Lesson 6: Stationary Points, Concavity & Curve Sketching
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Stationary points (f'(x) = 0), local maxima, local minima, horizontal points of inflection, second derivative test (f''(x) > 0 for local min, f''(x) < 0 for local max), and concavity changes (f''(x) = 0).
Locate and classify all stationary points using 1st and 2nd derivative tests, determine intervals of concavity, and sketch accurate polynomial/rational curves.
🎥 Media Anchor & Pedagogical Scaffold
Second Derivative Test — Khan Academy
Video (4 min 51 sec, Khan Academy): A visual guide to using the second derivative to classify stationary points as local maxima or minima, and to understand concavity. Clear, direct, and perfect for curve-sketching practice.
🧠 1. Before Viewing (Activate & Predict)
What is the physical and geometric meaning when f'(x) = 0 and f''(x) changes sign?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 4¾-minute clip. In your calculus logbook, record:
- The second derivative test: Write: If f''(x₀) > 0, local minimum; if f''(x₀) < 0, local maximum; if f''(x₀) = 0, test is inconclusive.
- A worked example: For f(x) = x³ - 3x, find f'(x) = 3x² - 3, set to 0 to get x = ±1. Then f''(x) = 6x. At x = -1: f''(-1) = -6 < 0 → local max. At x = 1: f''(1) = 6 > 0 → local min.
- Concavity: Note when f'' > 0 (curve is concave up) and when f'' < 0 (concave down).
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Draw concavity memory cues: f''(x) > 0 is 'happy cup' (concave up, local min); f''(x) < 0 is 'frown' (concave down, local max).
Immediate Task: Add to your Level 3 Calculus revision logbook (section 6): Curve Analysis & Stationary Point Classification.
⚡ Whakaoho | Do Now: Solving f′(x) = 0 (10 mins)
Solve 3x² − 12 = 0 and 2x³ − 8x = 0. Then differentiate f(x) = x³ − 3x and differentiate your answer again. Today the second derivative stops being an extra step and starts doing real work.
📖 Activity 1: Full Curve Sketch With Justified Nature (25 mins)
Take f(x) = x³ − 3x. Find the stationary points, then classify each one using f″(x): negative means a maximum, positive a minimum. Find where f″(x) = 0 and check whether concavity actually changes there before calling it a point of inflection. Sketch the curve using only what you derived, then check it against a graphing tool. Where your sketch and the tool disagree, the working tells you which one is wrong.
📝 Activity 2: Exam-Style Practice — Classify and Justify (20 mins)
For f(x) = x⁴ − 4x³: find and classify all stationary points. For Merit, use the second-derivative test and state its result at each point. For Excellence, explain what happens when f″(x) = 0 at a stationary point and why that case needs a different argument.
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 Alignment: Direct preparation for Level 3 Calculus (Apply differentiation methods in solving problems). Emphasise complete algebraic working and clear context conclusions for Merit/Excellence grades.