Lesson 5: Tangents, Normals & Rates of Change
Applying differentiation methods, rules, optimisation, and related rates for NCEA Level 3 Calculus.
🎯 Ngā Whāinga Akoranga | Learning Intentions
Geometric interpretation of derivative as tangent slope m_T = f'(x_0), normal slope m_N = -1 / f'(x_0), and physical rates of change (v = ds/dt, a = dv/dt).
Find linear equations of tangent y - y_0 = m_T (x - x_0) and normal lines to curves at given points, and calculate physical rates of change.
🎥 Media Anchor & Pedagogical Scaffold
Equation of a Tangent Line — Khan Academy
Video (8 min 7 sec, Khan Academy): A comprehensive visual and worked-example guide to finding the equation of a tangent line to a curve at a given point using the derivative as the slope. Multiple examples show the method clearly.
🧠 1. Before Viewing (Activate & Predict)
How does the perpendicular normal line to a curved boundary govern reflection optics and structural force vectors?
👁️ 2. During Viewing (Watch With a Job)
Watch the full 5¾-minute clip. In your calculus logbook, record:
- The tangent line formula: Write y - y₀ = m(x - x₀) where m = f'(x₀), the slope at the point.
- A worked example: For f(x) = x², find the tangent line at x = 3. (f'(x) = 2x, so m = 6 at x = 3. Point is (3, 9). Tangent: y - 9 = 6(x - 3) or y = 6x - 9.)
- Normal line concept: Note that the normal line (perpendicular to tangent) has slope m_N = -1/6 in the above example.
🗣️ 3. After Viewing & Kaiako Move (Process & Apply)
Kaiako Move: Connect normal lines to optical ray tracing and road curve camber design.
Immediate Task: Add to your Level 3 Calculus revision logbook (section 5): Tangent & Normal Line Equations & Kinematic Derivatives.
⚡ Whakaoho | Do Now: Gradient to Equation (10 mins)
Write the equation of the line through (2, 5) with gradient 3. Then write the equation of the line through the same point perpendicular to it. Recall what the derivative gives you at a single point, as opposed to what the original function gives you.
📖 Activity 1: Tangents, Normals and What dy/dx Means Here (25 mins)
For y = x³ − 4x at x = 2: find the gradient, then the tangent equation, then the normal equation. Sketch both on the curve and check by eye that the tangent's slope and the normal's slope look right. Then a context: if s(t) = 3t² − t gives displacement in metres after t seconds, find s′(2) and state its units and its physical meaning. A derivative with no units attached is an incomplete answer at this level.
📝 Activity 2: Exam-Style Practice — Points and Rates (20 mins)
Find the tangent to y = eˣ at x = 0, and the point on y = x² − 6x where the tangent is horizontal. For Merit, show the gradient calculation separately from the line equation. For Excellence, find where the normal to y = x² at x = 1 crosses the x-axis, and say what makes that a normal rather than a tangent.
🏫 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.