Lesson 7: Concavity & Curve Sketching

Lesson at a Glance | He Tirohanga Whakamua

Do NowWhat does second derivative f''(x) physically measure on a graph?10 min

Concavity & Second Derivative TestConcave Up (

f^{″} > 0

) vs Concave Down (

f^{″} < 0

)15 min

Points of Inflection ( $f^{″} (x) = 0$ )Distinguishing stationary vs non-stationary inflection points15 min

Portfolio EntryWrite Section 7: Concavity & Curve Sketching Guide10 min

Exit DrillFind point of inflection for

f (x) = x^{3} - 3 x^{2} + 4

5 min

Ngā Whāinga Ako | Learning Intentions

Students will know

Concavity: $f^{″} (x) > 0 \to$ Concave Up ( $\cup$ ), $f^{″} (x) < 0 \to$ Concave Down ( $\cap$ ).
Point of Inflection: A point where $f^{″} (x) = 0$ AND concavity changes sign.
Full Curve Sketching Protocol: Intercepts $\to$ Asymptotes $\to$ Stationary Points ( $f^{'} = 0$ ) $\to$ Inflection Points ( $f^{″} = 0$ ).

Students will demonstrate

By sketching a complete annotated graph of a cubic or rational function showing turning points and concavity changes.
By completing Section 7 of their Level 3 Calculus Differentiation Portfolio.

Do Now | Tīmatanga Whakaaro (10 min)

Curvature Prompt:

"If $f^{'} (x)$ tells us if a function is going UP or DOWN, what does $f^{″} (x)$ tell us?"

Unpack: $f^{″} (x)$ measures how fast the slope $f^{'} (x)$ itself is changing! It determines concavity—whether the curve is bending upwards like a cup ( $f^{″} > 0$ ) or bending downwards like an umbrella ( $f^{″} < 0$ ).

Points of Inflection & Concavity Test (15 min)

1. Non-Stationary Inflection ( $f^{″} = 0, f^{'} \neq 0$ )

$f (x) = x^{3} - 3 x \to f^{'} (x) = 3 x^{2} - 3 \to f^{″} (x) = 6 x$ .
Inflection point at $x = 0$ , but slope $f^{'} (0) = - 3 \neq 0$ !

2. Stationary Inflection ( $f^{″} = 0, f^{'} = 0$ )

$f (x) = x^{3} \to f^{'} (x) = 3 x^{2} \to f^{″} (x) = 6 x$ .
At $x = 0$ , both slope $f^{'} (0) = 0$ and concavity $f^{″} (0) = 0$ .

📁 Calculus Differentiation Portfolio — Section 7: Concavity & Curve Sketching

Students open their Level 3 Calculus Portfolio and complete Section 7:

Section 7 Requirements:

1. Concavity Summary Chart: Draw diagrams contrasting Concave Up ( $f^{″} > 0$ ) vs Concave Down ( $f^{″} < 0$ ) and inflection point transitions.

2. Full Curve Sketcher: Sketch $f (x) = 2 x^{3} - 3 x^{2} - 12 x + 5$ , annotating intercepts, turning points $(f^{'} = 0)$ , and inflection point $(f^{″} = 0)$ .

3. Extension: Second Derivative Test Failure Rationale: 1-paragraph explanation of why $f^{″} (x) = 0$ at a stationary point is inconclusive (e.g. $y = x^{4}$ vs $y = - x^{4}$ vs $y = x^{3}$ ), requiring a sign-table audit.

Exit Verification | Ka Mutu Hoki (5 min)

Exit Check:

"My Section 7 finds point of inflection at (1, 2) for f(x) = x^3 - 3x^2 + 4 where f''(x) = 6x - 6 = 0."

Teacher Planning & NCEA Alignment

NCEA Level 3 Calculus Alignment (6 Credits External):

Differentiation Methods: Apply second derivatives $f^{″} (x)$ to determine concavity, locate points of inflection, and sketch curves.

Vocabulary: Second derivative ( $f^{″} (x)$ ), concavity up, concavity down, point of inflection, stationary vs non-stationary inflection, curve sketching.

Lesson 7: Concavity, Second Derivatives & Curve Sketching