Finding stationary points, deciding whether they are maxima or minima using sign changes or the second derivative, and solving maximum and minimum problems.
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.
What size box holds the most? What shape of tin uses the least metal? Wherever you need a best answer, calculus finds it where the gradient is zero.
Narration uses your device’s built-in voice. No sound? Switch to “Captions only” — the captions advance on their own.
Move along the curve, then build the open box of greatest volume.
Copy these into your notebook in your own words — that is how they stick.
dy/dx = 0; find x, then y from the curve.
f″ < 0 maximum; f″ > 0 minimum; f″ = 0 inconclusive.
+ → − maximum; − → + minimum.
Variables → quantity and constraint → one variable → derivative = 0 → nature → answer.
State the allowed values; check end-points if needed.
Give the value asked (volume, area, dimension) with units.
Minimum surface area for fixed V: h = 2r.
Substitute back; compare with a value either side.
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 (22 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 − 6x² + 9x + 1.
(a) Find dy/dx and the coordinates of the stationary points of C. [5]
(b) Determine the nature of each stationary point. [3]
(c) Sketch the curve for −1 ≤ x ≤ 4, showing the stationary points and the y-intercept. (Check: cubic shape; (0, 1) marked; maximum (1, 5); minimum (3, 1).) [3]
Question 2 [11 marks]
An open box is made from a square sheet of card of side 12 cm by cutting a square of side x cm from each corner and folding up the sides.
(a) Show that the volume of the box is V = 4x³ − 48x² + 144x. [2]
(b) Find the value of x that makes V a maximum and justify your answer. [5]
(c) Find the maximum volume. [2]
(d) State the range of values of x for which the box can be made. [2]
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.