School / IB / MATH AA HL / Calculus Question practice
Calculus About this practice HL-only depth: product rule, quotient rule and integration by substitution.
Mathematics AA: Higher Level Calculus
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1 Differentiate y = x²·e^(3x) using the product rule. Differentiate y = x²·e(3x) using the product rule. Paper 1 style Medium 4 marks No calculator + 2 Marking analysis: A learner attempts the following task: “Differentiate y = x²·e^(3x) using the product rule.” Their response addresses only this point: “States the product rule dy/dx = u′v + uv′ with u = x², v = e^(3x).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis: A learner attempts the following task: “Differentiate y = x²·e(3x) using the product rule.” Their response addresses only this point: “States the product rule dy/dx = u′v + uv′ with u = x², v = e(3x) .” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 3 Use integration by substitution with u = x² + 1 to find ∫2x(x² + 1)⁵ dx. Paper 1 style Medium 5 marks No calculator + 4 Marking analysis: A learner attempts the following task: “Use integration by substitution with u = x² + 1 to find ∫2x(x² + 1)⁵ dx.” Their response addresses only this point: “States u = x² + 1 and finds du/dx = 2x, so du = 2x dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 5 Differentiate y = ln(x)/x using the quotient rule, and hence find the x-coordinate of the stationary point of the curve. Paper 1 style Hard 6 marks No calculator + 6 Marking analysis: A learner attempts the following task: “Differentiate y = ln(x)/x using the quotient rule, and hence find the x-coordinate of the stationary point of the curve.” Their response addresses only this point: “States the quotient rule dy/dx = (u′v − uv′)/v² with u = ln(x), v = x.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks No calculator + 7 The curve x² + y² = 25 passes through the point (3, 4). Use implicit differentiation to find dy/dx at this point. Paper 1 style Medium 4 marks No calculator + 8 Marking analysis: A learner attempts the following task: “The curve x² + y² = 25 passes through the point (3, 4). Use implicit differentiation to find dy/dx at this point.” Their response addresses only this point: “Differentiates both sides with respect to x, treating y as a function of x: 2x + 2y(dy/dx) = 0.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 9 A spherical balloon is inflated so that its volume increases at a constant rate of 100 cm³/s. Find the rate of increase of the radius when r = 5 cm. (V = (4/3)πr³) Paper 2 style Medium 4 marks Calculator + 10 Marking analysis: A learner attempts the following task: “A spherical balloon is inflated so that its volume increases at a constant rate of 100 cm³/s. Find the rate of increase of the radius when r = 5 cm. (V = (4/3)πr³)” Their response addresses only this point: “Differentiates V = (4/3)πr³ with respect to t using the chain rule: dV/dt = 4πr²(dr/dt).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 11 Solve the differential equation dy/dx = xy, given that y = 2 when x = 0. Paper 1 style Medium 5 marks No calculator + 12 Marking analysis: A learner attempts the following task: “Solve the differential equation dy/dx = xy, given that y = 2 when x = 0.” Their response addresses only this point: “Separates variables: (1/y) dy = x dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 13 Use integration by parts to find ∫x cos x dx. Paper 1 style Medium 5 marks No calculator + 14 Marking analysis: A learner attempts the following task: “Use integration by parts to find ∫x cos x dx.” Their response addresses only this point: “States the integration by parts formula ∫u dv = uv − ∫v du.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 15 Find the Maclaurin series for f(x) = eˣ up to and including the term in x³. Paper 1 style Medium 4 marks No calculator + 16 Marking analysis: A learner attempts the following task: “Find the Maclaurin series for f(x) = eˣ up to and including the term in x³.” Their response addresses only this point: “States the Maclaurin series formula f(x) = f(0) + f′(0)x + f″(0)x²/2! + f‴(0)x³/3! + …” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 17 Evaluate lim(x→0) (sin x)/x using L'Hôpital's rule, verifying that the limit is initially of the indeterminate form 0/0. Paper 1 style Medium 4 marks No calculator + 18 Marking analysis: A learner attempts the following task: “Evaluate lim(x→0) (sin x)/x using L'Hôpital's rule, verifying that the limit is initially of the indeterminate form 0/0.” Their response addresses only this point: “States that direct substitution of x = 0 gives sin(0)/0 = 0/0, an indeterminate form.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 19 The curve y = x³ − 6x² + 9x + 1 has two stationary points. Find their x-coordinates, and use the second derivative test to determine the nature of each. Paper 2 style Medium 5 marks Calculator + 20 Marking analysis: A learner attempts the following task: “The curve y = x³ − 6x² + 9x + 1 has two stationary points. Find their x-coordinates, and use the second derivative test to determine the nature of each.” Their response addresses only this point: “Finds dy/dx = 3x² − 12x + 9.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 21 Calculate the area enclosed between the curve y = 4 − x² and the x-axis. Paper 2 style Medium 5 marks Calculator + 22 Marking analysis: A learner attempts the following task: “Calculate the area enclosed between the curve y = 4 − x² and the x-axis.” Their response addresses only this point: “Finds the roots of 4 − x² = 0: x = −2 and x = 2, giving the limits of integration.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 23 The region bounded by y = x², the x-axis, and the line x = 2 is rotated 360° about the x-axis. Calculate the volume of the solid formed, using V = π∫y² dx. Paper 2 style Medium 5 marks Calculator + 24 Marking analysis: A learner attempts the following task: “The region bounded by y = x², the x-axis, and the line x = 2 is rotated 360° about the x-axis. Calculate the volume of the solid formed, using V = π∫y² dx.” Their response addresses only this point: “States V = π∫y² dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 25 Differentiate y = arctan(3x), using the standard result d/dx[arctan(u)] = u′/(1 + u²). Paper 1 style Easy 3 marks No calculator + 26 Marking analysis: A learner attempts the following task: “Differentiate y = arctan(3x), using the standard result d/dx[arctan(u)] = u′/(1 + u²).” Their response addresses only this point: “States the formula d/dx[arctan(u)] = u′/(1 + u²), with u = 3x.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks No calculator + 27 Given g(x) = ∫₁ˣ (t² + 1) dt, use the fundamental theorem of calculus to find g′(x), and hence evaluate g′(2). Paper 2 style Easy 3 marks Calculator + 28 Marking analysis: A learner attempts the following task: “Given g(x) = ∫₁ˣ (t² + 1) dt, use the fundamental theorem of calculus to find g′(x), and hence evaluate g′(2).” Their response addresses only this point: “States the fundamental theorem of calculus: if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks Calculator + 29 Water enters a recycling tank at 12 L min⁻¹. At time t minutes the tank contains V litres, and water leaves at a rate of 0.08V L min⁻¹. Initially V = 50. (a) Write a differential equation for V. (b) Find the equilibrium volume. (c) Solve for V in terms of t. (d) Determine the first time when V reaches 120 L. Paper 2 style Medium 5 marks Calculator + 30 Marking analysis: A learner attempts the following task: “Water enters a recycling tank at 12 L min⁻¹. At time t minutes the tank contains V litres, and water leaves at a rate of 0.08V L min⁻¹. Initially V = 50. (a) Write a differential equation for V. (b) Find the equilibrium volume. (c) Solve for V in terms of t. (d) Determine the first time when V reaches 120 L.” Their response addresses only this point: “Writes dV/dt = 12 − 0.08V.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + Self-assessed Not marked yet
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