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IB Maths: Calculus and Derivatives – Practice Paper 3 (10 Questions)
IB Maths: Calculus and Derivatives – Practice Paper 3 (10 Questions)
Paper 3 applies the derivative: velocity and acceleration, maxima and minima on a closed interval, minimising the surface of a cylinder, an optimal fence and a parallel tangent condition.
Quick answer Ten items, about twenty-five minutes. Items 01 to 04 cover stationary values, a tangent value and velocity and acceleration from displacement; items 05 to 07 cover a closed interval, a cylinder optimisation and parallel tangents; items 08 and 09 are a fence and a parallel condition; item 10 redoes the first derivative only decision.
01 Paper details (check answers afterwards)
| Item | Detail |
|---|---|
| Questions | 10 questions |
| Suggested time | 25 minutes |
| Difficulty mix | Foundation 4 · Intermediate 3 · Challenge 2 · Error redo 1 |
| Chapter | Chapter: IB Maths: Calculus and Derivatives – Tutor Lesson Plan |
| Level | IB SL |
02 Questions (10 items)
- FoundationGiven f(x) = x to the fourth minus 32x, solve f dash equals zero and give the value there.
- FoundationFor y = 5 minus 2x plus x squared, give the derivative and the value at x = 3.
- FoundationA displacement s(t) = t cubed minus 3t squared plus 2t in metres. Find the velocity at t = 2 seconds.
- FoundationA displacement s(t) = t cubed minus 6t squared plus 9t in metres. Find the acceleration at t = 2 seconds.
- IntermediateFind the maximum and minimum of f(x) = x cubed minus 3x squared plus 1 on the interval from 0 to 3.
- IntermediateA closed cylinder has a fixed volume of 250 pi cubic centimetres. Find the radius and height that minimise its surface area.
- IntermediateFind every point on y = x cubed minus 3x where the tangent has slope nine.
- ChallengeA rectangular field uses a wall on one side and 40 metres of fence on the other three. Find the largest area and the dimensions.
- ChallengeFind the point on y = x squared whose tangent is parallel to the line y = 4x minus 7.
- Error redoA student solves only the derivative for x cubed minus 3x and calls both stationary points minima. Give the correct answer.
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | x is 2, where the function equals minus 48 | The derivative gives x cubed equals eight, so x is two and the value is sixteen minus sixty four. |
| 2 | the derivative is 4 and the value is 8 | The derivative is minus two plus two x, giving four, and the function value is eight. |
| 3 | the velocity is 2 metres per second | Velocity is the derivative of displacement, and substituting two gives two metres per second. |
| 4 | the acceleration is zero | Acceleration is six t minus twelve, which vanishes at two, exactly where velocity is least. |
| 5 | the maximum is 1 and the minimum is minus 3 | Compare the stationary points at zero and two with the endpoint three, giving one, minus three and one. |
| 6 | the radius is 5 centimetres and the height is 10 centimetres | The height is 250 over r squared, and setting the surface derivative to zero gives a radius of five and a height of ten. |
| 7 | the points two comma two and minus two comma minus two | The derivative equals nine when x squared is four, and substituting gives two and minus two. |
| 8 | the largest area is 200 square metres, with sides of 20 and 10 metres | Fence length gives x plus two y equals forty, and the area derivative vanishes at twenty by ten. |
| 9 | the point two comma four | Parallel tangents share a gradient, so two x equals four and the point is two comma four. |
| 10 | a maximum of two at minus one and a minimum of minus two at one | The second derivative six x is negative at minus one and positive at one, so the two points differ. |
04 What to do after marking
- After marking, redo only the wrong items and write down which step went wrong (formula, unit, or misread number).
- Repeat the same item two days later; move to the next paper only when both attempts are correct.
- Parents only need the answer and note columns; there is no need to explain every item.
Questions and answers are written originally by the TutorZone team and are not taken from any school, publisher or examination authority paper. Every value is checked by computer.
