Tangents
- The tangent to a curve at a point has the same gradient as the curve there.
- Method: find the y-coordinate, find dy/dx at the point, then use y − y₁ = m(x − x₁).
- Example: y = x² − 3x + 4 at x = 2. y = 4 − 6 + 4 = 2. dy/dx = 2x − 3 = 1. The tangent is y − 2 = 1(x − 2), so y = x.
- A tangent with gradient 0 is horizontal.
Normals
- The normal is perpendicular to the tangent at the same point, so its gradient is −1 ÷ (tangent gradient).
- For y = x² − 3x + 4 at (2, 2), the normal has gradient −1, so y − 2 = −(x − 2), giving y = −x + 4.
- For y = x³ at (1, 1): the tangent gradient is 3, so the tangent is y = 3x − 2. The normal has gradient −1/3: y − 1 = −(1/3)(x − 1), giving x + 3y = 4.
Common mistakes
- Use the gradient at the point, not the gradient function itself.
- Substitute into the original equation to find y, not into dy/dx.
- Where lines cross an axis, set x = 0 or y = 0.
Key terms
- Tangent to a curve
- A straight line touching the curve with the same gradient at that point.
- Normal
- A line perpendicular to the tangent at the same point.
- Point of contact
- The point where a tangent touches a curve.
- Perpendicular gradient
- The negative reciprocal of a gradient.