36 exam-style questions on Power rule for polynomials (integer exponents) — topic 5.3 of Calculus in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
Find the derivative of f(x) = 3x⁵ − 7x² + 4.
Answer: 15x⁴ − 14x
Differentiate term by term: 3·5x⁴ = 15x⁴, −7·2x = −14x, and the constant 4 differentiates to 0.
For y = 2x³ − 5x, calculate dy/dx when x = −2.
Answer: 19
dy/dx = 6x² − 5. At x = −2: 6(4) − 5 = 24 − 5 = 19.
The gradient of the curve y = x⁴ − 4x is zero at which value of x?
Answer: x = 1
dy/dx = 4x³ − 4 = 0 gives x³ = 1, so x = 1.
Given f(x) = 6/x² = 6x⁻², determine f′(x).
Answer: −12x⁻³
Write 6x⁻²; the power rule gives 6·(−2)x⁻³ = −12x⁻³.
A curve has equation y = x³ − 6x² + 9x. State the number of stationary points (where dy/dx = 0).
Answer: 2
dy/dx = 3x² − 12x + 9 = 3(x² − 4x + 3) = 3(x − 1)(x − 3), which is zero at x = 1 and x = 3: two stationary points.
The second derivative of f(x) = x⁴ − 2x³ is:
Answer: 12x² − 12x
f'(x) = 4x³ − 6x², then f''(x) = 12x² − 12x.
That was 6 of 36 Power rule for polynomials (integer exponents) questions.
Practise all 36 freeFindAO2 · 2–6 marks
Obtain the answer and show the method; if the calculator does it, say what was entered.
StateAO1 · 1–2 marks
One word or one short sentence. No explanation, no working: the mark is for the fact itself.
ExplainAO3 · 1–3 marks
Give reasons or causes: say what happens AND why, linking each step with 'because' or 'so that'.
36 Power rule for polynomials (integer exponents) questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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