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Further Calculus PM2

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📚 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...
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Vectors PM2

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📐 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...
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Nuclear Model of the Atom

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Quiz:   Fullscreen Mode Other note source ⚛️ NUCLEAR MODEL OF THE ATOM 19.1 THE ATOM What is an Atom? ATOM = The smallest unit of a chemical element Structure of Atom ATOM = NUCLEUS + ELECTRONS Part Charge Location Mass Nucleus Positive (+) Center Almost all mass Electrons Negative (−) Orbit around nucleus Very small KEY FACT: Most of atom is EMPTY SPACE! Formation of IONS Neutral Atom: Number of Protons = Number of Electrons LOSE electrons → POSITIVE ION (+) GAIN electrons → NEGATIVE ION (−) Example: - Atom with 11 protons & 11 electrons = neutral - Loses 1 electron → 11 protons & 10 electrons = positive ion (+1) - Gains 1 electron → 11 protons & 12 electrons = negative ion (−1) Gold Foil Experiment (1911) Scientists: Geiger, Marsden, Rutherford Method: Shot alpha (α) particles at thin gold foil Results: Observation Conclusion Most α...
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SAMPLING - Theory & Formulas

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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...
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An Introduction to Organic Chemistry

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Quiz: Fullscreen Mode AN INTRODUCTION TO ORGANIC CHEMISTRY Complete Study Notes for Cambridge IGCSE Chemistry SECTION 16.1: Understanding Organic Chemistry Basics What Are Organic Compounds? Organic compounds are special chemicals that contain the element carbon . You can find them everywhere in your daily life! The food you eat (like bread, rice, and meat), your hair, plastic bottles, soap, and even medicines are all made from organic compounds. There are two types of organic compounds: Natural organic compounds - These come from nature, like the proteins in your hair or the sugar in fruits Synthetic organic compounds - These are man-made in factories, like plastics, detergents, and modern medicines Important Note: Not ALL compounds with carbon are organic! Carbon dioxide (CO₂) and calcium carbonate (CaCO₃) contain carbon but are NOT organic compounds. Wha...
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