Finding the equations of tangents and normals to curves, and calculating second derivatives.
5 narrated scenes. Press play, listen and read along — the points appear as they are explained. Use ‹ Prev / Next › or the scene bar to jump to any scene.
A tangent just touches a curve; a normal meets it at right angles. Both come from one thing — the derivative at the point.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Move the point and show the tangent and the normal.
Copy these into your notebook in your own words — that is how they stick.
y − y₁ = f′(x₁)(x − x₁).
y − y₁ = −1/f′(x₁) (x − x₁).
y-value → dy/dx → gradient at x₁ → equation.
Horizontal tangent ↔ vertical normal; gradient 0.
f″(x) = d²y/dx² = d/dx(dy/dx).
Parallel to y = mx + c: f′ = m. Perpendicular: f′ = −1/m.
Substitute the point into the curve and the tangent.
y″, f″(x), d²y/dx² all mean the second derivative.
3 quick games on this lesson. Earn XP and badges as you play — your best score for each game is saved. Games are for practice and do not change your exam readiness. More games in the Additional Mathematics Arcade.
8 items modelled on real CSEC Paper 1 questions. Choose one answer for each, then check. Aim for 80% or better. Work each item on paper first.
Write full answers, then reveal the mark scheme and tick the points you earned (23 marks in total). Show all your working — method marks count. Your answers are saved on this device.
Question 1 [11 marks]
The curve C has equation y = x³ − 3x + 2.
(a) Find dy/dx and the gradient of C at the point P where x = 2. [3]
(b) Find the equation of the tangent to C at P. [2]
(c) Find the equation of the normal to C at P. [3]
(d) Find the coordinates of the points on C where the tangent is horizontal. [3]
Question 2 [12 marks]
The curve D has equation y = x² − 4x + 7.
(a) Find the coordinates of the point on D where the gradient is 2. [3]
(b) Find the equations of the tangent and normal at this point. [4]
(c) Find d²y/dx² and hence state whether the curve is concave up or down. [2]
(d) The tangent meets the axes at A and B. Find the area of triangle OAB. [3]
Find these in your own past-paper booklets for extra timed practice. Items above are written in the style of these papers; any adapted from a specific question say so.