Lesson 5: Tangents, Normals & Rates of Change
The derivative as a slope you can actually use: the tangent at a point, the normal perpendicular to it, and velocity as the derivative of displacement.
🎯 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 8 min 7 sec 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 (5 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 (20 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 (15 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.
🎫 Exit Ticket: Slope, Normal, Motion (5 mins)
(a) Find the equation of the tangent to y = x² − 4x at x = 3. (b) State the gradient of the normal at that same point. (c) For s(t) = t³ − 6t², when is the object at rest?
🏫 Kaiako Planning & Pedagogy Notes
NCEA Level 3 alignment: Direct preparation for NCEA Level 3 Achievement Standard AS 91578, Apply differentiation methods in solving problems (external, 6 credits), against The New Zealand Curriculum (2007) Mathematics and Statistics Level 8 — Calculus. Emphasise complete algebraic working and a conclusion written in the context of the question, which is what separates Merit from Achieved.
Materials: Graphics calculator, Desmos (graph the curve and the tangent together — a tangent that visibly cuts through the curve is instant feedback), ruler and grid paper for hand sketches, calculus logbook.
Pacing (58 mins): Do Now 5 · media anchor 13 (the clip plus the before- and after-viewing prompts) · Activity 1 20 · Activity 2 15 · exit ticket 5.
Formative assessment — what to look for: Part (c) is aimed at a misconception this unit names explicitly: “at rest” means v = 0, not s = 0. An ākonga who solves s(t) = 0 has answered a different question fluently, which is the hardest kind of error to see in your own marking.
Differentiation. Entry: Supply the gradient m and the point, and ask only for the completed equation y − y₀ = m(x − x₀). On level: Full tangent and normal problems, plus the kinematics set. Extension: Find every point on y = x³ − 3x where the tangent is parallel to the line y = 9x, then confirm both points in Desmos.