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IB Maths: Calculus and Derivatives – Practice Paper 2 (10 Questions)
IB Maths: Calculus and Derivatives – Practice Paper 2 (10 Questions)
Paper 2 concentrates on differentiation technique: the product rule, the quotient rule, the chain rule and root notation, then tangents and normals, and the second derivative test for stationary points and inflections.
Quick answer Ten items, about twenty-five minutes. Items 01 to 04 cover polynomials, roots, the product rule and a tangent slope; items 05 to 07 cover the quotient rule, the chain rule and an exponential; items 08 and 09 are stationary points with an inflection and a normal; item 10 redoes the omitted inner derivative.
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 6x to the fifth minus 2x cubed, find the derivative and its value at 1.
- FoundationGiven y equals the square root of x for positive x, find the derivative and its value at 4.
- FoundationFor f(x) = (x squared plus 1)(x minus 3), use the product rule to find the derivative and its value at 2.
- FoundationFor f(x) = 2x cubed, give the tangent slope at x = minus 1.
- IntermediateFor y = (3x plus 2) over (x minus 1), use the quotient rule to find the derivative.
- IntermediateFor y = (2x minus 5) to the fourth, find the derivative and its value at 3.
- IntermediateFor y = e to the power 3x, find the derivative and its value at zero.
- ChallengeFor y = x cubed minus 6x squared plus 9x, find the stationary points and the point of inflection.
- ChallengeFind the equation of the normal to y = x squared minus 4x plus 1 at x = 3.
- Error redoA student writes the derivative of (x squared plus 1) cubed as three times the square. Give the correct derivative and its value at 1.
03 Answers and one-line notes
| # | Answer | One-line note |
|---|---|---|
| 1 | thirty x to the fourth minus six x squared, which is 24 at 1 | Differentiate term by term and substitute one to obtain twenty four. |
| 2 | one over two root x, which is one quarter at 4 | Writing the root as a half power gives one over two root x, and at four that is one quarter. |
| 3 | three x squared minus six x plus one, which is 1 at 2 | Two x times x minus three plus x squared plus one, the same as expanding first and then differentiating. |
| 4 | the slope is 6 | The derivative is six x squared, and squaring removes the negative sign. |
| 5 | minus five over (x minus one) squared | The numerator of the quotient rule collapses to minus five, so the function decreases on both branches. |
| 6 | eight times (2x minus five) cubed, which is 8 at x = 3 | Four times the cube times two, and the bracket equals one at three. |
| 7 | three e to the 3x, which is 3 at zero | Differentiating an exponential multiplies by the coefficient three, and at zero the value is three. |
| 8 | a maximum of four at one, a minimum of zero at three, and an inflection at two comma two | The derivative factors into three times x minus one times x minus three, and the second derivative changes sign at two. |
| 9 | the normal is y equals minus x over two minus one half | The tangent gradient is two, so the normal gradient is minus one half, through the point three comma minus two. |
| 10 | six x times the square, which is 24 at x = 1 | The inner derivative two x was omitted, and including it gives twenty four at one. |
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.
