中文（繁體）
中文（简体）
English
IB Maths: Calculus and Derivatives – Practice Paper 4 (10 Questions)
IB Maths: Calculus and Derivatives – Practice Paper 4 (10 Questions)
Paper 4 is a mixed review of the chapter: the power rule, constants and polynomials, stationary points, the second derivative, tangent gradients, extrema and a fence optimisation, plus velocity and acceleration from displacement.
Quick answer Ten items, about twenty-five minutes. Items 01 to 04 cover basic differentiation and a constant; items 05 to 07 cover stationary points, a second derivative and a tangent gradient; items 08 and 09 are extrema and a fence; item 10 redoes giving only the velocity.
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 of x equals x to the fifth, find f dash of x.
- FoundationGiven f of x equals seven x cubed minus two x, find f dash of x.
- FoundationGiven f of x equals x squared plus three x plus one, find f dash at one.
- FoundationGiven f of x equals six, find f dash of x.
- IntermediateFind the stationary points of f of x equals x cubed minus twenty-seven x.
- IntermediateWith f of x equals x to the fourth minus eight x squared, find the second derivative.
- IntermediateWhat is the gradient of the tangent to y equals x squared plus two x at x equals one?
- ChallengeFor f of x equals x cubed minus six x squared plus nine x, find the maximum and the minimum.
- ChallengeSixty metres of fence enclose a rectangle against a wall using three sides. Find the largest area.
- Error redoA particle has s of t equals t cubed minus three t in metres. A student gives only the velocity at two. What are the velocity and the acceleration?
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | f dash of x is five x to the fourth | Bring five down and reduce the index to four. |
| 2 | f dash of x is twenty-one x squared minus two | Differentiate term by term: seven x cubed gives twenty-one x squared, and minus two x gives minus two. |
| 3 | f dash at one is five | The derivative is two x plus three, which is five at one. |
| 4 | the derivative is zero | A constant graph is horizontal, so its gradient is zero. |
| 5 | x equals three and x equals minus three | Setting three x squared minus twenty-seven to zero gives x squared equal to nine, so x is plus or minus three. |
| 6 | the second derivative is twelve x squared minus sixteen | The first derivative is four x cubed minus sixteen x, so the second is twelve x squared minus sixteen. |
| 7 | the gradient is four | The derivative is two x plus two, which is four at one. |
| 8 | a maximum of four at one and a minimum of zero at three | The derivative factors as three times x minus one times x minus three, giving four and zero. |
| 9 | the largest area is four hundred and fifty square metres, fifteen by thirty | With x the side perpendicular to the wall, the area derivative vanishes at fifteen. |
| 10 | the velocity is nine metres per second and the acceleration twelve | Velocity is the first derivative and acceleration the second, so both must be given. |
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.
