Mathematics AA: Higher Level
Calculus — Topic 5
- 1.
Differentiate y = x²·e^(3x) using the product rule.
[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.
[4 marks] · no calculator - 3.
Use integration by substitution with u = x² + 1 to find ∫2x(x² + 1)⁵ dx.
[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.
[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.
[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.
[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.
[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.
[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)
[4 marks] - 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.
[4 marks] - 11.
Solve the differential equation dy/dx = xy, given that y = 2 when x = 0.
[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.
[5 marks] · no calculator - 13.
Use integration by parts to find ∫x cos x dx.
[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.
[5 marks] · no calculator - 15.
Find the Maclaurin series for f(x) = eˣ up to and including the term in x³.
[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.
[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.
[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.
[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.
[5 marks] - 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.
[5 marks] - 21.
Calculate the area enclosed between the curve y = 4 − x² and the x-axis.
[5 marks] - 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.
[5 marks] - 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.
[5 marks] - 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.
[5 marks] - 25.
Differentiate y = arctan(3x), using the standard result d/dx[arctan(u)] = u′/(1 + u²).
[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.
[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).
[3 marks] - 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.
[3 marks] - 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.
[5 marks] - 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.
[5 marks]