Lesson 6: Tangents & Turning Points

Lesson at a Glance | He Tirohanga Whakamua

Do NowFinding the tangent and normal line equations to y = x³ - 3x at x = 210 min

Tangent & Normal GeometryApplying

m_{T} = f^{'} (x_{1})

and perpendicular

m_{N} = - \frac{1}{m_{T}}

15 min

Stationary Points (f'(x) = 0)Locating local maxima, local minima, and stationary points of inflection15 min

Portfolio EntryWrite Section 6: Tangent-Normal & Stationary Point Analysis10 min

Exit DrillFind coordinates and nature of turning points for

y = 2 x^{3} - 9 x^{2} + 12 x

5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

Tangent Line: Equation $y - y_{1} = m_{T} (x - x_{1})$ where $m_{T} = f^{'} (x_{1})$ .
Normal Line: Line perpendicular to tangent ( $m_{N} = - \frac{1}{m_{T}}$ ), equation $y - y_{1} = m_{N} (x - x_{1})$ .
Stationary Turning Points: Points where $f^{'} (x) = 0$ . Classifying maxima (slope shifts $+ \to 0 \to -$ ) vs minima (slope shifts $- \to 0 \to +$ ).

Students will demonstrate

By deriving tangent/normal equations and classifying turning points for cubic and exponential functions.
By completing Section 6 of their Level 3 Calculus Differentiation Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Perpendicular Normal Prompt:

"At $x = 2$ , the curve $y = x^{2}$ has tangent slope $m_{T} = 4$ passing through $(2, 4)$ . What is the equation of the normal line that cuts perpendicularly through the exact same point?"

Unpack: Normal slope $m_{N} = - \frac{1}{4}$ . Using point-slope formula: $y - 4 = - \frac{1}{4} (x - 2) \to y = - \frac{1}{4} x + \frac{9}{2}$ !

Turning Point Classification (15 min)

1. Local Maximum ( $f^{'} (x) = 0$ )

$f^{'} (x)$ changes sign from positive to negative. Second derivative test: $f^{″} (x) < 0$ .

2. Local Minimum ( $f^{'} (x) = 0$ )

$f^{'} (x)$ changes sign from negative to positive. Second derivative test: $f^{″} (x) > 0$ .

📁 Calculus Differentiation Portfolio — Section 6: Tangents & Turning Points

Students open their Level 3 Calculus Portfolio and complete Section 6:

Section 6 Requirements:

1. Tangent & Normal Geometry Sketch: Draw a curve $y = f (x)$ with tangent line, normal line, and right-angle symbol at point of contact $(x_{1}, y_{1})$ .

2. Turning Point Solver: Find the exact coordinates and classify turning points for $f (x) = x^{3} - 6 x^{2} + 9 x + 2$ .

3. Extension: Horizontal Normal Paradox: 1-paragraph explanation of what happens to the normal line equation at a stationary turning point where $m_{T} = 0$ .

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 6 finds local maximum (1, 6) and local minimum (3, 2) for f(x) = x^3 - 6x^2 + 9x + 2 via f'(x) = 0."

Teacher Planning & NCEA Alignment

NCEA Level 3 Calculus Alignment (6 Credits External):

Differentiation Methods: Derive equations of tangent and normal lines, locate stationary points ( $f^{'} (x) = 0$ ), and determine their nature.

Vocabulary: Tangent line, normal line, perpendicular slope ( $m_{N} = - 1 / m_{T}$ ), stationary point, local maximum, local minimum, turning point.

Lesson 6: Tangents, Normals & Turning Points