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IB Maths: Calculus and Derivatives – Practice Paper 1 (10 Questions)
IB Maths: Calculus and Derivatives – Practice Paper 1 (10 Questions)
Paper 1 starts from polynomial differentiation: the power rule, constant terms, negative indices and roots, then tangent slopes, the chain rule, first principles and the second derivative test for stationary points.
Quick answer Ten items, about twenty-five minutes. Items 01 to 04 cover the power rule and negative indices; items 05 to 07 cover stationary equations, tangent slopes and the chain rule; items 08 and 09 are first principles and the second derivative test; item 10 redoes the forgotten power reduction.
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 cubed, find f dash of x.
- FoundationGiven f(x) = 4x squared minus 3x plus 7, find f dash of x and its value at 2.
- FoundationGiven f(x) = 5x to the fourth, find f dash at 2.
- FoundationGiven y = 1 over x for x not equal to zero, find the derivative.
- IntermediateGiven f(x) = x cubed minus 12x, solve f dash of x equals zero.
- IntermediateFor y = 3x squared plus 2x, give the slope and equation of the tangent at x = 1.
- IntermediateGiven f(x) = (2x + 1) cubed, find f dash of x and its value at 1.
- ChallengeUsing first principles, show that the derivative of x squared is 2x, then find its value at 3.
- ChallengeFor y = x cubed minus 3x squared plus 4, find both stationary points and classify them.
- Error redoA student writes the derivative of x cubed as 3x. Give the correct derivative and its value at 2.
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | f dash of x is three x squared | Bring the power down as a multiplier and reduce the power by one. |
| 2 | f dash is 8x minus 3, which is 13 at x = 2 | The constant term differentiates to zero, so seven disappears and the value at two is thirteen. |
| 3 | f dash is 20 x cubed, which gives 160 at x = 2 | Differentiate to twenty x cubed, then substitute two to obtain one hundred sixty. |
| 4 | the derivative is minus one over x squared | Write it as x to the minus one, then the power rule gives minus x to the minus two. |
| 5 | x is 2 or minus 2 | The derivative is three x squared minus twelve, so x squared is four and x is plus or minus two. |
| 6 | slope 8, tangent y = 8x minus 3 | The derivative gives eight, the point is one comma five, and the point slope form gives the line. |
| 7 | six times (2x plus 1) squared, which is 54 at x = 1 | The chain rule gives three times the square times two, and at one that is fifty four. |
| 8 | the derivative is 2x, and at 3 it is 6 | The difference quotient simplifies to two x plus h, and letting h tend to zero leaves two x. |
| 9 | a local maximum of four at zero and a local minimum of zero at two | The derivative factors as three x times x minus two, and the second derivative is negative at zero and positive at two. |
| 10 | the derivative is 3x squared, which is 12 at x = 2 | The mistake is forgetting to reduce the power, so the correct value at two is twelve, not six. |
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.
