Golden Education

📚 Further Calculus - Complete Guide Pure Mathematics 3| Cambridge AS & A Level 8.1 Derivative of tan⁻¹(x) Understanding tan⁻¹(x): The inverse tangent function (arctan) reverses the tangent operation. If tan(θ) = x, then θ = tan⁻¹(x) Derivation Let y = tan⁻¹(x) Then: tan(y) = x Differentiate both sides: sec²(y) × dy/dx = 1 Therefore: dy/dx = 1/sec²(y) Using identity: sec²(y) = 1 + tan²(y) = 1 + x² Result: dy/dx = 1/(1 + x²) KEY FORMULA: d/dx[tan⁻¹(x)] = 1/(1 + x²) With chain rule: d/dx[tan⁻¹(f(x))] = f'(x)/(1 + [f(x)]²) Example 1: Differentiate tan⁻¹(3x) Solution: Let f(x) = 3x, so f'(x) = 3 d/dx[tan⁻¹(3x)] = 3/(1 + 9x²) Example 2: Differentiate tan⁻¹(√x) Solution: f(x) = x^(1/2), f'(x) = 1/(2√x) d/dx[tan⁻¹(√x)] = [1/(2√x)] / [1 + x] = 1/[2√x(1 + x)] 8.2 Integration of 1/(x² + a²) The Reverse Process: Since d/dx[tan⁻¹(x)] = 1/(1 + x²), we can integrate backwards! KEY FORMULAS: ∫ 1...

📐 VECTORS: Complete Study Guide Cambridge AS & A Level Mathematics - Pure Mathematics 3 1. INTRODUCTION TO VECTORS What is a Vector? A vector is a quantity that has two essential properties: 📏 Magnitude (Size/Length) 🧭 Direction Scalars vs Vectors Scalars (Number only) Vectors (Number + Direction) Temperature (25°C) Velocity (60 km/h North) Mass (5 kg) Force (10 N Downward) Distance (10m) Displacement (10m East) 2. DISPLACEMENT VECTORS (Section 9.1) 2.1 Writing Vectors Method 1: Column Vector ⎛ x ⎞ v = ⎜ y ⎟ ⎝ z ⎠ Where x is movement along the x-axis, y al...

Other note source 📊 SAMPLING - Theory & Formulas Cambridge AS & A Level Mathematics 📖 Part 1: Introduction to Sampling Key Definitions Population: Complete set of ALL items of interest Sample: Part of the population (size = n) Representative Sample: Accurately reflects population characteristics Biased Sample: Does NOT properly represent population Random Sample: ALL possible samples of size n have equal probability of selection 💡 Why Use Samples? Reason Example 💰 Cost-Effective Test 50 products vs 10,000 ⏰ Time-Saving Survey 100 people vs millions 🔨 Destructive Testing Crash testing helmets 🌍 Impossible to Survey All All fish in the ocean 🎲 Random Sampling Methods Using Random Number Tables: Number population: 000 to 499 (for 500 items) Pick starting point in table Read digits matching your numbering Ignore numbers outside range Ignore repeats Using Excel: =RAN...

Other note source PDF Source 📐 Perimeter, Area and Volume Complete Guide for Grade 7 - Cambridge Curriculum 📏 Section 1: Perimeter and Area in Two Dimensions 1.1 Understanding Perimeter Perimeter is the total distance around the outside of a shape. To find the perimeter, add up the lengths of all sides. For any polygon: Perimeter = Sum of all sides Circle - Circumference The perimeter of a circle is called circumference . r d C = πd or C = 2πr where π ≈ 3.14 or 22/7 Example 1: Circle Circumference Find the circumference of a circle with radius 7 cm. Solution: C = 2πr C = 2 × 3.14 × 7 C = 43.96 cm 1.2 Understanding Area Area is the total space contained...

Other note source NUMERICAL SOLUTIONS OF EQUATIONS Concise Theory & Formula Guide Cambridge AS & A Level Mathematics 1. INTRODUCTION Numerical methods find approximate solutions to equations that cannot be solved algebraically. Examples include x³ + x - 4 = 0 , eˣ = 2x + 1 , and sin(x) = x - 1 . Historical Fact: Quintic equations (degree 5 and higher) generally have no algebraic solutions, making numerical methods essential. 2. LOCATING ROOTS (Section 6.1) Root Definition α is a root of f(x) = 0 if f(α) = 0 Method 1: Graphical Approach Rearrange equation as g(x) = h(x) , sketch both graphs, and find intersection points. Each intersection represents a root. Method 2: Change of Sign Method Change of Sign Principle: If f(x) is continuous and f(a) · f(b) If f(a) 0 → Root exists between a and b Example: Change of Sign Problem: Show f(x) = x⁵ + x - 1 = 0 has a root between 0 ...

Quiz 1:   Fullscreen Mode

Algebra Guide: Equations, Factors & Formulae 📐 Algebra Essentials: Equations, Factors & Formulae Welcome! This guide covers fundamental algebra concepts: solving equations (finding unknowns), factorization (breaking expressions into factors), and rearranging formulae (changing the subject). These skills apply to real-world problems from cooking times to physics calculations. 1️⃣ Solving Equations Linear Equations A linear equation has variables with power ≤ 1 (e.g., 3x + 1 = 13 ). Golden Rule: Whatever you do to one side, do to the other side to keep it balanced. Method 1: Function Machine x → [×3] → 3x → [+1] → 3x+1 = 13 ↓ Work backwards ↓ 4 ← [÷3] ← 12 ← [-1] ← 13 Result: x = 4 Method 2: Algebraic Steps Solve: 3x + 1 = 13 Step 1: 3x + 1 - 1 = 13 - 1 → 3x = 12 Step 2: 3x ÷ 3 = 12 ÷ 3 → x = 4 Type 1: Variable on ...

Other note source 📐 INTEGRATION - Theory & Formulas Cambridge AS & A Level Mathematics - Pure Mathematics 2 1. Introduction to Integration Integration is the reverse process of differentiation . Symbol: ∫ f(x) dx means "integrate f(x) with respect to x" ⚠️ Always add + c for indefinite integrals (constant of integration) Example: • Differentiate x² → get 2x • Integrate 2x → get x² + c 2. Integration of Exponential Functions Basic Rules ∫ e x dx = e x + c ∫ e (ax+b) dx = (1/a) e (ax+b) + c Example 1: ∫ e (3x) dx a = 3, so answer = (1/3)e (3x) + c Example 2: ∫ 6e (3x) dx = 6 × (1/3)e (3x) + c = 2e (3x) + c Example 3: Evaluate ∫₀² e (3x) dx Step 1: Integrate: [(1/3)e (3x) ]₀² Step 2: Upper limit: (1/3)e⁶ Step 3: Lower limit: (1/3)e⁰ = 1/3 Answer: (1/3)(e⁶ - 1) 3. Integration of 1/(ax+b) Basic Rules ∫ (1/x) dx = ln|x| + c ∫ 1/(ax+b) dx = (1/a) ln|ax+b| + c ⚠️ Always use |x| (absolute value) b...