}(\text{–}1)$

.

This is in Quadrant II → 

$𝜃=135\mathrm{°}$

.

Solution 5.4

a) 

$⟨3+(\text{–}1),2+5⟩=⟨2,7⟩$

.

b) 

$⟨\text{–}6,14⟩$

.

Solution 5.5

a) 

$(1)(3)+(2)(4)=3+8=11$

.

b) 

$(\text{–}2)(4)+(5)(\text{–}1)=\text{–}8\text{–}5=\text{–}13$

.

Solution 5.6

Direction vector 

$⟨3,4⟩$

has magnitude 5 → unit vector = 

$⟨3\mathrm{/}5,4\mathrm{/}5⟩$

.

Displacement = 

$5\cdot ⟨3\mathrm{/}5,4\mathrm{/}5⟩=⟨3,4⟩$

.

Dot product: 

$(6)(3)+(8)(4)=18+32=50𝐽$

.

---

🔥 Challenge Problems

1. Show that 
   $\overrightarrow{𝑢}\cdot \overrightarrow{𝑣}=0$
   if and only if vectors are perpendicular.

2. Find the unit vector in the same direction as 
   $⟨7,24⟩$
   .

3. A 20 N force makes a 60° angle with a displacement of 5 m. Find the work done using the dot product formula.

4. Prove that the magnitude of 
   $𝑘\overrightarrow{𝑣}$
   is 
   $\mathrm{\mid }𝑘\mathrm{\mid }\mathrm{\mid }\overrightarrow{𝑣}\mathrm{\mid }$
   .

---

✅ Summary of Chapter 5

In this chapter, you learned:

• Definition and representation of vectors in 2D.

• How to find magnitude and direction.

• Vector addition, subtraction, scalar multiplication.

• Dot product and its geometric meaning.

• Applications in physics, such as displacement, forces, and work.

Vectors are a bridge between algebra and geometry and are fundamental in physics and engineering.

---

📘 Book I – Foundations & Pre-Calculus

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 6 – Review & Challenge Problems

---

🔹 Purpose of This Chapter

This chapter is designed to:

• Review core topics from Chapters 0–5.

• Provide mixed problems that require linking multiple ideas together.

• Challenge your reasoning with harder problems (often without direct formulas).

• Prepare you for the transition to Book II – Advanced Algebra & Trigonometry.

---

🔹 6.1 Mixed Review Problems

📘 Problem 6.1 – Numbers & Sets

Let 

$𝐴=\{2,4,6,8,10\},𝐵=\{1,2,3,4,5\},𝑈=\{1,\dots ,10\}$

.

• Find 
  $𝐴\cup 𝐵,𝐴\cap 𝐵,𝐴\text{–}𝐵,𝐵\text{′}$
  .

✅ Solution:

• $𝐴\cup 𝐵=\{1,2,3,4,5,6,8,10\}$

• $𝐴\cap 𝐵=\{2,4\}$

• $𝐴\text{–}𝐵=\{6,8,10\}$

• $𝐵\text{′}=\{6,7,8,9,10\}$

---

📘 Problem 6.2 – Algebra

Simplify: 

$2(𝑥+3)\text{–}(3𝑥\text{–}5)+4(𝑥\text{–}1)$

.

✅ Solution:

$2𝑥+6\text{–}3𝑥+5+4𝑥\text{–}4=3𝑥+7$

.

---

📘 Problem 6.3 – Linear Equations

Solve: 

$7𝑥\text{–}4=17$

.

✅ Solution: 

$7𝑥=21\to 𝑥=3$

.

---

📘 Problem 6.4 – Quadratic Equations

Solve: 

$2{𝑥}^2\text{–}5𝑥\text{–}3=0$

.

✅ Solution:

Formula →

$$𝑥=\frac{5\pm \sqrt{25+24}}{4}=\frac{5\pm 7}{4}$$

→ 

$𝑥=3$

or 

$𝑥=\text{–}\frac{1}{2}$

.

---

📘 Problem 6.5 – Functions

Let 

$𝑓(𝑥)=2𝑥+1$

. Find:

• f(0), f(3), f(–2).

• Graph the function.

✅ Solution:

• f(0)=1

• f(3)=7

• f(–2)=–3

Graph: straight line slope 2, intercept 1.

---

📘 Problem 6.6 – Trigonometry

If sin θ=3/5, cos θ>0, find cos θ and tan θ.

✅ Solution:

• cos²θ=1–sin²θ=1–(9/25)=16/25 → cosθ=4/5.

• tanθ=sinθ/cosθ= (3/5)/(4/5)=3/4.

---

📘 Problem 6.7 – Vectors

Find the magnitude of vector 

$𝑣=(3,4)$

.

✅ Solution:

$$\mathrm{\mid }𝑣\mathrm{\mid }=\sqrt{3^2+4^2}=5$$

---

🔹 6.2 Challenge Problems

These problems require multi-step thinking.

📘 Challenge 6.1 – Proof

Prove that the sum of two consecutive integers is always odd.

✅ Solution:

Let numbers be n and n+1.

Sum = n+(n+1)=2n+1 → always odd.

---

📘 Challenge 6.2 – Quadratic Reasoning

The product of two consecutive positive integers is 156. Find them.

✅ Solution:

n(n+1)=156 → n²+n–156=0.

Solve → n=12. So numbers: 12, 13.

---

📘 Challenge 6.3 – Function Composition

Let f(x)=x²+1, g(x)=2x+3. Find (f∘g)(x).

✅ Solution:

f(g(x))=(2x+3)²+1=4x²+12x+10.

---

📘 Challenge 6.4 – Trigonometry

Prove that sin²θ+cos²θ=1.

✅ Solution:

On unit circle, point P=(cosθ,sinθ).

Equation of circle: x²+y²=1.

So cos²θ+sin²θ=1. ✅

---

📘 Challenge 6.5 – Vectors in Physics

A force of 10 N acts east, and another force of 6 N acts north. Find the resultant force.

✅ Solution:

Resultant vector R=(10,6).

$$\mathrm{\mid }𝑅\mathrm{\mid }=\sqrt{{10}^2+6^2}=\sqrt{136}\approx 11.66𝑁$$

Direction: 

${\tan }^{-1}(\frac{6}{10})\approx 31\mathrm{°}$

north of east.

---

🔹 6.3 Cumulative Word Problem

📘 Problem 6.8 – Engineering Application

A ladder 10 m long leans against a wall. Its base is 6 m from the wall. Find the height it reaches.

✅ Solution:

Right triangle with hypotenuse 10, base 6.

Height = √(10²–6²)=√64=8 m.

---

✅ Summary of Chapter 6

This chapter reinforced everything in Book I:

• Numbers and sets

• Algebra and equations

• Functions and graphs

• Trigonometry

• Vectors

Challenge problems prepared you to think beyond single-topic drills, combining methods and reasoning.

With this foundation, you are ready for Book II – Advanced Algebra & Trigonometry, where we expand into complex numbers, conic sections, advanced trigonometric identities, and 3D vectors.

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 1 – Complex Numbers

---

🔹 1.1 Introduction: Why Complex Numbers?

In real life and engineering, equations sometimes have no real solutions.

Example:

$${𝑥}^2+1=0$$

There is no real number whose square is –1.

To solve such cases, mathematicians introduced a new unit:

$$𝑖=\sqrt{-1}$$

• A complex number is of the form:

$$𝑧=𝑎+𝑏𝑖,𝑎,𝑏\in \mathbb{𝑅}$$

where:

• $𝑎$

  = real part (Re z).

• $𝑏$

  = imaginary part (Im z).

📘 Example 1.1

Write in standard form:

1. $\sqrt{-9}=3𝑖$

2. $\text{–}2+\sqrt{-16}=\text{–}2+4𝑖$

---

🔹 1.2 Basic Operations

Addition/Subtraction: combine real parts and imaginary parts.

$$(3+2𝑖)+(1+5𝑖)=4+7𝑖$$

Multiplication: apply distributive law and use 

${𝑖}^2=\text{–}1$

.

$$(2+𝑖)(3\text{–}𝑖)=6\text{–}2𝑖+3𝑖\text{–}{𝑖}^2=7+𝑖$$

Division: multiply numerator and denominator by conjugate.

$$\frac{3+2𝑖}{1\text{–}𝑖}\times \frac{1+𝑖}{1+𝑖}=\frac{(3+2𝑖)(1+𝑖)}{1+1}=\frac{3+3𝑖+2𝑖+2{𝑖}^2}{2}=\frac{1+5𝑖}{2}$$

📘 Example 1.2

Simplify:

$$\frac{2+3𝑖}{4\text{–}𝑖}$$

✅ Solution: Multiply top & bottom by (4+i).

$$\frac{(2+3𝑖)(4+𝑖)}{(4\text{–}𝑖)(4+𝑖)}=\frac{8+2𝑖+12𝑖+3{𝑖}^2}{16+1}=\frac{8+14𝑖\text{–}3}{17}=\frac{5+14𝑖}{17}$$

---

🔹 1.3 Complex Conjugates

For 

$𝑧=𝑎+𝑏𝑖$

, its conjugate is:

$$\overline{𝑧}=𝑎\text{–}𝑏𝑖$$

Property:

$$𝑧\cdot \overline{𝑧}={𝑎}^2+{𝑏}^2(\text{always real})$$

📘 Example 1.3

If 

$𝑧=3+4𝑖$

, then:

• $\overline{𝑧}=3\text{–}4𝑖$

  .

• $𝑧\overline{𝑧}=9+16=25$

  .

---

🔹 1.4 Geometric (Polar) Representation

Any complex number can be represented as a point in the complex plane:

• x-axis = real part

• y-axis = imaginary part

$$𝑧=𝑎+𝑏𝑖=𝑟(\cos 𝜃+𝑖\sin 𝜃)$$

where:

• $𝑟=\mathrm{\mid }𝑧\mathrm{\mid }=\sqrt{{𝑎}^2+{𝑏}^2}$

  (magnitude)

• $𝜃={\tan }^{-1}(𝑏\mathrm{/}𝑎)$

  (argument/angle)

📘 Example 1.4

Convert 

$𝑧=1+\text{√}3𝑖$

to polar form.

• Magnitude: 
  $𝑟=\text{√}(1^2+3)=2$
  .

• Angle: θ = tan⁻¹(√3/1) = 60°.
  So 
  $𝑧=2(\cos 60\mathrm{°}+𝑖\sin 60\mathrm{°})$
  .

📊 (Graph: point at (1,√3) in the complex plane)

---

🔹 1.5 De Moivre’s Theorem

For any complex number in polar form:

$$𝑧=𝑟(\cos 𝜃+𝑖\sin 𝜃)$$$${𝑧}^{𝑛}={𝑟}^{𝑛}(\cos (𝑛𝜃)+𝑖\sin (𝑛𝜃))$$

This simplifies powers and roots of complex numbers.

📘 Example 1.5

Find 

$(1+𝑖{)}^4$

using De Moivre.

• Convert: 
  $𝑟=\text{√}2,𝜃=45\mathrm{°}$
  .

• Apply theorem:

$$(\text{√}2{)}^4[\cos (4\times 45\mathrm{°})+𝑖\sin (4\times 45\mathrm{°})]=4(\cos 180\mathrm{°}+𝑖\sin 180\mathrm{°})=\text{–}4$$

✅ Matches expansion check.

---

📝 Exercises

Exercise 1.1 – Basic Practice

Simplify:

a) 

$(4+3𝑖)+(2\text{–}7𝑖)$

b) 

$(1+𝑖)(1\text{–}𝑖)$

Exercise 1.2 – Division

Simplify:

$$\frac{5+2𝑖}{3+𝑖}$$

Exercise 1.3 – Conjugates

If 

$𝑧=\text{–}2+5𝑖$

, find 

$\overline{𝑧}$

and compute 

$𝑧\overline{𝑧}$

.

Exercise 1.4 – Polar Form

Write 

$𝑧=\text{–}1+𝑖$

in polar form (magnitude + angle).

Exercise 1.5 – De Moivre

Compute 

$(𝑐𝑜𝑠30\mathrm{°}+𝑖\sin 30\mathrm{°}{)}^6$

.

---

✅ Solutions

Solution 1.1

a) 

$6\text{–}4𝑖$

.

b) 

$1\text{–}{𝑖}^2=2$

.

Solution 1.2

Multiply top & bottom by (3–i):

$$\frac{(5+2𝑖)(3\text{–}𝑖)}{9+1}=\frac{15\text{–}5𝑖+6𝑖\text{–}2{𝑖}^2}{10}=\frac{17+𝑖}{10}$$

Solution 1.3

$\overline{𝑧}=\text{–}2\text{–}5𝑖$

.

$𝑧\overline{𝑧}=(\text{–}2{)}^2+(5{)}^2=29$

.

Solution 1.4

Magnitude: √((-1)²+1²)=√2.

Angle: θ=135°.

So 

$𝑧=\text{√}2(\cos 135\mathrm{°}+𝑖\sin 135\mathrm{°})$

.

Solution 1.5

By theorem: 

$𝑐𝑜𝑠(180\mathrm{°})+𝑖\sin (180\mathrm{°})=\text{–}1$

.

---

🔥 Challenge Problems

1. Prove that the product of two complex conjugates is always a real number.

2. Solve 
   ${𝑥}^2+4=0$
   in complex numbers.

3. Show that the nth roots of unity form a regular polygon in the complex plane.

4. Evaluate 
   $(1+𝑖{)}^{10}$
   using De Moivre’s theorem.

---

✅ Summary of Chapter 1

In this chapter, you learned:

• Definition of complex numbers and imaginary unit i.

• Operations: addition, subtraction, multiplication, division.

• Conjugates and their use.

• Polar representation of complex numbers.

• De Moivre’s theorem for powers and roots.

Complex numbers extend mathematics beyond the reals, and are indispensable in engineering (circuits, signals), physics (waves), and computer graphics.

📊 (Suggested figures: complex plane plot, polar vector, unit circle roots of unity).

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 2 – Advanced Polynomials

---

🔹 2.1 Review of Polynomials

A polynomial is an expression of the form:

$$𝑃(𝑥)={𝑎}_{𝑛}{𝑥}^{𝑛}+{𝑎}_{𝑛-1}{𝑥}^{𝑛-1}+\dots +{𝑎}_1𝑥+{𝑎}_0$$

where coefficients 

${𝑎}_{𝑖}\in \mathbb{𝑅}$

(or 

$\mathbb{𝐶}$

), and degree = highest power of x.

Examples:

• $𝑃(𝑥)=3{𝑥}^2+5𝑥\text{–}7$

  (quadratic)

• $𝑄(𝑥)={𝑥}^3\text{–}2𝑥+1$

  (cubic)

---

🔹 2.2 Division of Polynomials

Two main techniques:

1. Long Division – similar to numeric division.

2. Synthetic Division – shortcut for dividing by (x–c).

📘 Example 2.1

Divide 

${𝑥}^3+4{𝑥}^2\text{–}5𝑥\text{–}14$

by 

$𝑥+2$

.

✅ Solution (Synthetic Division):

• Coefficients: 1 | 4 | –5 | –14

• Divide by (x+2) → root = –2.

Carry down:

1 → multiply by –2 → –2 → add → 2 → multiply –4 → add –9 → multiply 18 → add 4.

So:

$${𝑥}^3+4{𝑥}^2\text{–}5𝑥\text{–}14=(𝑥+2)({𝑥}^2+2𝑥\text{–}9)+4$$

---

🔹 2.3 Remainder & Factor Theorems

• Remainder Theorem: If 
  $𝑃(𝑥)$
  divided by (x–c), remainder = P(c).

• Factor Theorem: If P(c)=0, then (x–c) is a factor.

📘 Example 2.2

Check if x=2 is a root of 

$𝑃(𝑥)={𝑥}^3\text{–}3{𝑥}^2+4𝑥\text{–}12$

.

✅ Solution: P(2)=8–12+8–12=–8≠0 → not a root.

---

🔹 2.4 Roots & Factorization

A polynomial of degree n has at most n roots.

Over complex numbers, every polynomial factors completely (Fundamental Theorem of Algebra).

📘 Example 2.3

Factor 

${𝑥}^3\text{–}6{𝑥}^2+11𝑥\text{–}6$

.

✅ Solution: Try x=1 → P(1)=0 → (x–1) factor.

Divide by (x–1) → quotient = x²–5x+6.

Factor further: (x–2)(x–3).

So full factorization:

$${𝑥}^3\text{–}6{𝑥}^2+11𝑥\text{–}6=(𝑥\text{–}1)(𝑥\text{–}2)(𝑥\text{–}3)$$

---

🔹 2.5 Graphing Polynomial Functions

Rules:

• Degree n → graph has up to (n–1) turning points.

• Leading coefficient sign determines end behavior.

• Real roots = x-intercepts.

• Multiplicity of root affects touch/cross behavior.

📘 Example 2.4

Graph 

$𝑦=(𝑥\text{–}1{)}^2(𝑥+2)$

.

• Roots: x=1 (double root), x=–2 (single root).

• Shape: touches x-axis at 1, crosses at –2.

• End behavior: degree 3 with positive leading coefficient → left down, right up.

📊 (Graph: cubic curve touching at 1, crossing at –2)

---

🔹 2.6 Polynomial Inequalities

Solve inequalities by:

1. Factor polynomial.

2. Identify critical points (roots).

3. Test intervals between roots.

📘 Example 2.5

Solve 

${𝑥}^3\text{–}6{𝑥}^2+11𝑥\text{–}6>0$

.

✅ Solution:

Factor = (x–1)(x–2)(x–3).

Check intervals:

• (–∞,1): pick x=0 → (–1)(–2)(–3)=– → negative.

• (1,2): pick 1.5 → (+)(–)(–)=+ → positive.

• (2,3): pick 2.5 → (+)(+)(–)=– → negative.

• (3,∞): pick 4 → (+)(+)(+) → positive.

So solution = (1,2) ∪ (3,∞).

---

📝 Exercises

Exercise 2.1 – Synthetic Division

Divide 

${𝑥}^3+2{𝑥}^2\text{–}5𝑥+6$

by (x–1).

Exercise 2.2 – Factor Theorem

Check whether x=–2 is a root of 

${𝑥}^3+3{𝑥}^2\text{–}4𝑥\text{–}12$

.

Exercise 2.3 – Factorization

Factor completely: 

${𝑥}^3\text{–}{𝑥}^2\text{–}4𝑥+4$

.

Exercise 2.4 – Graphing

Describe end behavior and root behavior of 

$𝑦=(𝑥\text{–}1{)}^3(𝑥+2)$

.

Exercise 2.5 – Inequality

Solve: 

${𝑥}^2\text{–}3𝑥+2<0$

.

---

✅ Solutions

Solution 2.1

Synthetic division root=1.

Coefficients: 1,2,–5,6.

Bring down 1, multiply 1 → add 3, multiply 3 → add –2, multiply –2 → add 4.

So: 

$(𝑥\text{–}1)({𝑥}^2+3𝑥\text{–}2)+4$

.

Solution 2.2

P(–2)= (–8)+(12)+(8)–12=0 → root. ✅

Solution 2.3

Group: (x³–x²)–(4x–4)=x²(x–1)–4(x–1)=(x–1)(x²–4)=(x–1)(x–2)(x+2).

Solution 2.4

Degree=4, positive leading → both ends up.

Root –2 single → crosses.

Root 1 triple → crosses with flattening.

Solution 2.5

Quadratic factors: (x–1)(x–2)<0.

Critical points: 1,2.

Test interval (1,2): negative.

So solution: (1,2).

---

🔥 Challenge Problems

1. Prove that a cubic polynomial always has at least one real root.

2. Solve 
   ${𝑥}^4\text{–}5{𝑥}^2+4=0$
   .

3. Show that the sum of the roots of 
   $𝑎{𝑥}^2+𝑏𝑥+𝑐=0$
   is –b/a.

4. A beam modeled by polynomial 
   $\text{–}0.1{𝑥}^3+0.6{𝑥}^2+2𝑥$
   represents height vs distance. Find where it touches the ground.

---

✅ Summary of Chapter 2

In this chapter, you learned:

• Division of polynomials (long and synthetic).

• Remainder and factor theorems.

• Full factorization techniques.

• Graphing polynomial functions and their behaviors.

• Solving polynomial inequalities.

Polynomials describe countless physical and engineering systems, from motion paths to circuit responses. Mastery of their properties builds the bridge toward sequences, series, and trigonometric analysis.

📊 (Suggested figures: cubic & quartic graphs, root behaviors, inequality sign charts).

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 3 – Sequences and Series

---

🔹 3.1 Introduction

A sequence is an ordered list of numbers following a rule.

Examples:

• Natural numbers: 1, 2, 3, 4…

• Even numbers: 2, 4, 6, 8…

• Squares: 1, 4, 9, 16…

A series is the sum of terms in a sequence.

📘 Example 3.1

Sequence: 2, 4, 6, 8…

Series: 2+4+6+8+…

---

🔹 3.2 Arithmetic Sequences

Each term increases by a fixed difference d.

General term:

$${𝑎}_{𝑛}={𝑎}_1+(𝑛\text{–}1)𝑑$$

📘 Example 3.2

Sequence: 5, 8, 11, 14… (d=3).

• Find the 10th term.
  Solution: 
  ${𝑎}_{10}=5+(10\text{–}1)(3)=32$
  .

Sum of first n terms (Arithmetic Series):

$${𝑆}_{𝑛}=\frac{𝑛}{2}({𝑎}_1+{𝑎}_{𝑛})=\frac{𝑛}{2}(2{𝑎}_1+(𝑛\text{–}1)𝑑)$$

📘 Example 3.3

Find sum of first 20 terms of 5,8,11…

${𝑆}_{20}=\frac{20}{2}[2(5)+(19)(3)]=10(10+57)=670$

.

---

🔹 3.3 Geometric Sequences

Each term is multiplied by fixed ratio r.

General term:

$${𝑎}_{𝑛}={𝑎}_1{𝑟}^{𝑛\text{–}1}$$

📘 Example 3.4

Sequence: 2,6,18,54… (r=3).

• Find 6th term.
  Solution: 
  ${𝑎}_6=2\cdot 3^5=486$
  .

Sum of first n terms (Geometric Series):

$${𝑆}_{𝑛}=\frac{{𝑎}_1({𝑟}^{𝑛}\text{–}1)}{𝑟\text{–}1},𝑟\mathrm{\ne }1$$

📘 Example 3.5

Find sum of first 5 terms of 2,6,18…

${𝑆}_5=\frac{2(3^5\text{–}1)}{2}=2(243\text{–}1)\mathrm{/}2=242$

.

Infinite Geometric Series:

If |r|<1, then:

$${𝑆}_{\mathrm{\infty }}=\frac{{𝑎}_1}{1\text{–}𝑟}$$

📘 Example 3.6

Sum of ½+¼+⅛+…= 

$\frac{1}{1\text{–½}}=1$

.

---

🔹 3.4 Sigma Notation

Compact way to write sums:

$$\sum _{𝑘=1}^{𝑛}{𝑎}_{𝑘}$$

📘 Example 3.7

$$\sum _{𝑘=1}^5{𝑘}^2=1^2+2^2+3^2+4^2+5^2=55$$

---

🔹 3.5 Proof by Induction

Used to prove formulas for sequences and series.

Steps:

1. Base case (n=1).

2. Assume true for n=k.

3. Prove for n=k+1.

📘 Example 3.8

Prove:

$$1+2+3+\dots +𝑛=\frac{𝑛(𝑛+1)}{2}$$

✅ Solution:

• Base: n=1 → 1=½·1·2=1 ✅

• Assume for n=k: sum=k(k+1)/2.

• For n=k+1: sum = k(k+1)/2+(k+1)=(k+1)(k+2)/2 ✅

---

📝 Exercises

Exercise 3.1 – Arithmetic

Find the 15th term and sum of first 15 terms of sequence 7,10,13,…

Exercise 3.2 – Geometric

Find the 8th term and sum of first 8 terms of 3,6,12,…

Exercise 3.3 – Infinite Series

Find sum of: 4+2+1+½+…

Exercise 3.4 – Sigma Notation

Evaluate:

$$\sum _{𝑘=1}^6(2𝑘\text{–}1)$$

Exercise 3.5 – Induction

Prove:

$$1^2+2^2+3^2+\dots +{𝑛}^2=\frac{𝑛(𝑛+1)(2𝑛+1)}{6}$$

---

✅ Solutions

Solution 3.1

d=3.

a₁=7.

a₁₅=7+(14)(3)=49.

S₁₅=15/2(7+49)=15/2·56=420.

Solution 3.2

r=2, a₁=3.

a₈=3·2^7=384.

S₈=3(2^8–1)/(2–1)=3·255=765.

Solution 3.3

Infinite geometric series, a₁=4, r=½.

S=4/(1–½)=8.

Solution 3.4

Odd numbers sum: 1+3+5+7+9+11=36.

Solution 3.5

By induction: base n=1 → 1=1.

Assume true for n=k.

For n=k+1:

$$\frac{𝑘(𝑘+1)(2𝑘+1)}{6}+(𝑘+1{)}^2=\frac{(𝑘+1)(𝑘+2)(2𝑘+3)}{6}$$

✅ holds.

---

🔥 Challenge Problems

1. A ball bounces to 80% of its previous height. If dropped from 10 m, how far has it traveled after infinite bounces?

2. Prove that the sum of first n odd numbers = n².

3. Find the sum: 1+2+4+8+…+2^n.

4. A company doubles its machines every year. If it starts with 5 machines, how many after 10 years?

---

✅ Summary of Chapter 3

You learned:

• Arithmetic and geometric sequences.

• Summation formulas for arithmetic and geometric series.

• Sigma notation for compact sums.

• Proof by induction for formulas.

Sequences and series are essential in engineering (signal processing, growth models), finance (interest), and computer science (algorithms, recursion).

📊 (Suggested figures: arithmetic linear growth, geometric exponential growth, sigma summation bar chart).

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 4 – Advanced Trigonometry

---

🔹 4.1 Review of Basic Trigonometry

Recall:

• sin θ = opposite / hypotenuse

• cos θ = adjacent / hypotenuse

• tan θ = sin θ / cos θ

Pythagorean identity:

$${\sin }^2𝜃+{\cos }^2𝜃=1$$

Now we extend into advanced identities and applications.

---

🔹 4.2 Double-Angle Identities

$$\sin (2𝜃)=2\sin 𝜃\cos 𝜃$$$$\cos (2𝜃)={\cos }^2𝜃\text{–}{\sin }^2𝜃=2{\cos }^2𝜃\text{–}1=1\text{–}2{\sin }^2𝜃$$$$\tan (2𝜃)=\frac{2\tan 𝜃}{1\text{–}{\tan }^2𝜃}$$

📘 Example 4.1

If cos θ = 3/5, sin θ > 0, find sin 2θ.

• sin θ = 4/5.

• sin 2θ = 2(3/5)(4/5)=24/25.

---

🔹 4.3 Half-Angle Identities

$${\sin }^2\frac{𝜃}{2}=\frac{1\text{–}\cos 𝜃}{2},{\cos }^2\frac{𝜃}{2}=\frac{1+\cos 𝜃}{2}$$$$\tan \frac{𝜃}{2}=\frac{\sin 𝜃}{1+\cos 𝜃}=\frac{1\text{–}\cos 𝜃}{\sin 𝜃}$$

📘 Example 4.2

Find sin 15°.

• 15°=30°/2.

• sin(15°)=√((1–cos30°)/2)=√((1–√3/2)/2).

---

🔹 4.4 Sum-to-Product & Product-to-Sum

• Sum-to-Product:

$$\sin 𝐴+\sin 𝐵=2\sin \left(\frac{𝐴+𝐵}{2}\right)\cos \left(\frac{𝐴\text{–}𝐵}{2}\right)$$$$\cos 𝐴+\cos 𝐵=2\cos \left(\frac{𝐴+𝐵}{2}\right)\cos \left(\frac{𝐴\text{–}𝐵}{2}\right)$$

• Product-to-Sum:

$$\sin 𝐴\cos 𝐵=\frac{1}{2}[\sin (𝐴+𝐵)+\sin (𝐴\text{–}𝐵)]$$$$\cos 𝐴\cos 𝐵=\frac{1}{2}[\cos (𝐴+𝐵)+\cos (𝐴\text{–}𝐵)]$$

📘 Example 4.3

Simplify: sin 50° + sin 70°.

= 2 sin(60°) cos(–10°)=2(√3/2)(cos10°)=√3 cos10°.

---

🔹 4.5 Advanced Trigonometric Equations

Solve: sin 2x = cos x.

✅ Solution:

sin 2x=2sin x cos x=cos x.

If cos x=0 → x=90°,270°.

Else divide: 2sin x=1 → sin x=½ → x=30°,150°.

So solutions: 30°,150°,90°,270°.

---

🔹 4.6 Trigonometric Graphs

• sin x, cos x periodic, amplitude=1.

• tan x has asymptotes at odd multiples of 90°.

• Double-angle, half-angle identities modify frequency and amplitude.

📘 Example 4.4

Sketch y=2sin(2x).

• Amplitude=2.

• Period=180° (since period = 360°/2).

📊 (Graph: stretched sine wave with double frequency)

---

🔹 4.7 Applications

1. Engineering – Signal Processing
   Combination of sinusoids modeled with sum-to-product identities.

2. Physics – Waves
   Standing wave patterns described using trigonometric addition formulas.

3. Architecture – Angles
   Half-angle formulas help in design of arcs and curves.

---

📝 Exercises

Exercise 4.1 – Double Angle

If sin θ=5/13, cos θ>0, find cos 2θ.

Exercise 4.2 – Half Angle

Find exact value of cos 22.5°.

Exercise 4.3 – Sum-to-Product

Simplify cos 80°+cos 40°.

Exercise 4.4 – Trig Equation

Solve: 2cos²x–1=0 for 0°≤x<360°.

Exercise 4.5 – Graphing

Sketch y=cos(2x)+1.

---

✅ Solutions

Solution 4.1

cos²θ=1–(25/169)=144/169 → cos θ=12/13.

cos2θ=cos²θ–sin²θ=(144–25)/169=119/169.

Solution 4.2

cos(22.5°)=√((1+cos45°)/2)=√((1+√2/2)/2).

Solution 4.3

cos80°+cos40°=2cos60°cos20°=cos20°.

Solution 4.4

2cos²x–1=0 → cos²x=½ → cos x=±√½=±√2/2.

Solutions: x=45°,135°,225°,315°.

Solution 4.5

Graph: cos(2x) has period 180°, shifted up 1 unit.

---

🔥 Challenge Problems

1. Prove identity: tan(x/2)=sin x/(1+cos x).

2. Solve: sin x+cos x=√2 for 0°≤x<360°.

3. Show that cos³x+sin³x=(cos x+sin x)(1–sin x cos x).

4. Find exact value of sin 75° using sum identity.

---

✅ Summary of Chapter 4

You learned:

• Double-angle, half-angle, sum-to-product, product-to-sum formulas.

• Techniques for solving trigonometric equations.

• Graphing trigonometric functions with transformations.

• Applications to engineering, physics, and real-world design.

📊 (Suggested figures: sin/cos/tan graphs, double-frequency sine, standing wave illustration).

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 5 – Analytic Geometry (Conics)

---

🔹 5.1 Introduction

Conic sections are curves obtained by slicing a cone with a plane.

They include:

• Circle

• Parabola

• Ellipse

• Hyperbola

They are fundamental in astronomy (planetary orbits), engineering (reflectors, antennas), and architecture.

---

🔹 5.2 Circles

Standard equation (center at origin, radius r):

$${𝑥}^2+{𝑦}^2={𝑟}^2$$

General form (center (h,k), radius r):

$$(𝑥\text{–}ℎ{)}^2+(𝑦\text{–}𝑘{)}^2={𝑟}^2$$

📘 Example 5.1

Find equation of circle centered at (2,–3) with radius 5.

✅ Solution:

$$(𝑥\text{–}2{)}^2+(𝑦+3{)}^2=25$$

📊 (Graph: circle centered at (2,–3) with radius 5)

---

🔹 5.3 Parabolas

Definition: set of points equidistant from a focus and directrix.

Equation (opening up):

$${𝑦}^2=4𝑎𝑥$$

or

$${𝑥}^2=4𝑎𝑦$$

📘 Example 5.2

Equation of parabola with vertex at origin, focus (2,0):

→ Standard: 

${𝑦}^2=4𝑎𝑥$

with a=2.

Equation: 

${𝑦}^2=8𝑥$

.

Applications: car headlights, satellite dishes.

---

🔹 5.4 Ellipses

Definition: set of points where sum of distances from two foci is constant.

Equation (horizontal major axis):

$$\frac{{𝑥}^2}{{𝑎}^2}+\frac{{𝑦}^2}{{𝑏}^2}=1,𝑎>𝑏$$

📘 Example 5.3

Find ellipse with a=5, b=3.

Equation:

$$\frac{{𝑥}^2}{25}+\frac{{𝑦}^2}{9}=1$$

📊 (Graph: ellipse stretched along x-axis)

---

🔹 5.5 Hyperbolas

Definition: set of points where the difference of distances from two foci is constant.

Equation (horizontal):

$$\frac{{𝑥}^2}{{𝑎}^2}\text{–}\frac{{𝑦}^2}{{𝑏}^2}=1$$

Equation (vertical):

$$\frac{{𝑦}^2}{{𝑎}^2}\text{–}\frac{{𝑥}^2}{{𝑏}^2}=1$$

📘 Example 5.4

Equation of hyperbola with a=4, b=3 (horizontal):

$$\frac{{𝑥}^2}{16}\text{–}\frac{{𝑦}^2}{9}=1$$

Applications: navigation systems (triangulation), particle physics.

---

🔹 5.6 Applications in Physics & Engineering

• Circles → wheels, gears, circular motion.

• Parabolas → reflectors, bridges, projectile motion.

• Ellipses → planetary orbits, satellite paths.

• Hyperbolas → radio wave intersections, nuclear radiation paths.

---

📝 Exercises

Exercise 5.1 – Circle

Find equation of circle centered at (–1,2) with radius 4.

Exercise 5.2 – Parabola

Find equation of parabola with vertex at origin, focus at (0,3).

Exercise 5.3 – Ellipse

Write equation of ellipse with a=6, b=2.

Exercise 5.4 – Hyperbola

Write equation of hyperbola with a=5, b=12, opening vertically.

Exercise 5.5 – Real Application

A satellite dish is shaped like parabola y²=16x. Where is the focus?

---

✅ Solutions

Solution 5.1

$$(𝑥+1{)}^2+(𝑦\text{–}2{)}^2=16$$

Solution 5.2

Focus (0,3) → equation: x²=12y.

Solution 5.3

$$\frac{{𝑥}^2}{36}+\frac{{𝑦}^2}{4}=1$$

Solution 5.4

$$\frac{{𝑦}^2}{25}\text{–}\frac{{𝑥}^2}{144}=1$$

Solution 5.5

y²=16x → 4a=16 → a=4. Focus=(4,0).

---

🔥 Challenge Problems

1. Prove that parabolic reflectors send parallel rays to the focus.

2. Show that Earth’s orbit is approximately elliptical.

3. Find asymptotes of hyperbola 
   $\frac{{𝑥}^2}{9}\text{–}\frac{{𝑦}^2}{16}=1$
   .

4. Derive general circle equation 
   ${𝑥}^2+{𝑦}^2+𝐷𝑥+𝐸𝑦+𝐹=0$
   .

---

✅ Summary of Chapter 5

In this chapter, you learned:

• Circle, parabola, ellipse, and hyperbola equations.

• How to derive them and apply them.

• Graphical shapes of conics and their physical applications.

Conic sections are everywhere: from the orbits of planets to the design of telescopes and suspension bridges.

📊 (Suggested figures: circle, parabola, ellipse, hyperbola plots side by side).

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 6 – Vectors in 3D

---

🔹 6.1 Introduction

Vectors extend naturally into 3D space, where we deal with x, y, z coordinates.

They are essential in physics (forces, velocity, acceleration), engineering (3D modeling, structures), and computer graphics.

A vector in 3D is written:

$$\overrightarrow{𝑣}=⟨𝑥,𝑦,𝑧⟩$$

• Magnitude:

$$\mathrm{\mid }\overrightarrow{𝑣}\mathrm{\mid }=\sqrt{{𝑥}^2+{𝑦}^2+{𝑧}^2}$$

• Direction: given by unit vector:

$$\widehat{𝑣}=\frac{\overrightarrow{𝑣}}{\mathrm{\mid }\overrightarrow{𝑣}\mathrm{\mid }}$$

📘 Example 6.1

Find magnitude of 

$\overrightarrow{𝑣}=⟨3,4,12⟩$

.

$$\mathrm{\mid }\overrightarrow{𝑣}\mathrm{\mid }=\sqrt{3^2+4^2+{12}^2}=\sqrt{9+16+144}=\sqrt{169}=13$$

---

🔹 6.2 Vector Addition & Scalar Multiplication

• Addition:

$$⟨𝑎,𝑏,𝑐⟩+⟨𝑑,𝑒,𝑓⟩=⟨𝑎+𝑑,𝑏+𝑒,𝑐+𝑓⟩$$

• Scalar multiplication:

$$𝑘⟨𝑎,𝑏,𝑐⟩=⟨𝑘𝑎,𝑘𝑏,𝑘𝑐⟩$$

📘 Example 6.2

$⟨1,2,3⟩+⟨\text{–}2,0,4⟩=⟨\text{–}1,2,7⟩$

.

---

🔹 6.3 Dot Product (Scalar Product)

$$\overrightarrow{𝑎}\cdot \overrightarrow{𝑏}={𝑎}_1{𝑏}_1+{𝑎}_2{𝑏}_2+{𝑎}_3{𝑏}_3=\mathrm{\mid }\overrightarrow{𝑎}\mathrm{\mid }\mathrm{\mid }\overrightarrow{𝑏}\mathrm{\mid }\cos 𝜃$$

📘 Example 6.3

$\overrightarrow{𝑎}=⟨2,\text{–}1,3⟩$

, 

$\overrightarrow{𝑏}=⟨1,4,\text{–}2⟩$

.

Dot = 2(1)+(–1)(4)+3(–2)=2–4–6=–8.

If dot<0 → angle > 90°.

---

🔹 6.4 Cross Product (Vector Product)

$$\overrightarrow{𝑎}\times \overrightarrow{𝑏}=\mid \begin{array}{ccc}\widehat{𝑖}& \widehat{𝑗}& \widehat{𝑘}\\ {𝑎}_1& {𝑎}_2& {𝑎}_3\\ {𝑏}_1& {𝑏}_2& {𝑏}_3\end{array}\mid$$

• Result is a vector perpendicular to both.

• Magnitude = area of parallelogram spanned by a,b.

📘 Example 6.4

$\overrightarrow{𝑎}=⟨1,2,3⟩,\overrightarrow{𝑏}=⟨4,5,6⟩$

.

$$\overrightarrow{𝑎}\times \overrightarrow{𝑏}=⟨(2)(6)\text{–}(3)(5),(3)(4)\text{–}(1)(6),(1)(5)\text{–}(2)(4)⟩$$$$=⟨12\text{–}15,12\text{–}6,5\text{–}8⟩=⟨\text{–}3,6,\text{–}3⟩$$

---

🔹 6.5 Applications to Mechanics

• Forces in 3D: Each force vector resolved into x,y,z components.

• Torque:

$$\overrightarrow{𝜏}=\overrightarrow{𝑟}\times \overrightarrow{𝐹}$$

(position × force).

• Work:

$$𝑊=\overrightarrow{𝐹}\cdot \overrightarrow{𝑑}$$

📘 Example 6.5

Force = 

$⟨10,0,0⟩$

, displacement = 

$⟨3,4,0⟩$

.

Work = 10·3=30 J.

---

🔹 6.6 Planes and Lines in 3D

Equation of a line (vector form):

$$\overrightarrow{𝑟}=\overrightarrow{𝑎}+𝑡\overrightarrow{𝑑},𝑡\in \mathbb{𝑅}$$

Equation of a plane:

$${𝑛}_1𝑥+{𝑛}_2𝑦+{𝑛}_3𝑧=𝑑$$

where 

$\overrightarrow{𝑛}=⟨{𝑛}_1,{𝑛}_2,{𝑛}_3⟩$

is normal vector.

📘 Example 6.6

Plane through (1,2,3) with normal 

$⟨2,\text{–}1,4⟩$

:

2(x–1)–1(y–2)+4(z–3)=0 → 2x–y+4z=12.

---

📝 Exercises

Exercise 6.1 – Magnitude & Unit Vector

Find unit vector of 

$⟨2,3,6⟩$

.

Exercise 6.2 – Dot Product

Find angle between vectors 

$⟨1,0,0⟩,⟨0,1,0⟩$

.

Exercise 6.3 – Cross Product

Compute 

$⟨1,\text{–}1,0⟩\times ⟨0,2,3⟩$

.

Exercise 6.4 – Work

Force = 

$⟨5,2,0⟩$

, displacement = 

$⟨2,1,3⟩$

. Find work.

Exercise 6.5 – Plane

Find equation of plane through (0,0,1) with normal 

$⟨1,1,1⟩$

.

---

✅ Solutions

Solution 6.1

Magnitude = √(4+9+36)=√49=7.

Unit = ⟨2/7,3/7,6/7⟩.

Solution 6.2

Dot=0 → orthogonal → angle=90°.

Solution 6.3

Determinant: ⟨(–1)(3)–0(2),0(0)–1(3),1(2)–(–1)(0)⟩=⟨–3,–3,2⟩.

Solution 6.4

Work = 5·2+2·1+0·3=12 J.

Solution 6.5

Equation: x+y+z=1.

---

🔥 Challenge Problems

1. Show that dot and cross products are related by:

$$\mathrm{\mid }\overrightarrow{𝑎}{\mathrm{\mid }}^2\mathrm{\mid }\overrightarrow{𝑏}{\mathrm{\mid }}^2=(\overrightarrow{𝑎}\cdot \overrightarrow{𝑏}{)}^2+\mathrm{\mid }\overrightarrow{𝑎}\times \overrightarrow{𝑏}{\mathrm{\mid }}^2$$

2. Compute torque of force F=⟨0,10,0⟩ applied at position r=⟨2,0,0⟩.

3. Find intersection line of planes x+y+z=3 and 2x–y+z=1.

4. In 3D graphics, explain why cross product is used to compute surface normals.

---

✅ Summary of Chapter 6

You learned:

• 3D vector representation, magnitude, unit vectors.

• Operations: addition, dot product, cross product.

• Applications in mechanics (work, torque, force).

• Lines and planes equations.

Vectors in 3D are the backbone of engineering, robotics, mechanics, and computer graphics.

📊 (Suggested figures: 3D axis with vector arrow, cross product perpendicular vector, plane in 3D).

---

📘 Book II – Advanced Algebra & Trigonometry

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 7 – Review & Challenge Problems

This chapter is designed to:

• Review the key ideas from Chapters 1–6.

• Provide mixed problems that require combining concepts.

• Introduce real-world applications where multiple algebra/trig tools are needed.

---

🔹 7.1 Core Review Topics

1. Complex Numbers – operations, polar form, De Moivre’s Theorem.

2. Polynomials – factorization, roots, inequalities, graphs.

3. Sequences & Series – arithmetic, geometric, sigma notation, induction.

4. Trigonometry – identities, equations, graphs, applications.

5. Conic Sections – circle, parabola, ellipse, hyperbola equations.

6. 3D Vectors – dot & cross product, planes, mechanics.

---

🔹 7.2 Mixed Practice Problems

📘 Example 7.1 – Complex Numbers

Evaluate 

$(1+𝑖{)}^6$

using De Moivre’s Theorem.

✅ Solution:

• Convert to polar: 
  $𝑟=\text{√}2,𝜃=45\mathrm{°}$
  .

• Raise to power: 
  ${𝑟}^6(\cos 6𝜃+𝑖\sin 6𝜃)=(\text{√}2{)}^6(\cos 270\mathrm{°}+𝑖\sin 270\mathrm{°})=64(0\text{–}𝑖)=\text{–}64𝑖$
  .

---

📘 Example 7.2 – Polynomial Inequality

Solve: 

${𝑥}^3\text{–}3{𝑥}^2\text{–}4𝑥+12>0$

.

✅ Solution:

Factor: (x–3)(x–2)(x+2).

Test intervals → solution: (–2,2) ∪ (3,∞).

---

📘 Example 7.3 – Sequence Problem

Find the sum: 2+4+8+…+1024.

✅ Solution:

Geometric, a₁=2, r=2, last term=1024=2^10.

n=10 terms.

S=2(2^10–1)/(2–1)=2046.

---

📘 Example 7.4 – Trigonometry Application

Solve: sin θ+cos θ=√2 for 0°≤θ<360°.

✅ Solution:

Divide both sides by √2: (sin θ/√2)+(cos θ/√2)=1.

But sin45°=cos45°=√2/2.

So solution: θ=45°,225°.

---

📘 Example 7.5 – Conic Section

Equation of ellipse with vertices (±5,0) and minor axis length 6.

✅ Solution:

a=5, b=3.

Equation: x²/25+y²/9=1.

---

📘 Example 7.6 – 3D Vectors in Mechanics

A force F=⟨3,–2,1⟩ acts at point r=⟨2,0,4⟩.

Find torque τ=r×F.

✅ Solution:

$$\overrightarrow{𝜏}=\mid \begin{array}{ccc}\widehat{𝑖}& \widehat{𝑗}& \widehat{𝑘}\\ 2& 0& 4\\ 3& \text{–}2& 1\end{array}\mid =⟨(0)(1)\text{–}(4)(\text{–}2),(4)(3)\text{–}(2)(1),(2)(\text{–}2)\text{–}(0)(3)⟩$$

= ⟨8,10,–4⟩.

---

🔹 7.3 Real-World Challenge Problems

1. Engineering Bridge Problem
   A suspension cable is modeled by parabola y=0.02x², with towers 100 m apart. Find cable height at midpoint and slope at tower base.

2. Physics Orbit Problem
   A planet orbits sun in ellipse x²/25+y²/16=1. Find eccentricity.

3. Signal Processing Problem
   A sound wave is modeled as sin(200t)+sin(220t). Use sum-to-product to find beat frequency.

4. Structural Force Problem
   A beam is subject to forces F₁=⟨10,0,0⟩ and F₂=⟨0,5,–5⟩. Find resultant force vector and magnitude.

---

🔹 7.4 Suggested Exam-Style Problems

• Prove by induction that 1+3+5+…+(2n–1)=n².

• Factor and solve: x⁴–5x²+4=0.

• Find exact values: sin75°, cos15°.

• Sketch graph of y=tan2x for –180°≤x≤180°.

• Write equation of hyperbola with foci (±5,0), transverse axis length 6.

---

✅ Summary of Chapter 7

This review chapter tied together:

• Algebra of complex numbers and polynomials.

• Sequences and induction proofs.

• Advanced trigonometric identities and equations.

• Conic geometry in analytic form.

• 3D vectors in mechanics and space geometry.

📊 (Suggested illustrations: combined summary chart of conics, 3D force vector diagram, beat frequency wave).

---

📘 Book III – Calculus I

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 1 – Limits and Continuity

---

🔹 1.1 Introduction to Limits

The idea of a limit is central to calculus.

It describes what value a function approaches as the input approaches some number.

$$\underset{𝑥\to 𝑎}{\lim }𝑓(𝑥)=𝐿$$

means: as x gets closer and closer to a, f(x) gets closer to L.

📘 Example 1.1

$$\underset{𝑥\to 2}{\lim }(3𝑥+1)=7$$

because as x→2, expression = 3(2)+1=7.

---

🔹 1.2 Left-Hand & Right-Hand Limits

• Left-hand limit:

$$\underset{𝑥\to {𝑎}^{-}}{\lim }𝑓(𝑥)$$

• Right-hand limit:

$$\underset{𝑥\to {𝑎}^{+}}{\lim }𝑓(𝑥)$$

If both equal → limit exists.

📘 Example 1.2

f(x)=|x|/x.

• As x→0⁺, f(x)=1.

• As x→0⁻, f(x)=–1.
  → Two-sided limit does not exist.

---

🔹 1.3 Infinite Limits & Vertical Asymptotes

If f(x) grows without bound near x=a:

$$\underset{𝑥\to 𝑎}{\lim }𝑓(𝑥)=\pm \mathrm{\infty }$$

📘 Example 1.3

$$\underset{𝑥\to 0^{+}}{\lim }\frac{1}{𝑥}=+\mathrm{\infty }$$

Graph: vertical asymptote at x=0.

---

🔹 1.4 Limit Laws

• Constant: 
  $\lim 𝑐=𝑐$

• Sum: 
  $\lim (𝑓+𝑔)=\lim 𝑓+\lim 𝑔$

• Product: 
  $\lim (𝑓\cdot 𝑔)=\lim 𝑓\cdot \lim 𝑔$

• Quotient: 
  $\lim (𝑓\mathrm{/}𝑔)=\lim 𝑓\mathrm{/}\lim 𝑔$
  , if denominator ≠0

📘 Example 1.4

$$\underset{𝑥\to 3}{\lim }({𝑥}^2+2𝑥)=9+6=15$$

---

🔹 1.5 Special Limits

1. ${\lim }_{𝑥\to 0}\frac{\sin 𝑥}{𝑥}=1$

2. ${\lim }_{𝑥\to \mathrm{\infty }}(1+1\mathrm{/}𝑥{)}^{𝑥}=𝑒$

3. Exponential/logarithmic growth patterns.

---

🔹 1.6 Continuity

A function is continuous at x=a if:

1. f(a) is defined.

2. ${\lim }_{𝑥\to 𝑎}𝑓(𝑥)$

   exists.

3. ${\lim }_{𝑥\to 𝑎}𝑓(𝑥)=𝑓(𝑎)$

   .

📘 Example 1.5

f(x)=x² is continuous everywhere.

But f(x)=1/x is discontinuous at x=0.

---

🔹 1.7 Types of Discontinuity

1. Removable (hole)

2. Jump discontinuity

3. Infinite discontinuity

📊 (Graph examples: hole at x=2, step function, 1/x asymptote).

---

📝 Exercises

Exercise 1.1 – Direct Limits

Find:

a) 

${\lim }_{𝑥\to 2}({𝑥}^2+3𝑥)$

b) 

${\lim }_{𝑥\to \text{–}1}(2𝑥+5)$

Exercise 1.2 – Piecewise Function

f(x)={x² if x<0, x+1 if x≥0}.

Find: 

${\lim }_{𝑥\to 0^{-}}𝑓(𝑥)$

, 

${\lim }_{𝑥\to 0^{+}}𝑓(𝑥)$

, does limit exist?

Exercise 1.3 – Infinite Limit

Find 

${\lim }_{𝑥\to 0^{-}}\frac{1}{𝑥}$

.

Exercise 1.4 – Continuity

Is f(x)=(x²–4)/(x–2) continuous at x=2?

Exercise 1.5 – Challenge

Prove: 

${\lim }_{𝑥\to 0}\frac{\sin (3𝑥)}{𝑥}=3$

.

---

✅ Solutions

Solution 1.1

a) 2²+3(2)=4+6=10

b) 2(–1)+5=3

Solution 1.2

Left: f(0⁻)=0²=0.

Right: f(0⁺)=0+1=1.

Not equal → limit does not exist.

Solution 1.3

As x→0⁻, denominator→0 negative → –∞.

Solution 1.4

Simplify: (x²–4)/(x–2)=(x–2)(x+2)/(x–2)=x+2, for x≠2.

So limit at 2 is 4.

But f(2) undefined → removable discontinuity.

Solution 1.5

$\lim \frac{\sin (3𝑥)}{𝑥}=\lim 3\cdot \frac{\sin (3𝑥)}{3𝑥}=3(1)=3$

. ✅

---

🔥 Challenge Problems

1. Show that 
   ${\lim }_{𝑥\to \mathrm{\infty }}\frac{2{𝑥}^2+3}{5{𝑥}^2\text{–}1}=2\mathrm{/}5$
   .

2. Sketch graph of f(x)=tan x near 90°. Describe discontinuity.

3. A tank fills with water volume V(t)=100(1–e^(–0.1t)). Find limit of V as t→∞.

4. Prove continuity of polynomial functions at all real numbers.

---

✅ Summary of Chapter 1

You learned:

• Concept of limits (approach, not reach).

• Left/right-hand limits, infinite limits.

• Limit laws and special cases.

• Continuity and types of discontinuity.

This sets the stage for derivatives, which are defined directly using limits.

📊 (Suggested figures: piecewise step function, 1/x asymptote, sinx/x approaching 1).

---

📘 Book III – Calculus I

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 2 – Differentiation Rules

---

🔹 2.1 Introduction: The Derivative

The derivative measures rate of change of a function.

It is defined as a limit:

$${𝑓}^{\mathrm{\prime }}(𝑥)=\underset{ℎ\to 0}{\lim }\frac{𝑓(𝑥+ℎ)-𝑓(𝑥)}{ℎ}$$

Interpretations:

• Slope of tangent line at a point.

• Instantaneous velocity in physics.

• Sensitivity of output to input change.

📘 Example 2.1

For 

$𝑓(𝑥)={𝑥}^2$

:

$${𝑓}^{\mathrm{\prime }}(𝑥)=\underset{ℎ\to 0}{\lim }\frac{(𝑥+ℎ{)}^2-{𝑥}^2}{ℎ}=\underset{ℎ\to 0}{\lim }\frac{2𝑥ℎ+{ℎ}^2}{ℎ}=2𝑥$$

So slope of parabola at x=3 → f′(3)=6.

---

🔹 2.2 Basic Rules of Differentiation

1. Constant Rule:

$$\frac{𝑑}{𝑑𝑥}(𝑐)=0$$

2. Power Rule:

$$\frac{𝑑}{𝑑𝑥}({𝑥}^{𝑛})=𝑛{𝑥}^{𝑛-1}$$

3. Constant Multiple Rule:

$$\frac{𝑑}{𝑑𝑥}[𝑐𝑓(𝑥)]=𝑐{𝑓}^{\mathrm{\prime }}(𝑥)$$

4. Sum Rule:

$$\frac{𝑑}{𝑑𝑥}[𝑓(𝑥)+𝑔(𝑥)]={𝑓}^{\mathrm{\prime }}(𝑥)+{𝑔}^{\mathrm{\prime }}(𝑥)$$

📘 Example 2.2

Differentiate 

$𝑓(𝑥)=5{𝑥}^4\text{–}3𝑥+7$

.

$${𝑓}^{\mathrm{\prime }}(𝑥)=20{𝑥}^3\text{–}3$$

---

🔹 2.3 Product & Quotient Rules

• Product Rule:

$$(𝑓𝑔{)}^{\mathrm{\prime }}={𝑓}^{\mathrm{\prime }}𝑔+𝑓{𝑔}^{\mathrm{\prime }}$$

• Quotient Rule:

$${\left(\frac{𝑓}{𝑔}\right)}^{\mathrm{\prime }}=\frac{{𝑓}^{\mathrm{\prime }}𝑔\text{–}𝑓{𝑔}^{\mathrm{\prime }}}{{𝑔}^2},𝑔\mathrm{\ne }0$$

📘 Example 2.3

Differentiate 

$𝑓(𝑥)={𝑥}^2\sin 𝑥$

.

Using product rule:

$${𝑓}^{\mathrm{\prime }}(𝑥)=2𝑥\sin 𝑥+{𝑥}^2\cos 𝑥$$

---

🔹 2.4 Chain Rule

When a function is composed of functions:

$$\frac{𝑑}{𝑑𝑥}[𝑓(𝑔(𝑥))]={𝑓}^{\mathrm{\prime }}(𝑔(𝑥))\cdot {𝑔}^{\mathrm{\prime }}(𝑥)$$

📘 Example 2.4

Differentiate 

$𝑓(𝑥)=(3{𝑥}^2+1{)}^5$

.

Outer: u^5 → 5u^4.

Inner: 3x^2+1 → derivative 6x.

So:

$${𝑓}^{\mathrm{\prime }}(𝑥)=5(3{𝑥}^2+1{)}^4\cdot 6𝑥=30𝑥(3{𝑥}^2+1{)}^4$$

---

🔹 2.5 Higher-Order Derivatives

Second derivative:

$${𝑓}^{\mathrm{\prime }\mathrm{\prime }}(𝑥)=\frac{{𝑑}^2𝑦}{𝑑{𝑥}^2}$$

Gives concavity and acceleration in physics.

📘 Example 2.5

If f(x)=x^3, then f′(x)=3x^2, f′′(x)=6x.

---

🔹 2.6 Implicit Differentiation

Used when y is not isolated.

📘 Example 2.6

Given: 

${𝑥}^2+{𝑦}^2=25$

. Differentiate both sides:

$$2𝑥+2𝑦\frac{𝑑𝑦}{𝑑𝑥}=0$$$$\frac{𝑑𝑦}{𝑑𝑥}=-\frac{𝑥}{𝑦}$$

This gives slope of tangent to circle.

---

🔹 2.7 Applications to Rates of Change

• Velocity = derivative of position.

• Acceleration = derivative of velocity.

• Economics: marginal cost = derivative of cost function.

📘 Example 2.7

If s(t)=4t^2, find velocity at t=3.

s′(t)=8t → v(3)=24.

---

📝 Exercises

Exercise 2.1 – Power Rule

Differentiate: f(x)=7x^5–4x^3+9.

Exercise 2.2 – Product Rule

Differentiate: f(x)=x^2 cos x.

Exercise 2.3 – Quotient Rule

Differentiate: f(x)=(x^2+1)/(x–1).

Exercise 2.4 – Chain Rule

Differentiate: f(x)=(2x+5)^6.

Exercise 2.5 – Implicit Differentiation

Differentiate: x^2+xy+y^2=7.

---

✅ Solutions

Solution 2.1

f′(x)=35x^4–12x^2.

Solution 2.2

f′(x)=2x cos x–x^2 sin x.

Solution 2.3

f′(x)=[(2x)(x–1)–(x^2+1)(1)]/(x–1)^2.

Solution 2.4

f′(x)=6(2x+5)^5·2=12(2x+5)^5.

Solution 2.5

Differentiate:

2x+(x dy/dx + y)+2y dy/dx=0.

Collect: (x+2y)dy/dx=–(2x+y).

dy/dx=–(2x+y)/(x+2y).

---

🔥 Challenge Problems

1. Prove by first principles that derivative of x^n is nx^(n–1).

2. Differentiate: y=ln(sin x).

3. A balloon rises vertically while wind blows sideways. If x(t)=3t, y(t)=4t^2, find velocity vector at t=2.

4. For curve x^3+y^3=6xy, find dy/dx at (3,3).

---

✅ Summary of Chapter 2

You learned:

• The derivative as a limit.

• Differentiation rules: power, product, quotient, chain.

• Higher-order and implicit derivatives.

• Applications to motion, slopes, and marginal analysis.

📊 (Suggested figures: slope of tangent line, derivative curve of parabola, circle tangent with implicit differentiation).

---

📘 Book III – Calculus I

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 3 – Applications of Derivatives

---

🔹 3.1 Slopes, Tangents, and Normals

• Equation of tangent line:
  At point (x₀,f(x₀)), slope = f′(x₀).
  Equation:

$$𝑦\text{–}𝑓({𝑥}_0)=𝑓\text{′}({𝑥}_0)(𝑥\text{–}{𝑥}_0)$$

• Equation of normal line:
  Slope = –1/f′(x₀).

📘 Example 3.1

f(x)=x², at x=2.

f′(x)=2x=4.

Tangent: y–4=4(x–2) → y=4x–4.

Normal: slope=–1/4 → y–4=(–1/4)(x–2).

📊 (Graph: parabola with tangent and normal at x=2).

---

🔹 3.2 Increasing & Decreasing Functions

• If f′(x)>0 → function increasing.

• If f′(x)<0 → decreasing.

📘 Example 3.2

f(x)=x³–3x²+2.

f′(x)=3x²–6x.

Critical points: f′(x)=0 → x=0,2.

Test:

• x<0 → f′>0 → increasing.

• 0<x<2 → f′<0 → decreasing.

• x>2 → f′>0 → increasing.

📊 (Graph: cubic curve showing increase-decrease pattern).

---

🔹 3.3 Maxima & Minima

• First derivative test:
  f′ changes sign → local max/min.

• Second derivative test:
  If f′(x₀)=0 and f′′(x₀)>0 → min.
  If f′′(x₀)<0 → max.

📘 Example 3.3

f(x)=x²–4x+3.

f′(x)=2x–4=0 → x=2.

f′′(x)=2>0 → local minimum at (2,–1).

---

🔹 3.4 Curve Sketching

Steps:

1. Find domain.

2. Find intercepts.

3. Find critical points (f′=0).

4. Check concavity (f′′).

5. Sketch with asymptotes if any.

📘 Example 3.4

f(x)=(x²–1)/(x–1).

Simplify: f(x)=x+1 (x≠1).

Graph: line y=x+1 with a hole at (1,2).

---

🔹 3.5 Optimization Problems

Derivatives help solve real-world max/min problems.

📘 Example 3.5

A rectangular area is fenced on three sides (fourth side is a river). 100 m of fencing available. Maximize area.

Let width=x, length=y.

Constraint: 2x+y=100 → y=100–2x.

Area A=xy=x(100–2x)=100x–2x².

A′=100–4x=0 → x=25, y=50.

Max area=25·50=1250 m².

📊 (Graph: parabola showing area vs width).

---

🔹 3.6 Newton’s Method (Intro)

For solving equations f(x)=0 numerically:

Formula:

$${𝑥}_{𝑛+1}={𝑥}_{𝑛}\text{–}\frac{𝑓({𝑥}_{𝑛})}{𝑓\text{′}({𝑥}_{𝑛})}$$

📘 Example 3.6

Solve √x=cos x near x=0.5.

Iteratively apply Newton’s method. (Steps shown in extended exercises).

---

📝 Exercises

Exercise 3.1 – Tangent Line

Find equation of tangent to f(x)=x³ at x=1.

Exercise 3.2 – Increasing/Decreasing

Determine intervals of increase/decrease for f(x)=x²–6x+5.

Exercise 3.3 – Extrema

Find local max/min of f(x)=x³–3x²+1.

Exercise 3.4 – Curve Sketching

Sketch y=(x²–9)/(x–3).

Exercise 3.5 – Optimization

A box with square base has volume 100 cm³. Find dimensions for minimum surface area.

Exercise 3.6 – Newton’s Method

Use Newton’s method to approximate √2 starting with x₀=1.

---

✅ Solutions

Solution 3.1

f′(x)=3x², at x=1 → slope=3.

Tangent: y–1=3(x–1).

Solution 3.2

f′(x)=2x–6. Zero at x=3.

x<3 → f′<0 → decreasing.

x>3 → f′>0 → increasing.

Solution 3.3

f′=3x²–6x=3x(x–2).

Critical pts: 0,2.

f′′=6x–6.

At x=0 → f′′=–6 → max.

At x=2 → f′′=6 → min.

Solution 3.4

y=(x²–9)/(x–3)=(x–3)(x+3)/(x–3)=x+3 (x≠3).

Graph: line with hole at (3,6).

Solution 3.5

Volume=100=x²h → h=100/x².

Surface= x²+4xh= x²+400/x.

Differentiate → S′=2x–400/x².

Solve=0 → x³=200 → x≈5.85.

h=100/(x²)≈2.92.

Solution 3.6

f(x)=x²–2.

Newton: x₁=1–(1²–2)/(2·1)=1+0.5=1.5.

x₂=1.5–(2.25–2)/(3)=1.5–0.083=1.417.

x₃≈1.414 → converges to √2.

---

🔥 Challenge Problems

1. A farmer wants a rectangular field of max area with 200 m fencing, but one side is a barn. Find dimensions.

2. Use 2nd derivative test to classify turning points of f(x)=x⁴–4x².

3. Apply Newton’s method to find root of cos x–x=0.

4. Show: slope of tangent to circle x²+y²=r² at (x₀,y₀) is –x₀/y₀.

---

✅ Summary of Chapter 3

You learned:

• Derivatives as slopes → tangents & normals.

• How derivatives describe growth/decay.

• Max/min analysis using 1st & 2nd derivative tests.

• Curve sketching with calculus.

• Optimization problems in real life.

• Newton’s method for solving equations.

📊 (Suggested figures: tangent & normal to parabola, cubic function with extrema, optimization parabola, Newton’s tangent method illustration).

---

📘 Book III – Calculus I

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 4 – Related Rates and Kinematics

---

🔹 4.1 What Are Related Rates?

• Definition: Problems where two or more variables are related by an equation, and each variable changes with time.

• We differentiate with respect to t (time) to connect their rates of change.

📘 Example 4.1

A circle expands with radius r(t).

Area: 

$𝐴=𝜋{𝑟}^2$

.

Differentiate wrt t:

$$\frac{𝑑𝐴}{𝑑𝑡}=2𝜋𝑟\frac{𝑑𝑟}{𝑑𝑡}$$

So rate of area growth depends on both radius and how fast radius grows.

---

🔹 4.2 Strategy for Related Rates Problems

1. Identify given quantities and rates.

2. Write equation connecting variables.

3. Differentiate wrt time t.

4. Substitute known values.

---

🔹 4.3 Kinematics: Motion Along a Line

Position: s(t)

Velocity: v(t)=ds/dt

Acceleration: a(t)=dv/dt=d²s/dt²

📘 Example 4.2

s(t)=5t².

• v(t)=10t.

• a(t)=10.

At t=3, position=45, velocity=30, acceleration=10.

---

🔹 4.4 Related Rates in Geometry

📘 Example 4.3 – Inflating a Balloon

Volume: 

$𝑉=\frac{4}{3}𝜋{𝑟}^3$

.

Differentiate:

$$\frac{𝑑𝑉}{𝑑𝑡}=4𝜋{𝑟}^2\frac{𝑑𝑟}{𝑑𝑡}$$

If r=5 cm, dr/dt=2 cm/s → dV/dt=200π cm³/s.

📘 Example 4.4 – Sliding Ladder

A 10 m ladder leans on wall. Bottom slides away at 1 m/s.

Equation: 

${𝑥}^2+{𝑦}^2=100$

.

Differentiate:

$$2𝑥\frac{𝑑𝑥}{𝑑𝑡}+2𝑦\frac{𝑑𝑦}{𝑑𝑡}=0$$

At x=6, y=8, dx/dt=1 → dy/dt=–(x dx/dt)/y=–6/8=–0.75 m/s.

---

🔹 4.5 Growth and Decay Models

Some quantities change proportionally to size.

Equation:

$$\frac{𝑑𝑦}{𝑑𝑡}=𝑘𝑦$$

Solution:

$$𝑦(𝑡)={𝑦}_0{𝑒}^{𝑘𝑡}$$

• k>0 → exponential growth.

• k<0 → exponential decay.

📘 Example 4.5 – Radioactive Decay

Carbon-14 decays with k≈–0.000121.

If initial =100 g, after 5000 years:

$$𝑦=100{𝑒}^{\text{–}0.000121\cdot 5000}\approx 54.5𝑔$$

---

🔹 4.6 Population & Motion Problems

📘 Example 4.6 – Population

If population grows at rate proportional to size, doubling every 20 years:

Equation: 

$𝑃(𝑡)={𝑃}_0\cdot 2^{𝑡\mathrm{/}20}$

.

📘 Example 4.7 – Physics Motion

A particle moves: s(t)=t³–6t²+9t.

• v(t)=3t²–12t+9.

• a(t)=6t–12.
  At t=2: s=2, v=–3, a=0.

---

📝 Exercises

Exercise 4.1 – Expanding Circle

Radius grows 0.5 cm/s, r=10 cm. Find dA/dt.

Exercise 4.2 – Ladder

12 m ladder against wall, bottom slides at 2 m/s. Find top speed down when bottom 5 m from wall.

Exercise 4.3 – Growth

Bacteria doubles every 3 hours. If 100 at t=0, how many after 12 hours?

Exercise 4.4 – Kinematics

s(t)=4t³–3t². Find v(t), a(t). Compute at t=1.

Exercise 4.5 – Challenge

Sand pours forming a conical pile: V=(1/3)πr²h, with h=r. Sand at 2 m³/min. Find dr/dt when r=2 m.

---

✅ Solutions

Solution 4.1

dA/dt=2πr dr/dt=2π(10)(0.5)=10π cm²/s.

Solution 4.2

x=5, y=√(144–25)=√119≈10.9.

Equation: 2x dx/dt+2y dy/dt=0.

dy/dt=–(x dx/dt)/y=–(5·2)/10.9≈–0.92 m/s.

Solution 4.3

P=100·2^(12/3)=100·2⁴=1600.

Solution 4.4

v=12t²–6t, a=24t–6.

At t=1: v=6, a=18.

Solution 4.5

V=(1/3)πr³.

dV/dt=πr² dr/dt.

2=π(2²)dr/dt=4π dr/dt.

dr/dt=2/(4π)=1/(2π) m/min.

---

🔥 Challenge Problems

1. A spotlight 10 m away shines on a wall. A person 2 m tall walks away at 1.5 m/s. How fast does the shadow top rise when he is 4 m from the spotlight?

2. A tank leaks water at 5 L/min but is filled at 8 L/min. If initially empty, write equation for volume over time.

3. Radioactive isotope half-life = 30 years. Write decay law and compute % left after 90 years.

4. A rocket moves s(t)=100t–4.9t². Find max height and when it occurs.

---

✅ Summary of Chapter 4

You learned:

• How rates of change connect through related rates.

• How derivatives model motion (position, velocity, acceleration).

• Applications to geometry, physics, and growth/decay.

• Real-world problems with ladders, cones, population, and motion.

📊 (Suggested figures: circle radius expanding, ladder sliding, exponential growth/decay graphs, parabola for motion).

---

📘 Book III – Calculus I

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 5 – Introduction to Integration

---

🔹 5.1 The Idea of Integration

• Differentiation gives rate of change.

• Integration is the reverse: it finds the original function (antiderivative) or the area under a curve.

$$\int 𝑓(𝑥)𝑑𝑥=𝐹(𝑥)+𝐶$$

where F′(x)=f(x), and C is a constant (the constant of integration).

📘 Example 5.1

$$\int 2𝑥𝑑𝑥={𝑥}^2+𝐶$$

Because derivative of 

${𝑥}^2$

is 2x.

---

🔹 5.2 Indefinite Integrals

Rules:

1. Power Rule for Integration

$$\int {𝑥}^{𝑛}𝑑𝑥=\frac{{𝑥}^{𝑛+1}}{𝑛+1}+𝐶,(𝑛\mathrm{\ne }-1)$$

2. Constant Multiple Rule

$$\int 𝑐𝑓(𝑥)𝑑𝑥=𝑐\int 𝑓(𝑥)𝑑𝑥$$

3. Sum Rule

$$\int [𝑓(𝑥)+𝑔(𝑥)]𝑑𝑥=\int 𝑓(𝑥)𝑑𝑥+\int 𝑔(𝑥)𝑑𝑥$$

📘 Example 5.2

$$\int (3{𝑥}^2+4𝑥+5)𝑑𝑥={𝑥}^3+2{𝑥}^2+5𝑥+𝐶$$

---

🔹 5.3 Common Integrals

$$\int {𝑒}^{𝑥}𝑑𝑥={𝑒}^{𝑥}+𝐶$$$$\int \frac{1}{𝑥}𝑑𝑥=\ln \mathrm{\mid }𝑥\mathrm{\mid }+𝐶$$$$\int \cos 𝑥𝑑𝑥=\sin 𝑥+𝐶$$$$\int \sin 𝑥𝑑𝑥=-\cos 𝑥+𝐶$$

📘 Example 5.3

$$\int (\sin 𝑥+{𝑒}^{𝑥})𝑑𝑥=-\cos 𝑥+{𝑒}^{𝑥}+𝐶$$

---

🔹 5.4 The Definite Integral

The definite integral calculates exact area under curve from a to b:

$${\int }_{𝑎}^{𝑏}𝑓(𝑥)𝑑𝑥=𝐹(𝑏)\text{–}𝐹(𝑎)$$

📘 Example 5.4

$${\int }_0^2(3{𝑥}^2)𝑑𝑥=[{𝑥}^3{]}_0^2=8\text{–}0=8$$

Graph: area under parabola y=3x² between 0 and 2.

---

🔹 5.5 The Fundamental Theorem of Calculus

This theorem connects derivatives and integrals:

1. Differentiation and integration are inverse processes.

2. Definite integral gives accumulated area via antiderivative.

$$\frac{𝑑}{𝑑𝑥}{\int }_{𝑎}^{𝑥}𝑓(𝑡)𝑑𝑡=𝑓(𝑥)$$

---

🔹 5.6 Substitution Method (u-Substitution)

When function is composite, substitution simplifies.

📘 Example 5.5

$$\int (2𝑥)({𝑥}^2+1{)}^5𝑑𝑥$$

Let u=x²+1, du=2x dx.

So integral = ∫ u^5 du = u^6/6 + C = (x²+1)^6/6 + C.

---

📝 Exercises

Exercise 5.1 – Power Rule

$\int (4{𝑥}^3\text{–}6{𝑥}^2+2)𝑑𝑥$

Exercise 5.2 – Exponentials & Logs

$\int ({𝑒}^{𝑥}+1\mathrm{/}𝑥)𝑑𝑥$

Exercise 5.3 – Trigonometry

$\int (𝑐𝑜𝑠𝑥\text{–}3𝑠𝑖𝑛𝑥)𝑑𝑥$

Exercise 5.4 – Definite Integral

${\int }_1^3(2𝑥)𝑑𝑥$

Exercise 5.5 – Substitution

$\int (3{𝑥}^2)({𝑥}^3+5{)}^4𝑑𝑥$

---

✅ Solutions

Solution 5.1

= x^4 – 2x^3 + 2x + C

Solution 5.2

= e^x + ln|x| + C

Solution 5.3

= sin x + 3cos x + C

Solution 5.4

= [x²]_1^3 = 9–1=8

Solution 5.5

u=x³+5 → du=3x² dx.

Integral = ∫ u^4 du = u^5/5 + C = (x³+5)^5/5 + C

---

🔥 Challenge Problems

1. Compute: 
   ${\int }_0^{𝜋}𝑠𝑖𝑛𝑥𝑑𝑥$
   .

2. Show by substitution: 
   $\int (𝑐𝑜𝑠𝑥)(𝑠𝑖𝑛𝑥)𝑑𝑥=𝑠𝑖{𝑛}^2𝑥\mathrm{/}2+𝐶$
   .

3. A car’s velocity is v(t)=6t (m/s). Find distance traveled in first 5 seconds.

4. Prove: derivative of ∫f(x) dx = f(x).

---

✅ Summary of Chapter 5

You learned:

• Indefinite integrals as antiderivatives.

• Definite integrals as areas.

• Fundamental theorem of calculus.

• Substitution method.

📊 (Suggested figures: parabola with shaded area, substitution diagram showing u-transform, sin curve with shaded area under one period).

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📘 Book III – Calculus I

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 6 – Definite Integrals & Area Problems

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🔹 6.1 The Definite Integral

The definite integral measures net area under a curve between two points:

$${\int }_{𝑎}^{𝑏}𝑓(𝑥)𝑑𝑥=𝐹(𝑏)-𝐹(𝑎)$$

where 

$𝐹$

is any antiderivative of 

$𝑓$

.

• If f(x)≥0 → area above x-axis.

• If f(x)≤0 → area below x-axis (counts negative).

📘 Example 6.1

$${\int }_0^3(2𝑥)𝑑𝑥=[{𝑥}^2{]}_0^3=9$$

Graph: straight line y=2x with area as triangle under curve.

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🔹 6.2 Properties of Definite Integrals

1. Reversing limits:

$${\int }_{𝑎}^{𝑏}𝑓(𝑥)𝑑𝑥=-{\int }_{𝑏}^{𝑎}𝑓(𝑥)𝑑𝑥$$

2. Additivity:

$${\int }_{𝑎}^{𝑐}𝑓(𝑥)𝑑𝑥+{\int }_{𝑐}^{𝑏}𝑓(𝑥)𝑑𝑥={\int }_{𝑎}^{𝑏}𝑓(𝑥)𝑑𝑥$$

3. Zero-width:

$${\int }_{𝑎}^{𝑎}𝑓(𝑥)𝑑𝑥=0$$

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🔹 6.3 Area Between Curves

If f(x)≥g(x) on [a,b]:

$$\text{Area}={\int }_{𝑎}^{𝑏}[𝑓(𝑥)-𝑔(𝑥)]𝑑𝑥$$

📘 Example 6.2

Find area between y=x² and y=x on [0,1].

Area=∫₀¹ (x–x²) dx = [x²/2 – x³/3]₀¹=½–⅓=⅙.

📊 Graph: parabola and line crossing at x=0,1.

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🔹 6.4 Applications to Physics

1. Work Done by a Force
   If force F(x) moves an object along x-axis:

$$𝑊={\int }_{𝑎}^{𝑏}𝐹(𝑥)𝑑𝑥$$

2. Distance from Velocity
   If v(t) is velocity:

$$𝑠={\int }_{𝑎}^{𝑏}𝑣(𝑡)𝑑𝑡$$

📘 Example 6.3

Car velocity v(t)=10+2t (m/s) for 0≤t≤5.

Distance=∫₀⁵ (10+2t) dt = [10t+t²]_0⁵=50+25=75 m.

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🔹 6.5 Average Value of a Function

Average of f(x) on [a,b]:

$${𝑓}_{𝑎𝑣𝑔}=\frac{1}{𝑏-𝑎}{\int }_{𝑎}^{𝑏}𝑓(𝑥)𝑑𝑥$$

📘 Example 6.4

f(x)=x² on [0,2].

f_avg=(1/2)∫₀² x² dx=(1/2)[x³/3]₀²=(1/2)(8/3)=4/3.

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🔹 6.6 Improper Integrals (Intro)

Sometimes limits are infinite:

$${\int }_1^{\mathrm{\infty }}\frac{1}{{𝑥}^2}𝑑𝑥=\underset{𝑏\to \mathrm{\infty }}{\lim }{\int }_1^{𝑏}\frac{1}{{𝑥}^2}𝑑𝑥=\underset{𝑏\to \mathrm{\infty }}{\lim }[-\frac{1}{𝑥}{]}_1^{𝑏}=1$$

📘 Example 6.5

∫₀^∞ e^(–x) dx = 1.

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📝 Exercises

Exercise 6.1 – Definite Integral

∫₀⁴ (3x²) dx

Exercise 6.2 – Properties

Evaluate ∫₁⁵ f(x) dx given ∫₁³ f(x) dx=7, ∫₃⁵ f(x) dx=–2.

Exercise 6.3 – Area Between Curves

Find area between y=x and y=√x on [0,1].

Exercise 6.4 – Physics

Force F(x)=5x moves object from 0→3 m. Work done?

Exercise 6.5 – Average Value

Find average of f(x)=sin x on [0,π].

Exercise 6.6 – Improper Integral

∫₁^∞ 1/x dx (show divergence).

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✅ Solutions

Solution 6.1

∫₀⁴ 3x² dx=[x³]_0⁴=64.

Solution 6.2

∫₁⁵ f(x) dx=7+(–2)=5.

Solution 6.3

Area=∫₀¹ (√x–x) dx=[(2/3)x^(3/2)–x²/2]_0¹=2/3–1/2=1/6.

Solution 6.4

Work=∫₀³ 5x dx=[(5/2)x²]_0³=(5/2)(9)=22.5 J.

Solution 6.5

f_avg=(1/π)∫₀^π sin x dx=(1/π)(2)=2/π.

Solution 6.6

∫₁^b 1/x dx=ln b. As b→∞, ln b→∞ → diverges.

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🔥 Challenge Problems

1. Compute ∫₀^π cos²x dx using identity cos²x=(1+cos2x)/2.

2. Find centroid of region bounded by y=x², y=0, x=0→1.

3. Evaluate ∫₀^∞ (1/(1+x²)) dx.

4. Prove that area under exponential decay ∫₀^∞ e^(–ax) dx=1/a (a>0).

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✅ Summary of Chapter 6

You learned:

• Definite integrals as exact accumulated area.

• Properties of integrals (additivity, reversal).

• Area between curves.

• Physics applications (work, distance).

• Average value of functions.

• Intro to improper integrals.

📊 (Suggested figures: parabola with shaded region, two intersecting curves, exponential decay graph for improper integral).

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📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 1 – Advanced Integration Techniques

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🔹 1.1 Integration by Parts

Derived from product rule:

$$\int 𝑢𝑑𝑣=𝑢𝑣\text{–}\int 𝑣𝑑𝑢$$

• Choose u (easy to differentiate).

• Choose dv (easy to integrate).

📘 Example 1.1

$$\int 𝑥{𝑒}^{𝑥}𝑑𝑥$$

Let u=x → du=dx.

dv=e^x dx → v=e^x.

= x e^x – ∫ e^x dx = x e^x – e^x + C.

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🔹 1.2 Trigonometric Integrals

Some integrals use trig identities:

1. ${\sin }^2𝑥=\frac{1\text{–}\cos 2𝑥}{2}$

2. ${\cos }^2𝑥=\frac{1+\cos 2𝑥}{2}$

📘 Example 1.2

$$\int {\sin }^2𝑥𝑑𝑥=\int \frac{1\text{–}\cos 2𝑥}{2}𝑑𝑥=\frac{𝑥}{2}\text{–}\frac{\sin 2𝑥}{4}+𝐶$$

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🔹 1.3 Trigonometric Substitution

Useful for integrals with 

$\sqrt{{𝑎}^2\text{–}{𝑥}^2},\sqrt{{𝑎}^2+{𝑥}^2},\sqrt{{𝑥}^2\text{–}{𝑎}^2}$

.

📘 Example 1.3

$$\int \frac{𝑑𝑥}{\sqrt{{𝑎}^2\text{–}{𝑥}^2}}$$

Let 

$𝑥=𝑎\sin 𝜃$

. Then dx=a cosθ dθ.

= ∫ dθ = θ + C = arcsin(x/a)+C.

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🔹 1.4 Partial Fractions

For rational functions:

$$\frac{𝑃(𝑥)}{𝑄(𝑥)}\to \sum \frac{𝐴}{(𝑙𝑖𝑛𝑒𝑎𝑟)}+\frac{𝐵𝑥+𝐶}{(𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐)}$$

📘 Example 1.4

$$\int \frac{2𝑥+3}{(𝑥+1)(𝑥+2)}𝑑𝑥$$

Decompose: (2x+3)/((x+1)(x+2)) = A/(x+1)+B/(x+2).

Solve → A=1, B=1.

So integral=∫ (1/(x+1)+1/(x+2)) dx=ln|x+1|+ln|x+2|+C.

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🔹 1.5 Improper Integrals

Limits at infinity or discontinuities.

📘 Example 1.5

$${\int }_1^{\mathrm{\infty }}\frac{1}{{𝑥}^2}𝑑𝑥=\underset{𝑏\to \mathrm{\infty }}{\lim }[\text{–}1\mathrm{/}𝑥{]}_1^{𝑏}=1$$

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🔹 1.6 Applications

• Probability density functions.

• Physics: work, energy.

• Engineering: signal processing integrals.

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📝 Exercises

Exercise 1.1 – Integration by Parts

$\int 𝑥\cos 𝑥𝑑𝑥$

Exercise 1.2 – Trig Integral

$\int {\cos }^2𝑥𝑑𝑥$

Exercise 1.3 – Substitution

$\int \frac{𝑥}{\sqrt{9\text{–}{𝑥}^2}}𝑑𝑥$

Exercise 1.4 – Partial Fractions

$\int \frac{3𝑥+5}{{𝑥}^2+3𝑥+2}𝑑𝑥$

Exercise 1.5 – Improper

${\int }_0^{\mathrm{\infty }}{𝑒}^{\text{–}𝑥}𝑑𝑥$

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✅ Solutions

Solution 1.1

u=x, dv=cos x dx → v=sin x.

= x sin x – ∫ sin x dx = x sin x + cos x + C.

Solution 1.2

∫ cos²x dx = ∫ (1+cos2x)/2 dx = x/2 + sin2x/4 + C.

Solution 1.3

Let u=9–x², du=–2x dx.

∫ x/√(9–x²) dx = –½ ∫ du/√u = –√(9–x²)+C.

Solution 1.4

Denominator=(x+1)(x+2).

Decompose: (3x+5)/(...) = A/(x+1)+B/(x+2).

Solve: A=2, B=1.

Integral=2ln|x+1|+ln|x+2|+C.

Solution 1.5

∫₀^∞ e^(–x) dx = [–e^(–x)]₀^∞ = 0–(–1)=1.

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🔥 Challenge Problems

1. Evaluate ∫ x² e^x dx by parts.

2. Compute ∫ tan²x dx using trig identities.

3. Show ∫₀^∞ (1/(1+x²)) dx = π/2.

4. Find ∫ (5x²+7x+3)/(x³+3x²+2x) dx.

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✅ Summary of Chapter 1

You learned:

• Integration by parts.

• Trigonometric integrals & substitution.

• Partial fractions decomposition.

• Improper integrals.

• Applications across physics, engineering, and probability.

📊 (Suggested figures: shaded area under curve for improper integral, trig substitution diagram with right triangle).

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📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 2 – Differential Equations (First-Order Models)

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🔹 2.1 What is a Differential Equation?

• An equation that involves a function and its derivatives.

• Order = highest derivative in equation.

• First-order = only first derivative appears.

📘 Example 2.1

$$\frac{𝑑𝑦}{𝑑𝑥}=3𝑦$$

says that rate of change of y is proportional to y.

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🔹 2.2 Separable Equations

Form:

$$\frac{𝑑𝑦}{𝑑𝑥}=𝑓(𝑥)𝑔(𝑦)$$

Steps:

1. Rearrange: dy/g(y) = f(x) dx.

2. Integrate both sides.

3. Solve for y.

📘 Example 2.2

$$\frac{𝑑𝑦}{𝑑𝑥}=𝑦$$

Separate: dy/y = dx.

Integrate: ln|y| = x + C.

So y = Ce^x.

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🔹 2.3 Linear First-Order Equations

Form:

$$\frac{𝑑𝑦}{𝑑𝑥}+𝑃(𝑥)𝑦=𝑄(𝑥)$$

Solve with integrating factor:

$$𝜇(𝑥)={𝑒}^{\int 𝑃(𝑥)𝑑𝑥}$$

Solution:

$$𝑦\cdot 𝜇(𝑥)=\int 𝑄(𝑥)𝜇(𝑥)𝑑𝑥+𝐶$$

📘 Example 2.3

$$\frac{𝑑𝑦}{𝑑𝑥}+𝑦={𝑒}^{𝑥}$$

Here P(x)=1 → μ(x)=e^x.

Multiply through: e^x dy/dx + e^x y = e^{2x}.

Left = d/dx (y e^x).

Integrate: y e^x = ½ e^{2x}+C.

So y = ½ e^x + Ce^(–x).

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🔹 2.4 Applications

a) Exponential Growth & Decay

$$\frac{𝑑𝑦}{𝑑𝑡}=𝑘𝑦$$

Solution: y(t)=y₀e^(kt).

📘 Example 2.4

Population of 100 doubles every 10 years. k=ln2/10.

After 30 years: y=100 e^(ln2·3)=100·8=800.

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b) Newton’s Law of Cooling

$$\frac{𝑑𝑇}{𝑑𝑡}=\text{–}𝑘(𝑇\text{–}{𝑇}_{𝑒𝑛𝑣})$$

Solution:

$$𝑇(𝑡)={𝑇}_{𝑒𝑛𝑣}+({𝑇}_0\text{–}{𝑇}_{𝑒𝑛𝑣}){𝑒}^{\text{–}𝑘𝑡}$$

📘 Example 2.5

Hot object at 100°C in room 20°C. After 10 min, T=60°C.

Find k.

60=20+(80)e^(–10k). → e^(–10k)=40/80=0.5 → k=(ln2)/10≈0.0693.

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c) RC Circuits

Equation:

$$\frac{𝑑𝑄}{𝑑𝑡}+\frac{1}{𝑅𝐶}𝑄=\frac{𝑉}{𝑅}$$

Solution:

$$𝑄(𝑡)=𝐶𝑉(1\text{–}{𝑒}^{\text{–}𝑡\mathrm{/}𝑅𝐶})$$

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🔹 2.5 Direction Fields (Intro)

• Graphical tool for visualizing solutions without solving.

• Small line segments show slope dy/dx at grid points.

• Useful for qualitative behavior.

📊 (Suggested diagram: slope field for dy/dx=y).

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📝 Exercises

Exercise 2.1 – Separable

Solve dy/dx=2xy.

Exercise 2.2 – Linear

Solve dy/dx+2y=4.

Exercise 2.3 – Growth

Bacteria grow proportional to size, doubling in 5 hours. Starting at 500, find size after 15 hours.

Exercise 2.4 – Cooling

A cup of coffee at 90°C cools to 70°C in 15 minutes in a room at 20°C. Find its temperature after 30 minutes.

Exercise 2.5 – RC Circuit

For R=10 Ω, C=2 F, V=12 V, find Q(t).

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✅ Solutions

Solution 2.1

dy/dx=2xy → dy/y=2x dx.

Integrate: ln|y|=x²+C → y=Ce^(x²).

Solution 2.2

Integrating factor= e^(∫2 dx)=e^(2x).

(y e^(2x))′=4 e^(2x).

Integrate: y e^(2x)=2 e^(2x)+C → y=2+Ce^(–2x).

Solution 2.3

Doubling time=5 → k=ln2/5.

After 15: y=500 e^(k·15)=500 e^(3ln2)=500·8=4000.

Solution 2.4

Newton’s law: T(t)=20+(70)e^(–kt).

At 15: 70=20+70 e^(–15k) → 50=70 e^(–15k) → e^(–15k)=5/7.

k=–(1/15)ln(5/7)≈0.0257.

At 30: T=20+70 e^(–30k)≈20+70·(0.507)≈55.5°C.

Solution 2.5

Q(t)=CV(1–e^(–t/RC))=24(1–e^(–t/20)).

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🔥 Challenge Problems

1. Solve logistic growth: dy/dt=ky(1–y/K).

2. A tank drains with dV/dt=–k√V. Solve for V(t).

3. Radioactive isotope with half-life 20 years: find decay constant and remaining after 100 years.

4. Show solution curves of dy/dx=–y/x are hyperbolas xy=C.

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✅ Summary of Chapter 2

You learned:

• How to solve separable and linear first-order DEs.

• Growth/decay models, cooling, and RC circuits.

• How direction fields give qualitative solutions.

• Differential equations as mathematical models of reality.

📊 (Suggested figures: slope field, cooling curve, RC charging graph, logistic growth curve).

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📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 3 – Multivariable Functions

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🔹 3.1 Functions of Two Variables

A function of two variables assigns a value for each ordered pair (x,y).

$$𝑧=𝑓(𝑥,𝑦)$$

• Example: f(x,y)=x²+y² is a paraboloid.

• Domain: set of (x,y) inputs.

• Range: set of z values.

📊 (Graph idea: surface plot of z=x²+y²)

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🔹 3.2 Limits and Continuity in 2 Variables

$$\underset{(𝑥,𝑦)\to (𝑎,𝑏)}{\lim }𝑓(𝑥,𝑦)$$

exists if approaching from any path gives the same value.

📘 Example 3.1

f(x,y)=x²+y².

As (x,y)→(0,0), limit=0 (independent of path).

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🔹 3.3 Partial Derivatives

Rate of change in one variable, keeping the other fixed.

$${𝑓}_{𝑥}(𝑥,𝑦)=\frac{\mathrm{\partial }𝑓}{\mathrm{\partial }𝑥},{𝑓}_{𝑦}(𝑥,𝑦)=\frac{\mathrm{\partial }𝑓}{\mathrm{\partial }𝑦}$$

📘 Example 3.2

f(x,y)=x²y+3y².

• f_x=2xy.

• f_y=x²+6y.

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🔹 3.4 Higher-Order Partials

Second derivatives: f_xx, f_yy, f_xy, f_yx.

For smooth functions: f_xy=f_yx (Clairaut’s theorem).

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🔹 3.5 Tangent Planes & Linear Approximation

For z=f(x,y) near (a,b):

$$𝑧\approx 𝑓(𝑎,𝑏)+{𝑓}_{𝑥}(𝑎,𝑏)(𝑥\text{–}𝑎)+{𝑓}_{𝑦}(𝑎,𝑏)(𝑦\text{–}𝑏)$$

📘 Example 3.3

f(x,y)=x²+y² at (1,2).

f(1,2)=1+4=5.

f_x=2x=2, f_y=2y=4.

Tangent plane: z≈5+2(x–1)+4(y–2).

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🔹 3.6 Gradients and Directional Derivatives

Gradient vector:

$$\mathrm{\nabla }𝑓=⟨{𝑓}_{𝑥},{𝑓}_{𝑦}⟩$$

Points in direction of steepest increase.

Directional derivative in direction unit vector u:

$${𝐷}_{𝑢}𝑓(𝑥,𝑦)=\mathrm{\nabla }𝑓(𝑥,𝑦)\cdot 𝑢$$

📘 Example 3.4

f(x,y)=x²+y².

∇f=⟨2x,2y⟩.

At (3,4): ∇f=⟨6,8⟩. Steepest increase direction is toward (3,4) from origin.

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🔹 3.7 Extrema of Functions of 2 Variables

Critical points: where f_x=f_y=0.

Use second derivative test with Hessian matrix:

$$𝐷={𝑓}_{𝑥𝑥}{𝑓}_{𝑦𝑦}\text{–}({𝑓}_{𝑥𝑦}{)}^2$$

• If D>0, f_xx>0 → local min.

• If D>0, f_xx<0 → local max.

• If D<0 → saddle point.

📘 Example 3.5

f(x,y)=x²–y².

f_x=2x, f_y=–2y. Critical point at (0,0).

f_xx=2, f_yy=–2, f_xy=0 → D=–4<0 → saddle point.

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📝 Exercises

Exercise 3.1

Find f_x and f_y: f(x,y)=3x²y–2xy².

Exercise 3.2

Find tangent plane to z=x²+y² at (2,1).

Exercise 3.3

Find gradient of f(x,y)=ln(x²+y²) at (1,1).

Exercise 3.4

Classify critical point of f(x,y)=x²+y²–4x–6y+13.

Exercise 3.5 – Challenge

Find max/min of f(x,y)=x²+y² subject to constraint x²+y²=1 (unit circle).

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✅ Solutions

Solution 3.1

f_x=6xy–2y², f_y=3x²–4xy.

Solution 3.2

f(2,1)=5.

f_x=2x=4, f_y=2y=2.

Plane: z=5+4(x–2)+2(y–1).

Solution 3.3

∇f=⟨2x/(x²+y²),2y/(x²+y²)⟩.

At (1,1): ⟨1,1⟩.

Solution 3.4

f_x=2x–4, f_y=2y–6. Critical at (2,3).

f_xx=2, f_yy=2, f_xy=0. D=4>0, f_xx>0 → local minimum.

Solution 3.5

Constraint circle x²+y²=1. On circle, f=x²+y²=1 (constant). So all points are equal value=1.

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🔥 Challenge Problems

1. Find critical points of f(x,y)=x³–3xy². Classify them.

2. Compute directional derivative of f(x,y)=x²y at (1,2) in direction of vector (3,4).

3. Use Lagrange multipliers to maximize f(x,y)=xy subject to x²+y²=10.

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✅ Summary of Chapter 3

You learned:

• Functions of multiple variables.

• Limits, continuity, partial derivatives.

• Tangent planes, gradients, directional derivatives.

• Extrema in 2 variables with Hessian test.

• Intro to constrained optimization.

📊 (Suggested diagrams: paraboloid surface, tangent plane, gradient vector arrows, saddle surface).

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📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 4 – Multiple Integrals

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🔹 4.1 Double Integrals over Rectangles

For a function of two variables:

$${\iint }_{𝑅}𝑓(𝑥,𝑦)𝑑𝐴={\int }_{𝑎}^{𝑏}{\int }_{𝑐}^{𝑑}𝑓(𝑥,𝑦)𝑑𝑦𝑑𝑥$$

• R is rectangular region [a,b]×[c,d].

• Integrate one variable at a time (iterated integral).

📘 Example 4.1

$${\int }_0^1{\int }_0^2(𝑥+2𝑦)𝑑𝑦𝑑𝑥$$

= ∫₀¹ [xy+y²]_0² dx = ∫₀¹ (2x+4) dx = [x²+4x]_0¹=5.

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🔹 4.2 Double Integrals over General Regions

• If R is bounded by curves:

$${\iint }_{𝑅}𝑓(𝑥,𝑦)𝑑𝐴={\int }_{𝑎}^{𝑏}{\int }_{{𝑔}_1(𝑥)}^{{𝑔}_2(𝑥)}𝑓(𝑥,𝑦)𝑑𝑦𝑑𝑥$$

📘 Example 4.2

Region: bounded by y=0, y=x, x∈[0,1].

∬ y dA = ∫₀¹ ∫₀^x y dy dx = ∫₀¹ [y²/2]_0^x dx = ∫₀¹ x²/2 dx = 1/6.

📊 (Diagram: triangle under line y=x)

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🔹 4.3 Double Integrals in Polar Coordinates

Use polar when region is circular:

$$𝑥=𝑟\cos 𝜃,𝑦=𝑟\sin 𝜃,𝑑𝐴=𝑟𝑑𝑟𝑑𝜃$$

📘 Example 4.3

Find area of unit circle:

$${\iint }_{𝑅}1𝑑𝐴={\int }_0^{2𝜋}{\int }_0^1𝑟𝑑𝑟𝑑𝜃={\int }_0^{2𝜋}(\text{½})𝑑𝜃=𝜋$$

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🔹 4.4 Applications of Double Integrals

1. Area of a Region
   Area=∬ 1 dA.

2. Mass of a Plate
   If density=ρ(x,y):

$$𝑀=\iint 𝜌(𝑥,𝑦)𝑑𝐴$$

3. Center of Mass

$$\overline{𝑥}=\frac{1}{𝑀}\iint 𝑥𝜌𝑑𝐴,\overline{𝑦}=\frac{1}{𝑀}\iint 𝑦𝜌𝑑𝐴$$

---

🔹 4.5 Triple Integrals

For f(x,y,z) in region V:

$${\iiint }_{𝑉}𝑓(𝑥,𝑦,𝑧)𝑑𝑉$$

📘 Example 4.4

Find volume of cube 0≤x,y,z≤1.

∭ 1 dV = ∫₀¹ ∫₀¹ ∫₀¹ 1 dz dy dx = 1.

---

🔹 4.6 Triple Integrals in Cylindrical & Spherical Coordinates

• Cylindrical: (r,θ,z), dV=r dr dθ dz.

• Spherical: (ρ,θ,φ), dV=ρ² sinφ dρ dφ dθ.

📘 Example 4.5

Volume of sphere radius R:

$$𝑉=\iiint 1𝑑𝑉={\int }_0^{2𝜋}{\int }_0^{𝜋}{\int }_0^{𝑅}{𝜌}^2\sin 𝜑𝑑𝜌𝑑𝜑𝑑𝜃=\frac{4}{3}𝜋{𝑅}^3$$

---

📝 Exercises

Exercise 4.1

Evaluate ∫₀² ∫₀¹ (x+3y) dy dx.

Exercise 4.2

Find area of region bounded by y=0, y=√x, x∈[0,1].

Exercise 4.3

Use polar coordinates to compute area of circle radius 2.

Exercise 4.4

Find mass of plate with density ρ(x,y)=x+y on unit square [0,1]×[0,1].

Exercise 4.5

Compute volume of sphere radius 2 using spherical coordinates.

---

✅ Solutions

Solution 4.1

= ∫₀² [xy+1.5y²]_0¹ dx=∫₀² (x+1.5) dx=[x²/2+1.5x]_0²=5.

Solution 4.2

Area=∫₀¹ ∫₀^√x dy dx=∫₀¹ √x dx=[(2/3)x^(3/2)]₀¹=2/3.

Solution 4.3

∫₀^{2π}∫₀² r dr dθ=∫₀^{2π} (2) dθ=8π.

Solution 4.4

M=∫₀¹∫₀¹ (x+y) dy dx=∫₀¹ [xy+y²/2]_0¹ dx=∫₀¹ (x+½) dx=1+½=1.5.

Solution 4.5

V=(4/3)π(2³)=(32/3)π.

---

🔥 Challenge Problems

1. Find centroid of triangle with vertices (0,0), (1,0), (0,2).

2. Evaluate ∫∫R (x²+y²) dA where R is unit circle.

3. Show by spherical coordinates: volume of hemisphere radius R=½(4/3 πR³).

4. Compute mass of circular plate radius 3 with density ρ=r.

---

✅ Summary of Chapter 4

You learned:

• Double integrals over rectangles and general regions.

• Polar coordinates for circular regions.

• Applications: area, mass, center of mass.

• Triple integrals in Cartesian, cylindrical, spherical.

• Volume formulas (cube, sphere, hemisphere).

📊 (Suggested figures: rectangle region, parabola boundary, circle in polar coords, sphere with spherical coords).

---

📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 5 – Introduction to Linear Algebra

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🔹 5.1 Matrices and Notation

A matrix is a rectangular array of numbers arranged in rows and columns.

$$𝐴=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]$$

• Rows: horizontal entries.

• Columns: vertical entries.

• Size: m×n (m rows, n columns).

📘 Example 5.1

Matrix A (2×3) has 2 rows and 3 columns.

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🔹 5.2 Matrix Operations

1. Addition: same-size matrices add elementwise.

2. Scalar Multiplication: multiply every entry by scalar.

3. Matrix Multiplication: if A is m×n and B is n×p, then AB is m×p.

📘 Example 5.2

$$𝐴=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],𝐵=\left[\begin{array}{cc}2& 0\\ 1& 2\end{array}\right]$$

AB= [[1·2+2·1 , 1·0+2·2],[3·2+4·1 , 3·0+4·2]]=[[4,4],[10,8]].

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🔹 5.3 Determinants

For a square matrix, determinant = scalar value useful for solving systems.

For 2×2:

$$\det \left[\begin{array}{cc}𝑎& 𝑏\\ 𝑐& 𝑑\end{array}\right]=𝑎𝑑\text{–}𝑏𝑐$$

For 3×3, expansion formula or rule of Sarrus.

📘 Example 5.3

$$\det \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]=1\cdot 4\text{–}2\cdot 3=\text{–}2$$

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🔹 5.4 Inverse of a Matrix

For A (2×2):

$${𝐴}^{-1}=\frac{1}{\det (𝐴)}\left[\begin{array}{cc}𝑑& -𝑏\\ -𝑐& 𝑎\end{array}\right]$$

Exists only if det(A)≠0.

📘 Example 5.4

$$𝐴=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],\det (𝐴)=\text{–}2$$$${𝐴}^{-1}=\frac{1}{\text{–}2}\left[\begin{array}{cc}4& \text{–}2\\ \text{–}3& 1\end{array}\right]=\left[\begin{array}{cc}\text{–}2& 1\\ 1.5& \text{–}0.5\end{array}\right]$$

---

🔹 5.5 Systems of Linear Equations

Can be written as matrix form:

$$𝐴𝑋=𝑏$$

• Solve using substitution, elimination, or matrix methods (inverse, Gaussian elimination).

📘 Example 5.5

x+2y=5

3x+4y=11

Matrix:

A=[[1,2],[3,4]], X=[x,y]^T, b=[5,11]^T.

Solve: X=A⁻¹b = [1,2]^T → solution x=1, y=2.

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🔹 5.6 Gaussian Elimination

Row operations reduce augmented matrix to row echelon form.

📘 Example 5.6

Solve system:

x+y+z=6

2y+5z=–4

2x+5y–z=27

Augmented matrix:

$$\left[\begin{array}{ccc}1& 1& 1\mathrm{\mid }6\\ 0& 2& 5\mathrm{\mid }\text{–}4\\ 2& 5& \text{–}1\mathrm{\mid }27\end{array}\right]$$

Row reduce → solution (x,y,z)=(5,3,–2).

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🔹 5.7 Applications

• Engineering statics: solving forces in trusses.

• Circuits: currents using Kirchhoff’s laws.

• Computer graphics: matrix transformations.

• Data science: systems of equations in regression.

📊 (Suggested diagram: vectors in plane forming solution intersection point).

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📝 Exercises

Exercise 5.1 – Determinant

Compute det([[2,3],[1,4]]).

Exercise 5.2 – Inverse

Find inverse of [[2,1],[5,3]].

Exercise 5.3 – System

Solve using matrices:

x+2y=7, 2x–y=1.

Exercise 5.4 – Gaussian Elimination

Solve:

x+2y+3z=10

2x+5y+z=8

3x+4y+2z=9

Exercise 5.5 – Application

Matrix multiplication: rotate vector (1,0) by 90° using rotation matrix.

---

✅ Solutions

Solution 5.1

det=2·4–3·1=8–3=5.

Solution 5.2

det=2·3–1·5=6–5=1.

Inverse=[[3,–1],[–5,2]].

Solution 5.3

Matrix form: [[1,2],[2,–1]] [x,y]^T=[7,1]^T.

Det=–5.

Solution: (x,y)=(3,2).

Solution 5.4

Row reduction → solution (x,y,z)=(1,2,3).

Solution 5.5

Rotation matrix 90°: [[0,–1],[1,0]].

[[0,–1],[1,0]]·[1,0]^T=[0,1]^T.

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🔥 Challenge Problems

1. Prove det(AB)=det(A)det(B).

2. Show that a system Ax=b has unique solution iff det(A)≠0.

3. Find inverse of 3×3 matrix [[1,0,1],[0,1,1],[1,1,1]].

4. Use matrices to solve truss equilibrium with 3 forces meeting at a joint.

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✅ Summary of Chapter 5

You learned:

• Matrix basics, operations, inverses.

• Determinants and their role.

• Systems of equations via matrices and elimination.

• Applications in physics, engineering, data, and graphics.

📊 (Suggested figures: 2×2 matrix transformation of square, truss equilibrium system).

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📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 6 – Eigenvalues and Eigenvectors

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🔹 6.1 Motivation

Eigenvalues and eigenvectors describe directions in which a linear transformation acts by only stretching or compressing (without changing direction).

Applications:

• Vibrations & stability (mechanical engineering).

• Quantum mechanics (operators and states).

• Computer science (PCA, Google PageRank).

📊 (Suggested figure: transformation of vectors by matrix, one direction preserved as eigenvector.)

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🔹 6.2 Definitions

Given a square matrix A, a nonzero vector v is an eigenvector if:

$$𝐴𝑣=𝜆𝑣$$

• λ is the eigenvalue.

• v is the eigenvector corresponding to λ.

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🔹 6.3 Characteristic Equation

To find eigenvalues:

$$\det (𝐴\text{–}𝜆𝐼)=0$$

📘 Example 6.1

A=[[2,1],[1,2]].

Characteristic equation: det([[2–λ,1],[1,2–λ]])=(2–λ)²–1=λ²–4λ+3=0.

Solutions: λ=1,3.

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🔹 6.4 Finding Eigenvectors

Plug λ back into (A–λI)v=0.

📘 Example 6.2

For λ=3 in A=[[2,1],[1,2]]:

(A–3I)=[[–1,1],[1,–1]].

Equation: –x+y=0 → eigenvector v=[1,1]^T.

For λ=1:

(A–I)=[[1,1],[1,1]] → x=–y → eigenvector v=[1,–1]^T.

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🔹 6.5 Properties of Eigenvalues

• Trace(A)=sum of eigenvalues.

• det(A)=product of eigenvalues.

• Diagonalizable matrices can be written as A=PDP⁻¹, where D is diagonal of eigenvalues.

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🔹 6.6 Applications

1. Mechanical Vibrations: Eigenvalues = natural frequencies.

2. Markov Chains: Steady state given by eigenvector of eigenvalue 1.

3. Principal Component Analysis: Eigenvectors of covariance matrix = directions of maximum variance.

4. Differential Equations: Solutions involve exponentials with eigenvalues.

---

📝 Exercises

Exercise 6.1

Find eigenvalues of A=[[4,2],[1,3]].

Exercise 6.2

Find eigenvectors of A=[[0,–1],[1,0]].

Exercise 6.3

Verify trace and determinant equal eigenvalue sum/product for A=[[2,1],[1,2]].

Exercise 6.4

Diagonalize A=[[5,4],[1,2]].

Exercise 6.5 – Application

In a Markov chain transition matrix

$$𝑃=\left[\begin{array}{cc}0.7& 0.3\\ 0.2& 0.8\end{array}\right],$$

find steady-state distribution.

---

✅ Solutions

Solution 6.1

Characteristic: det([[4–λ,2],[1,3–λ]])=(4–λ)(3–λ)–2=λ²–7λ+10=0.

λ=5,2.

Solution 6.2

Characteristic: det([–λ,–1],[1,–λ])=λ²+1=0 → λ=±i.

Eigenvectors are complex: for λ=i, v=[i,1]^T.

Solution 6.3

For A=[[2,1],[1,2]], eigenvalues=1,3. Sum=4=trace. Product=3=det. ✅

Solution 6.4

det(A–λI)=(5–λ)(2–λ)–4=λ²–7λ+6=0 → λ=1,6.

Eigenvectors: λ=1 → v=[–4,1]^T, λ=6 → v=[1,1]^T.

So P=[[–4,1],[1,1]], D=diag(1,6).

Solution 6.5

Eigenvalue λ=1 → (P–I)v=0. Solve: [–0.3,0.3;0.2,–0.2]v=0 → v=[3,2]^T.

Normalized: steady state=[0.6,0.4].

---

🔥 Challenge Problems

1. Find eigenvalues/eigenvectors of 3×3 matrix A=[[2,0,0],[0,3,4],[0,4,9]].

2. Show rotation matrix R(θ) has eigenvalues cosθ±i sinθ.

3. Compute steady-state distribution for 3-state Markov chain with given transition matrix.

4. In PCA, show first eigenvector corresponds to maximum variance direction.

---

✅ Summary of Chapter 6

You learned:

• Definitions of eigenvalues and eigenvectors.

• Characteristic polynomial method.

• Diagonalization.

• Applications in physics, probability, and data science.

📊 (Suggested figures: eigenvector directions under transformation, vibration modes, steady-state Markov distribution).

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📘 Book IV – Calculus II & Linear Algebra

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 7 – Review & Challenge Problems

This chapter reviews the core concepts of Book IV and provides mixed problems that combine techniques from integration, differential equations, multivariable calculus, and linear algebra.

---

🔹 7.1 Core Review Topics

• Advanced Integration: integration by parts, trig substitution, partial fractions, improper integrals.

• Differential Equations: separable, linear first-order, applications (growth/decay, cooling, circuits).

• Multivariable Functions: partial derivatives, gradients, tangent planes, optimization.

• Multiple Integrals: double/triple integrals, polar/cylindrical/spherical coordinates, applications to area, volume, mass.

• Linear Algebra: matrices, determinants, Gaussian elimination, eigenvalues/eigenvectors.

---

🔹 7.2 Mixed Review Problems

📘 Problem 7.1 – Integration & DEs

Solve the differential equation using integration:

$$\frac{𝑑𝑦}{𝑑𝑥}=2𝑥{𝑒}^{{𝑥}^2},𝑦(0)=1$$

---

📘 Problem 7.2 – Double Integral

Find the mass of a triangular plate bounded by x=0, y=0, x+y=1 with density ρ(x,y)=x+y.

---

📘 Problem 7.3 – Gradient & Tangent Plane

For f(x,y)=ln(x²+y²), find:

1. Gradient at (1,1).

2. Tangent plane approximation at (1,1).

---

📘 Problem 7.4 – Triple Integral (Spherical Coordinates)

Compute volume of a sphere of radius R using triple integrals.

---

📘 Problem 7.5 – Linear Algebra System

Solve:

x+2y+3z=14

2x+3y+z=10

3x+y+2z=11

---

📘 Problem 7.6 – Eigenvalues & Eigenvectors

Find eigenvalues and eigenvectors of A=[[4,1],[2,3]].

---

---

🔹 7.3 Solutions

Solution 7.1

dy/dx=2x e^(x²).

Integrate: y=∫2x e^(x²) dx= e^(x²)+C.

At x=0, y=1 → C=0.

So y=e^(x²). ✅

---

Solution 7.2

Mass=∬(x+y)dA. Region: triangle with vertices (0,0),(1,0),(0,1).

∫₀¹ ∫₀^{1–x} (x+y) dy dx.

=∫₀¹ [xy+y²/2]_0^{1–x} dx=∫₀¹ (x(1–x)+(1–x)²/2) dx=1/3. ✅

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Solution 7.3

∇f=⟨2x/(x²+y²),2y/(x²+y²)⟩. At (1,1): (1,1).

Tangent plane: z≈ln(2)+1(x–1)+1(y–1). ✅

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Solution 7.4

V=∭ dV=∫₀^{2π} ∫₀^π ∫₀^R ρ² sinφ dρ dφ dθ=4/3 πR³. ✅

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Solution 7.5

Augmented matrix row reduce → solution (x,y,z)=(2,3,1). ✅

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Solution 7.6

Characteristic: det([[4–λ,1],[2,3–λ]])=(4–λ)(3–λ)–2=λ²–7λ+10.

Roots: λ=5,2.

For λ=5: (A–5I)=[[–1,1],[2,–2]] → v=[1,1].

For λ=2: (A–2I)=[[2,1],[2,1]] → v=[–½,1]. ✅

---

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🔹 7.4 Challenge Problems

1. Evaluate:

$${\int }_0^{\mathrm{\infty }}\frac{1}{1+{𝑥}^4}𝑑𝑥$$

(hint: symmetry, advanced substitution).

2. Solve logistic growth DE:

$$\frac{𝑑𝑦}{𝑑𝑡}=0.5𝑦(1\text{–}𝑦\mathrm{/}100),𝑦(0)=10$$

3. Find centroid of quarter circle x²+y²≤R², x≥0,y≥0.

4. Diagonalize A=[[0,1],[–2,–3]]. Use it to solve system of DEs: dx/dt=Ax.

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🔹 7.5 Summary of Chapter 7

• Reviewed Book IV: advanced integration, DEs, multivariable functions, multiple integrals, linear algebra.

• Worked through interdisciplinary problems combining techniques.

• Introduced challenge problems to prepare for Book V.

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✅ Book IV Complete 🎉

You now have a full set of chapters covering:

• Advanced Calculus (Integration, DEs, Multivariable, Multiple Integrals).

• Linear Algebra (matrices, systems, eigenvalues).

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📘 Book V – Probability, Statistics & Applied Mathematics

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 1 – Combinatorics & the Binomial Theorem

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🔹 1.1 What is Combinatorics?

Combinatorics is the mathematics of counting and arrangements.

It answers questions like:

• How many ways can you arrange 5 books on a shelf?

• How many different 3-digit passwords exist?

• If a coin is tossed 10 times, in how many ways do 4 heads appear?

It provides the foundation for probability.

---

🔹 1.2 Factorials

The factorial function counts the number of ways to arrange n distinct objects:

$$𝑛!=𝑛\times (𝑛\text{–}1)\times (𝑛\text{–}2)\times \cdots \times 1$$

• 0! is defined as 1.

📘 Example 1.1

• 4! = 4×3×2×1 = 24.

• 0! = 1.

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🔹 1.3 Permutations

A permutation is an ordered arrangement of objects.

Number of permutations of n objects taken r at a time:

$$𝑃(𝑛,𝑟)=\frac{𝑛!}{(𝑛\text{–}𝑟)!}$$

📘 Example 1.2

How many 3-digit codes can be formed from digits 1–5, without repetition?

P(5,3)=5!/(5–3)! = 60.

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🔹 1.4 Combinations

A combination is a selection of objects without regard to order.

$$𝐶(𝑛,𝑟)=\left(\genfrac{}{}{0px}{}{𝑛}{𝑟}\right)=\frac{𝑛!}{𝑟!(𝑛\text{–}𝑟)!}$$

📘 Example 1.3

From 10 students, how many ways to select 3 for a committee?

C(10,3)=120.

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🔹 1.5 Binomial Theorem

Expansion of (a+b)^n:

$$(𝑎+𝑏{)}^{𝑛}=\sum _{𝑘=0}^{𝑛}\left(\genfrac{}{}{0px}{}{𝑛}{𝑘}\right){𝑎}^{𝑛\text{–}𝑘}{𝑏}^{𝑘}$$

📘 Example 1.4

Expand (x+1)^4.

= C(4,0)x^4+C(4,1)x^3+C(4,2)x^2+C(4,3)x+C(4,4)

= x^4+4x^3+6x^2+4x+1.

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🔹 1.6 Pascal’s Triangle

Binomial coefficients appear in Pascal’s Triangle:

       1
      1 1
     1 2 1
    1 3 3 1
   1 4 6 4 1
  1 5 10 10 5 1

• Each entry is the sum of the two above.

• Row n gives coefficients for (a+b)^n.

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🔹 1.7 Applications

1. Probability: number of outcomes.

2. Statistics: sampling without replacement.

3. Computer science: algorithms and coding.

4. Engineering: reliability analysis, arrangements of system components.

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📝 Exercises

Exercise 1.1 – Factorials

Compute: 6!, 8!/6!, 0!.

Exercise 1.2 – Permutations

How many ways can 4 letters be arranged from the alphabet (26 total), without repetition?

Exercise 1.3 – Combinations

How many ways to choose 5 cards from a standard deck of 52?

Exercise 1.4 – Binomial Expansion

Find coefficient of x³ in (x+2)^5.

Exercise 1.5 – Pascal’s Triangle

Write row 6 of Pascal’s Triangle.

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✅ Solutions

Solution 1.1

6!=720, 8!/6!=56, 0!=1.

Solution 1.2

P(26,4)=26×25×24×23=358,800.

Solution 1.3

C(52,5)=2,598,960.

Solution 1.4

(x+2)^5=∑ C(5,k)x^(5–k)2^k.

Coefficient of x³ is when k=2: C(5,2)·2²=10·4=40.

Solution 1.5

Row 6: 1,6,15,20,15,6,1.

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🔥 Challenge Problems

1. Prove: ∑_{k=0}^n C(n,k)=2^n.

2. Find number of 5-digit numbers using digits 1–9 with no repetition.

3. Expand (x–y)^6 using binomial theorem.

4. From 20 people, form a basketball team of 5 players and a coach. How many ways?

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✅ Summary of Chapter 1

You learned:

• Factorials, permutations, and combinations.

• Binomial theorem and Pascal’s triangle.

• Applications in probability, statistics, and engineering.

📊 (Suggested figures: Pascal’s triangle, probability tree, sample binomial distribution histogram).

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📘 Book V – Probability, Statistics & Applied Mathematics

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 2 – Probability Basics

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🔹 2.1 What is Probability?

Probability measures the likelihood of an event:

$$𝑃(𝐸)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

• 0 ≤ P(E) ≤ 1.

• P(E)=0 → impossible event.

• P(E)=1 → certain event.

📘 Example 2.1

Rolling a fair die:

• P(rolling a 4)=1/6.

• P(rolling an even)=3/6=½.

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🔹 2.2 Sample Space and Events

• Sample space (S): set of all possible outcomes.

• Event (E): subset of S.

📘 Example 2.2

Toss two coins: S={HH,HT,TH,TT}.

Event E={HH,TT} (“same on both coins”).

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🔹 2.3 Rules of Probability

1. Complement Rule:

$$𝑃({𝐸}^{𝑐})=1\text{–}𝑃(𝐸)$$

2. Addition Rule (Mutually Exclusive Events):

$$𝑃(𝐴\cup 𝐵)=𝑃(𝐴)+𝑃(𝐵)$$

3. General Addition Rule:

$$𝑃(𝐴\cup 𝐵)=𝑃(𝐴)+𝑃(𝐵)\text{–}𝑃(𝐴\cap 𝐵)$$

📘 Example 2.3

A card is drawn from a deck.

P(A=red)=26/52=½, P(B=king)=4/52=1/13.

P(A∩B)=2/52=1/26.

So P(A∪B)=½+1/13–1/26=29/52.

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🔹 2.4 Conditional Probability

Probability of A given B:

$$𝑃(𝐴\mathrm{\mid }𝐵)=\frac{𝑃(𝐴\cap 𝐵)}{𝑃(𝐵)}$$

📘 Example 2.4

Deck of 52 cards.

P(A=ace | B=red card)=2/26=1/13.

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🔹 2.5 Independent Events

A and B are independent if:

$$𝑃(𝐴\cap 𝐵)=𝑃(𝐴)𝑃(𝐵)$$

📘 Example 2.5

Roll two dice.

P(A=first die=6)=1/6, P(B=second die=5)=1/6.

P(A∩B)=1/36=(1/6)(1/6). ✅ Independent.

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🔹 2.6 Bayes’ Theorem

For events A₁,…,An partitioning sample space and event B:

$$𝑃({𝐴}_{𝑖}\mathrm{\mid }𝐵)=\frac{𝑃(𝐵\mathrm{\mid }{𝐴}_{𝑖})𝑃({𝐴}_{𝑖})}{\sum _{𝑗}𝑃(𝐵\mathrm{\mid }{𝐴}_{𝑗})𝑃({𝐴}_{𝑗})}$$

📘 Example 2.6

A medical test detects disease with 95% accuracy, false positive rate=2%. Prevalence=1%.

P(disease|positive)=?

P(D)=0.01, P(Pos|D)=0.95.

P(Pos|¬D)=0.02, P(¬D)=0.99.

$$𝑃(𝐷\mathrm{\mid }𝑃𝑜𝑠)=\frac{0.95\cdot 0.01}{0.95\cdot 0.01+0.02\cdot 0.99}\approx 0.324$$

So only ~32% chance of disease given positive test.

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🔹 2.7 Law of Total Probability

If events A₁,…,An partition S:

$$𝑃(𝐵)=\sum _{𝑖}𝑃(𝐵\mathrm{\mid }{𝐴}_{𝑖})𝑃({𝐴}_{𝑖})$$

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📝 Exercises

Exercise 2.1

A fair coin is tossed 3 times. Find probability of exactly 2 heads.

Exercise 2.2

From a deck of 52, probability of drawing a red queen.

Exercise 2.3

A box has 4 red, 6 blue balls. One drawn at random. P(red)?

Exercise 2.4

If P(A)=0.4, P(B)=0.5, P(A∩B)=0.2, find P(A∪B).

Exercise 2.5

A machine produces 95% good items, 5% defective. If one item is chosen and found defective, what is probability it came from Machine 1 if Machine 1 makes 60% of items and has defect rate 8%, Machine 2 makes 40% with defect rate 1%? (Bayes).

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✅ Solutions

Solution 2.1

Sample size=8. Favorable={HHT,HTH,THH}. P=3/8.

Solution 2.2

2 red queens out of 52. P=2/52=1/26.

Solution 2.3

P(red)=4/10=0.4.

Solution 2.4

P(A∪B)=0.4+0.5–0.2=0.7.

Solution 2.5

P(M1)=0.6, P(M2)=0.4.

P(D|M1)=0.08, P(D|M2)=0.01.

P(D)=0.6·0.08+0.4·0.01=0.048+0.004=0.052.

P(M1|D)=0.048/0.052≈0.923. ✅

---

🔥 Challenge Problems

1. Roll 2 dice. Probability sum=7 or 11?

2. A family has 2 children. What is P(both boys | at least one boy)?

3. A medical test gives 90% accuracy. Disease prevalence=5%. Compute P(disease|positive).

4. Using Bayes’ theorem, prove that as prevalence→0, false positives dominate.

---

✅ Summary of Chapter 2

You learned:

• Classical definition of probability.

• Sample spaces, events, and rules.

• Conditional probability and independence.

• Bayes’ theorem and law of total probability.

📊 (Suggested figures: Venn diagram of events, probability tree, Bayes’ flow diagram).

---

📘 Book V – Probability, Statistics & Applied Mathematics

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 3 – Random Variables and Distributions

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🔹 3.1 What is a Random Variable?

A random variable (RV) assigns a number to each outcome of a random experiment.

• Discrete RV → takes countable values (0,1,2,…).

• Continuous RV → takes any value in an interval.

📘 Example 3.1

• Toss 3 coins, X = number of heads. (Discrete).

• Measure time for a reaction to finish. (Continuous).

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🔹 3.2 Probability Mass Function (PMF)

For a discrete RV X:

$$𝑃(𝑋=𝑥)=𝑝(𝑥),\sum _{𝑥}𝑝(𝑥)=1$$

📘 Example 3.2

Roll a die: X=outcome.

p(x)=1/6 for x=1,…,6.

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🔹 3.3 Probability Density Function (PDF)

For a continuous RV X with PDF f(x):

$$𝑃(𝑎\le 𝑋\le 𝑏)={\int }_{𝑎}^{𝑏}𝑓(𝑥)𝑑𝑥,{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}𝑓(𝑥)𝑑𝑥=1$$

📘 Example 3.3

X uniformly distributed on [0,1].

f(x)=1 for 0≤x≤1.

P(0.2≤X≤0.5)=∫₀.²^0.5 1 dx=0.3.

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🔹 3.4 Cumulative Distribution Function (CDF)

$$𝐹(𝑥)=𝑃(𝑋\le 𝑥)$$

• For discrete: step function.

• For continuous: F(x)=∫_{–∞}^x f(t) dt.

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🔹 3.5 Expectation and Variance

• Expected value (mean):

$$𝐸[𝑋]=\sum 𝑥𝑝(𝑥)\text{(discrete)},𝐸[𝑋]=\int 𝑥𝑓(𝑥)𝑑𝑥\text{(continuous)}$$

• Variance:

$$𝑉𝑎𝑟(𝑋)=𝐸[(𝑋\text{–}𝐸[𝑋]{)}^2]=𝐸[{𝑋}^2]\text{–}(𝐸[𝑋]{)}^2$$

📘 Example 3.4

Roll a fair die: X=outcome.

E[X]=(1+2+…+6)/6=3.5.

Var(X)=E[X²]–E[X]²=(91/6)–(3.5)²≈2.92.

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🔹 3.6 Common Distributions

1. Bernoulli (success/failure, p=probability of success).

P(X=1)=p, P(X=0)=1–p.

2. Binomial (n trials, p success each).

$$𝑃(𝑋=𝑘)=\left(\genfrac{}{}{0px}{}{𝑛}{𝑘}\right){𝑝}^{𝑘}(1\text{–}𝑝{)}^{𝑛\text{–}𝑘}$$

📘 Example 3.5

Flip 10 coins, probability of 6 heads: C(10,6)(0.5)¹⁰=210/1024≈0.205.

3. Poisson (events in fixed time/space, mean λ).

$$𝑃(𝑋=𝑘)=\frac{{𝜆}^{𝑘}{𝑒}^{\text{–}𝜆}}{𝑘!}$$

📘 Example 3.6

Average 3 emails/hour. P(5 emails in an hour)=λ=3, k=5= (3⁵ e^–3)/5!≈0.1008.

4. Uniform (continuous).

f(x)=1/(b–a) for a≤x≤b.

5. Normal (Gaussian).

$$𝑓(𝑥)=\frac{1}{𝜎\sqrt{2𝜋}}{𝑒}^{\text{–}(𝑥\text{–}𝜇{)}^2\mathrm{/}(2{𝜎}^2)}$$

📘 Example 3.7

Heights ~ N(170,10²). Probability between 160 and 180 ≈ 0.68.

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🔹 3.7 Joint Distributions

For two RVs X,Y:

• Joint PMF: p(x,y).

• Joint PDF: f(x,y).

• Independence: f(x,y)=f₁(x)f₂(y).

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🔹 3.8 Covariance and Correlation

• Covariance:

$$𝐶𝑜𝑣(𝑋,𝑌)=𝐸[(𝑋\text{–}𝐸[𝑋])(𝑌\text{–}𝐸[𝑌])]$$

• Correlation coefficient (ρ):

$$𝜌=\frac{𝐶𝑜𝑣(𝑋,𝑌)}{{𝜎}_{𝑋}{𝜎}_{𝑌}},\text{–}1\le 𝜌\le 1$$

📘 Example 3.8

If Cov(X,Y)=0 → no linear relation.

If ρ=1 → perfect positive linear relation.

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📝 Exercises

Exercise 3.1 – PMF

Toss 2 coins. Let X=# of heads. Find PMF.

Exercise 3.2 – Expectation

Roll a die. Compute E[X], Var(X).

Exercise 3.3 – Binomial

Compute P(exactly 2 successes) for n=5, p=0.3.

Exercise 3.4 – Poisson

If average accidents/day=2, probability of exactly 3 accidents?

Exercise 3.5 – Normal Approximation

SAT scores ~ N(500,100²). P(score between 400 and 600)?

---

✅ Solutions

Solution 3.1

X={0,1,2}. P(0)=1/4, P(1)=1/2, P(2)=1/4.

Solution 3.2

E[X]=3.5, Var(X)=35/12≈2.92.

Solution 3.3

P(X=2)=C(5,2)(0.3)²(0.7)³=10·0.09·0.343≈0.3087.

Solution 3.4

λ=2, k=3 → (2³ e^–2)/3!=8·0.1353/6≈0.180.

Solution 3.5

Standardize: z=(600–500)/100=1, z=(400–500)/100=–1.

P(–1≤Z≤1)=0.6826≈68%.

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🔥 Challenge Problems

1. Prove: mean of binomial(n,p)=np, variance=np(1–p).

2. For Poisson(λ), show E[X]=λ.

3. Find covariance of two independent RVs (hint: should be 0).

4. Two RVs have correlation ρ=–1. What does this mean geometrically?

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✅ Summary of Chapter 3

You learned:

• Random variables (discrete/continuous).

• PMFs, PDFs, and CDFs.

• Expectation, variance.

• Common distributions (Bernoulli, Binomial, Poisson, Uniform, Normal).

• Joint distributions, covariance, correlation.

📊 (Suggested figures: die PMF bar chart, normal curve, Poisson distribution, scatterplot showing correlation).

---

📘 Book V – Probability, Statistics & Applied Mathematics

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 4 – Statistics

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🔹 4.1 What is Statistics?

• Probability: starts with a model → predicts outcomes.

• Statistics: starts with data → infers the model.

Engineering, science, economics, and medicine all rely on statistics to analyze measurements, detect trends, and make decisions.

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🔹 4.2 Descriptive Statistics

Measures of Central Tendency

• Mean (average):

$$\overline{𝑥}=\frac{1}{𝑛}\sum _{𝑖=1}^{𝑛}{𝑥}_{𝑖}$$

• Median: middle value after sorting.

• Mode: most frequent value.

📘 Example 4.1

Data: {2,3,3,5,10}.

Mean=23/5=4.6, Median=3, Mode=3.

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Measures of Dispersion

• Range: max–min.

• Variance:

$${𝑠}^2=\frac{1}{𝑛-1}\sum ({𝑥}_{𝑖}-\overline{𝑥}{)}^2$$

• Standard deviation: square root of variance.

📘 Example 4.2

Data: {2,4,4,4,5,5,7,9}, mean=5.

Variance=4, std dev=2.

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🔹 4.3 The Normal Distribution

The bell curve:

$$𝑓(𝑥)=\frac{1}{𝜎\sqrt{2𝜋}}{𝑒}^{\text{–}(𝑥\text{–}𝜇{)}^2\mathrm{/}(2{𝜎}^2)}$$

• Symmetric around mean μ.

• 68–95–99.7 Rule:

  • 68% within 1σ,

  • 95% within 2σ,

  • 99.7% within 3σ.

📊 (Graph: bell curve with shaded 1σ, 2σ, 3σ regions.)

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🔹 4.4 Standard Normal and z-Scores

Standard normal: μ=0, σ=1.

z-score:

$$𝑧=\frac{𝑥\text{–}𝜇}{𝜎}$$

📘 Example 4.3

Heights ~ N(170, 10²). Find P(height < 185).

z=(185–170)/10=1.5 → P≈0.9332.

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🔹 4.5 Sampling and Central Limit Theorem (CLT)

• Sample mean approximates population mean.

• CLT: distribution of sample means → normal as n grows, regardless of original distribution.

📘 Example 4.4

Sample 50 students’ heights. Even if heights not perfectly normal, sample mean ~ N(μ,σ²/n).

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🔹 4.6 Confidence Intervals

Estimate population mean with confidence level:

$$𝐶𝐼=\overline{𝑥}\pm {𝑧}^{\ast }\frac{𝜎}{\sqrt{𝑛}}$$

📘 Example 4.5

Sample mean height=172 cm, σ=10, n=100.

95% CI=172 ± 1.96·10/√100 = 172 ± 1.96 = (170.04,173.96).

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🔹 4.7 Hypothesis Testing (Intro)

Steps:

1. State null (H₀) and alternative (H₁).

2. Choose significance level (α=0.05 common).

3. Compute test statistic.

4. Compare to critical value or p-value.

📘 Example 4.6

Claim: μ=100. Sample mean=104, σ=8, n=64.

z=(104–100)/(8/√64)=4/1=4.

At α=0.05, z=4 > 1.96 → reject H₀.

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📝 Exercises

Exercise 4.1 – Central Tendency

Find mean, median, mode of {12,15,12,18,20,20,20}.

Exercise 4.2 – Dispersion

Compute variance and std dev of {5,7,3,7,5}.

Exercise 4.3 – z-Score

IQ ~ N(100,15²). Find probability IQ>130.

Exercise 4.4 – CLT

Population mean=50, σ=10. For n=25, find std dev of sample mean.

Exercise 4.5 – Confidence Interval

A machine fills bottles. Sample: mean=505 ml, σ=8 ml, n=64. Find 95% CI.

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✅ Solutions

Solution 4.1

Mean=117/7=16.7, Median=18, Mode=20.

Solution 4.2

Mean=5.4. Variance=2.3, std dev≈1.52.

Solution 4.3

z=(130–100)/15=2.

P(Z>2)=0.0228≈2.3%.

Solution 4.4

σ/√n=10/5=2.

Solution 4.5

95% CI=505±1.96·(8/8)=505±1.96= (503.04,506.96).

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🔥 Challenge Problems

1. A dataset of 100 test scores has mean=75, std dev=10. Approx how many scores between 65 and 85 (using 68–95–99.7 rule)?

2. Show variance is minimized at mean by proving: 
   $\sum ({𝑥}_{𝑖}\text{–}𝑎{)}^2$
   minimized at a=mean.

3. A factory claims μ=200 g. A sample of 40 has mean=196, σ=12. Perform z-test at α=0.05.

4. Prove CLT for Bernoulli(p) trials (binomial distribution → normal approximation).

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✅ Summary of Chapter 4

You learned:

• Descriptive statistics: mean, median, mode, variance.

• Normal distribution and z-scores.

• CLT and why sampling works.

• Confidence intervals and hypothesis testing.

📊 (Suggested figures: histogram vs normal curve, z-score diagram, CI bar).

---

📘 Book V – Probability, Statistics & Applied Mathematics

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

---

Chapter 5 – Mathematical Modeling Applications

This chapter shows how algebra, calculus, probability, and linear algebra combine in real-world applications:

• Trusses (structural engineering).

• Electrical circuits (differential equations).

• Motion & growth models (exponential/logistic).

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🔹 5.1 Modeling with Vectors: Trusses and Forces

Trusses are frameworks of bars connected at joints. Each joint must satisfy equilibrium:

$$\sum {𝐹}_{𝑥}=0,\sum {𝐹}_{𝑦}=0$$

We use vectors to model forces.

📘 Example 5.1

A joint has two bars:

• Force along x-axis = 500 N.

• Force at 60° = 300 N.

Resultant force:

$$\overrightarrow{𝑅}=(500+300\cos 60\mathrm{°},300\sin 60\mathrm{°})=(650,260)𝑁$$

Magnitude=√(650²+260²)≈700 N.

📊 (Diagram: truss joint with force arrows, vector components.)

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🔹 5.2 Modeling with Differential Equations: Electrical Circuits

Kirchhoff’s laws + calculus → model circuits.

For an RC circuit (resistor + capacitor):

$$𝑅\frac{𝑑𝑞}{𝑑𝑡}+\frac{𝑞}{𝐶}=𝐸(𝑡)$$

• q(t)=charge.

• E(t)=voltage source.

📘 Example 5.2

RC circuit, R=10Ω, C=0.1F, E(t)=5 V (constant).

Equation: 10 dq/dt + 10q=5.

Solution: q(t)=0.5(1–e^(–t)).

Current: i(t)=dq/dt=0.05e^(–t).

📊 (Graph: exponential charging curve of capacitor.)

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🔹 5.3 Exponential Growth and Decay

Many natural processes follow:

$$\frac{𝑑𝑦}{𝑑𝑡}=𝑘𝑦\Rightarrow 𝑦(𝑡)={𝑦}_0{𝑒}^{𝑘𝑡}$$

• k>0 → growth.

• k<0 → decay.

📘 Example 5.3

Population doubles every 5 years. Initial=1000.

Growth rate k=ln(2)/5≈0.139.

After 20 years: y=1000e^(0.139·20)=16,000.

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🔹 5.4 Logistic Growth Model

When resources limit growth:

$$\frac{𝑑𝑦}{𝑑𝑡}=𝑘𝑦(1-\frac{𝑦}{𝑀})$$

• M=carrying capacity.

📘 Example 5.4

Bacteria in lab: k=0.2, M=1000, initial=10.

Solution:

$$𝑦(𝑡)=\frac{1000}{1+99{𝑒}^{\text{–}0.2𝑡}}$$

Population grows fast, then levels at 1000.

📊 (Graph: logistic curve vs exponential curve.)

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🔹 5.5 Probabilistic Models in Reliability

Systems with independent components:

• Reliability in series:

$${𝑅}_{𝑠𝑦𝑠𝑡𝑒𝑚}=\prod {𝑅}_{𝑖}$$

• Reliability in parallel:

$${𝑅}_{𝑠𝑦𝑠𝑡𝑒𝑚}=1\text{–}\prod (1\text{–}{𝑅}_{𝑖})$$

📘 Example 5.5

System with two components: R₁=0.9, R₂=0.8.

Series: 0.72.

Parallel: 0.98.

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🔹 5.6 Summary Applications

• Trusses → equilibrium with vectors.

• Circuits → DEs model current/charge.

• Growth Models → exponential/logistic DEs.

• Reliability → probability + combinatorics.

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📝 Exercises

Exercise 5.1

Force 100 N at 45° + 200 N along x-axis. Find resultant magnitude.

Exercise 5.2

In RC circuit with R=5Ω, C=2F, voltage E=10V constant, solve for charge q(t).

Exercise 5.3

Radioactive decay: half-life=10 hours, initial=80 g. Find remaining after 30 hours.

Exercise 5.4

Logistic model: M=500, k=0.1, initial=50. Find y(t). What is population after 20 units of time?

Exercise 5.5

System with 3 components, reliabilities 0.9, 0.95, 0.99. Compute reliability in (a) series, (b) parallel.

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✅ Solutions

Solution 5.1

Fx=200+100cos45=200+70.7=270.7.

Fy=100sin45=70.7.

Magnitude=√(270.7²+70.7²)≈280 N.

Solution 5.2

Equation: 5 dq/dt+q/2=10 → dq/dt+0.1q=2.

Solution: q(t)=20(1–e^(–0.1t)).

Solution 5.3

Decay: y(t)=80(½)^(t/10). At t=30: 80(½)^3=10 g.

Solution 5.4

y(t)=500/(1+9e^(–0.1t)). At t=20: ≈339.

Solution 5.5

Series=0.9·0.95·0.99=0.846.

Parallel=1–(0.1·0.05·0.01)=0.99995.

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🔥 Challenge Problems

1. A beam supports forces of 1000 N at 30° and 800 N at 120°. Find resultant force vector and magnitude.

2. Solve RLC circuit differential equation: L d²q/dt² + R dq/dt + q/C=E(t).

3. Logistic growth: show that solution approaches M as t→∞.

4. For system of n identical components with reliability R, derive reliability formulas for series and parallel cases.

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✅ Summary of Chapter 5

You saw math in action:

• Vectors & trusses → mechanics of structures.

• Differential equations & circuits → current/voltage.

• Growth/decay models → biology, physics, finance.

• Reliability models → engineering systems.

📊 (Suggested figures: truss with forces, capacitor charging curve, exponential vs logistic graph, system reliability diagram.)

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📘 Book V – Probability, Statistics & Applied Mathematics

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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Chapter 6 – Capstone Challenge Problems

This chapter blends algebra, calculus, probability, statistics, and modeling. Each problem is designed to reflect real-world complexity.

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🔹 6.1 Mixed Problems

📘 Problem 6.1 – Optimization & Probability
A factory makes screws. The diameter follows N(5mm, 0.1²).
a) What percent are between 4.9 and 5.1 mm?
b) If 1000 screws are made, how many are expected to be within tolerance?

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📘 Problem 6.2 – Reliability & Linear Algebra
A system has 3 components in series with reliabilities R=[0.95, 0.9, 0.8].
a) Compute overall reliability.
b) Represent as vector multiplication with system matrix.

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📘 Problem 6.3 – Differential Equations & Growth
Population model: dy/dt=0.3y(1–y/1000), y(0)=50.
Find population at t=20.

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📘 Problem 6.4 – Multivariable Calculus in Physics
Temperature distribution T(x,y)=x²+y² (in °C).
At (2,1):
a) Find gradient ∇T.
b) In which direction does temperature increase fastest?

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📘 Problem 6.5 – Probability & Combinatorics
A password is 6 characters long, chosen from 26 letters + 10 digits (36 total).
a) How many possible passwords?
b) If one is chosen at random, probability it has no digits?

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📘 Problem 6.6 – Integration & Statistics
Compute expected value of X with PDF f(x)=2x, 0≤x≤1.

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📘 Problem 6.7 – Trusses & Vectors
Two forces act at a joint:

• 100 N at 30°.

• 150 N at 120°.
  Find resultant vector and magnitude.

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📘 Problem 6.8 – Circuit & DE
RC circuit: R=10Ω, C=0.2F, voltage=5V constant.
Equation: 10 dq/dt+5q=5.
Solve for q(t).

---

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🔹 6.2 Solutions

Solution 6.1
a) Standardize: z=(5.1–5)/0.1=1; z=(4.9–5)/0.1=–1.
P(–1<Z<1)=0.6826=68.3%.
b) Out of 1000 → 683 screws. ✅

Solution 6.2
a) R=0.95·0.9·0.8=0.684.
b) System vector model: [R1 R2 R3] multiplied in series (diag matrix).

Solution 6.3
Logistic solution: y(t)=1000/(1+19e^(–0.3t)).
At t=20 → ≈687. ✅

Solution 6.4
∇T=(2x,2y). At (2,1): (4,2).
Direction=unit vector (4,2)/√(20)= (0.894,0.447). ✅

Solution 6.5
a) 36^6≈2.18×10^9.
b) 26^6/36^6≈0.079 → 7.9%. ✅

Solution 6.6
E[X]=∫₀¹ x·2x dx=∫₀¹ 2x² dx=[2/3]≈0.667. ✅

Solution 6.7
Fx=100cos30+150cos120=86.6–75=11.6.
Fy=100sin30+150sin120=50+129.9=179.9.
Magnitude=√(11.6²+179.9²)≈180 N. ✅

Solution 6.8
Equation: dq/dt+0.5q=0.5.
Solution: q(t)=1–e^(–0.5t). ✅

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🔹 6.3 Challenge Problems

1. Bridge Design (Integration + Vectors):
   Compute total load on a parabolic arch y=10–x² from –3≤x≤3 with uniform load density=2 kN/m.

2. Machine Learning Probability (Statistics):
   A classifier predicts “positive” with accuracy 90% for true cases, 10% false positive. Prevalence=5%. Compute P(true positive | predicted positive).

3. Physics Modeling (DE):
   A cooling object obeys Newton’s law: dT/dt=–k(T–20). If T(0)=100, T(10)=60, find k and T(t).

4. Matrix & Eigenvalues:
   Find eigenvalues of A=[[2,1],[1,2]] and interpret them as principal axes of symmetry.

5. Reliability System (Combinatorics + Probability):
   System has 4 identical components with R=0.9.
   a) Reliability in series.
   b) Reliability in parallel.
   c) Compare.

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🔹 6.4 Summary of Chapter 6

• Mixed problems tested integration, DEs, probability, statistics, linear algebra, and applications.

• Challenge problems linked directly to engineering, physics, and computer science.

• This chapter closes the series by simulating how different math fields interact in practice.

📊 (Suggested diagrams: arch load curve, logistic growth graph, reliability block diagram, Bayes’ probability tree.)

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✅ Book V Complete 🎉

You now have a 5-volume full textbook:

• Book I: Foundations & Pre-Calculus

• Book II: Advanced Algebra & Trigonometry

• Book III: Calculus I

• Book IV: Calculus II & Linear Algebra

• Book V: Probability, Statistics & Applied Mathematics

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📚 Mathematics Foundations for Engineering and Science – Examples, Exercises & Solutions

By Ronen Kolton Yehuda (Messiah King RKY), with the assistance of ChatGPT AI

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🎓 Epilogue – The Journey Through Five Volumes

Mathematics is not just a subject—it is the language of science and engineering, the grammar of the universe, and the foundation for problem-solving across disciplines.
This five-volume series was designed to bridge high school knowledge to academic and engineering-level mathematics, providing not only theory but also worked examples, exercises, and challenge problems.

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📘 Book I – Foundations & Pre-Calculus

• You began at the ground level: numbers, arithmetic, sets, and logic.

• Progressed to algebra, functions, trigonometry, and vectors.

• Learned the “alphabet” of mathematics, establishing the building blocks for everything that follows.

📊 (Diagrams: number lines, trigonometric circles, vector arrows)

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📘 Book II – Advanced Algebra & Trigonometry

• Expanded into complex numbers, polynomials, sequences, series, and conics.

• Mastered advanced trigonometry and extended into 3D vectors.

• These tools gave you the mathematical maturity to transition smoothly into calculus.

📊 (Diagrams: complex plane, conic sections, vector products in 3D)

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📘 Book III – Calculus I

• Discovered the core ideas of calculus: limits, derivatives, and integrals.

• Learned to apply derivatives to optimization, motion, curve sketching, and rates of change.

• Entered the world of integration: areas, antiderivatives, and the fundamental theorem of calculus.

📊 (Diagrams: tangent lines, area under curve, motion graphs)

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📘 Book IV – Calculus II & Linear Algebra

• Moved into advanced integration, differential equations, and multivariable calculus.

• Explored double/triple integrals, gradients, and partial derivatives.

• Entered linear algebra: matrices, determinants, systems of equations, and eigenvalues—tools at the heart of engineering, physics, and computer science.

📊 (Diagrams: 3D surfaces, contour maps, matrix transformations)

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📘 Book V – Probability, Statistics & Applied Mathematics

• Completed the journey by connecting mathematics to uncertainty and real-world models.

• Learned probability, random variables, distributions, statistics, confidence intervals, and hypothesis testing.

• Applied everything in trusses, circuits, growth models, and reliability systems.

• Finished with capstone problems integrating the entire 5-book path.

📊 (Diagrams: probability trees, normal curves, logistic growth vs exponential curves)

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🌟 What You Can Do Now

With the completion of this series, you now have the mathematical toolbox to:

• Solve engineering problems (forces, circuits, reliability).

• Build mathematical models (population growth, logistic systems, financial decay).

• Perform data analysis (statistics, confidence intervals, hypothesis testing).

• Understand the mathematics behind computer algorithms, AI, and physics.

• Tackle advanced university-level mathematics with confidence.

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🔥 Final Thought

Mathematics is not about memorization—it is about thinking clearly, solving problems, and recognizing patterns.
These volumes are a beginning, not an end. With practice and curiosity, you can extend these foundations into modern science, engineering innovation, and beyond.

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✅ Series Complete 🎉
"Mathematics is the key and door to the sciences." – Galileo

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Computer Science Foundations with Java – Theory, Examples & Exercises 📚