[H2 MATH] Chapter 8 - Applications of Differentiation

SHALYN TAY COPYRIGHTED © A LEVEL H2 MATHEMATICS APPLICATIONS OF DIFFERENTIATION

CHAPTER ANALYSIS MASTERY EXAM WEIGHTAGE SHALYN TAY COPYRIGHTED ©

Strictly Increasing / Decreasing & Concavity SHALYN TAY COPYRIGHTED © Function 𝑓′(𝑥) Graph Increasing > 0 Decreasing < 0 Concavity 𝑓′′(𝑥) Graph Concave Up Minimum > 0 Concave Down Maximum < 0 Function is strictly increasing over the interval (a, b) Function is strictly decreasing over the interval (a, b) Function is concave up over the interval (a, b) Function is concave down over the interval (a, b)

Stationary Points, Turning Points & Points Of Inflection SHALYN TAY COPYRIGHTED © Method 1: First Derivative Test 𝑥 𝑑𝑦 𝑑𝑥 Shape Nature of Stationary Point Maximum Minimum Point of Inflection 𝑥𝑥− 𝑥+ 𝑥𝑥− 𝑥+ 𝑥𝑥− 𝑥+ +ve -ve0 -ve +ve0 +ve +ve0 -ve -ve0 Method 2: Second Derivative Test Nature of Stationary Point 𝑓′′(𝑥) Maximum > 0 Minimum < 0 In general, we know a point is stationary when 𝑑𝑦 𝑑𝑥 = 0 . However, questions usually ask for the nature of stationary points. This is also useful when drawing graphs in graphing techniques.

SHALYN TAY COPYRIGHTED © Tangents & Normals Equation of Tangent: 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 Equation of Tangent: 𝑦 − 𝑦1 = − 1 𝑚 (𝑥 − 𝑥1) Maximisation & Minimisation (Guidelines) 1. Denote each changing quantity by a variable 2. Write a formula for quantity to be maximized/minimized 3. Express formula from Step (2) in terms of 1 variable only 4. Differentiate and equate derivative to 0 for stationary values 5. Use first or second derivative test to justify if is maximum/minimum 6. Answer question Rate of Change 𝑑𝑦 𝑑𝑡 > 0 ⟹ y increases with time 𝑑𝑦 𝑑𝑡 < 0 ⟹ y decreases with time Chain Rule: 𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑡 × 𝑑𝑡 𝑑𝑥 Rate of Change (Guidelines) 1. Denote each changing quantity by a variable 2. Find equations relating the variables 3. Use chain rule to link up derivatives 4. Write down values of variables and given rates of change 5. Solve for unknown rate Note: Contextual questions, important to practice different types. Guidelines are generic.

6 For more notes & learning materials, visit: www.overmugged.com ‘A’ levels crash course program IG handle: @overmugged DARRELL ER (COPYRIGHTED) © Join our telegram channel: @overmuggedAlevels Need help? Shalyn Tay (Private tutor with 4 years of experience) 82014166 (Whatsapp) @shalyntay (telegram username) Professionally designed crash courseto help you get a condensed revision before your ‘A’ Levels! Each H2 subject will have 3 crash course modules which will cover their entire H2 syllabus. The 4 hour module focuses on going through key concepts and identifying commonly tested questions! The crash courses modules will begin in June 2021 and last till Oct 2021. Pre-register now on our website and secure your slots!