IB Math AI HL — Tangents and normals

Key Idea: The derivative f'(x) gives the gradient of the tangent line to a curve at any point. Topic 5.4 uses derivatives to write the equation of the tangent (touching the curve) and the equation of the normal (perpendicular to the tangent) at a specific point. These are standard IB exam question types.

✅ Tangent and normal formulas

Example: f(x) = x³ − 3x + 2. Find the tangent and normal at x = 2. f(2) = 8 − 6 + 2 = 4. Point: (2, 4). f'(x) = 3x² − 3. f'(2) = 12 − 3 = 9. Tangent: y − 4 = 9(x − 2) → y = 9x − 14 Normal gradient = −1/9. Normal: y − 4 = −(1/9)(x − 2) → y = −x/9 + 2/9 + 4 → y = −(1/9)x + (38/9)

The point (a, f(a)) lies on both the tangent and the normal. Always calculate f(a) — it gives the y-coordinate of the point of contact. The tangent touches the curve at one point; it does not cross it (at that point). If the question asks where else the tangent meets the curve, set the tangent equation equal to f(x) and solve.

Paper 1 (GDC allowed): Show all differentiation steps. Write f'(x) first, then substitute x = a. Then write the point-slope equation. Paper 2 (GDC allowed): Use GDC to evaluate f(a) and f'(a) if the function is complex. But still write the equation construction steps in your working.