Calculus: Kinematics and Practical Problems — Practice Questions
12 exam-style questions (39 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.
Section A — Warm-up (grades 6–7)
- (2 marks)s = 2t³ − 5t + 1. Find the velocity when t = 2.
- (2 marks)The velocity of a particle is v = 4t² − 3t m/s. Find its acceleration when t = 3.
- (3 marks)s = t³ − 6t² + 9t. Find the times when the particle is at rest.
- (3 marks)The height of a ball after t seconds is h = 20t − 5t² metres. Find the maximum height.
Section B — Core (grades 7–8)
- (3 marks)s = t³ − 3t² + 4, for t > 0. Find the acceleration at the moment when the particle is instantaneously at rest.
- (3 marks)v = 3t² − 18t + 24. Find the minimum velocity.
- (3 marks)A rectangle has perimeter 36 cm and width x cm.
(a) Show that its area is A = 18x − x².
(b) Use calculus to find the maximum area. - (4 marks)A farmer uses 40 m of fencing to make a rectangular pen against a wall. The two sides perpendicular to the wall are each x m. Find the maximum area of the pen.
Section C — Stretch (grades 8–9)
- (4 marks)The cost of a journey is C = 2x + 50x dollars, where x > 0. Find the value of x that gives the minimum cost, and the minimum cost.
- (4 marks)An open box is made from a square sheet of card 24 cm by 24 cm by cutting squares of side x cm from each corner.
(a) Show that V = x(24 − 2x)².
(b) Find the maximum volume. - (4 marks)A particle has displacement s = t³ − 9t² + 24t metres. Find the total distance it travels in the first 4 seconds.
Hint
Find when it changes direction (v = 0) first. - (4 marks)Find the coordinates of the stationary point of y = x³ − 12x + 5 where x > 0, and show that it is a minimum.
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