Advanced calculus and modelling

Paper labels are an unofficial, independently authored grouping, applied only where a question's own format genuinely matches a real paper convention (such as IB Mathematics AA's non-calculator/calculator split). They do not reproduce any exam board's real paper numbering or mark allocation, and uncertain questions are labelled general practice instead.

• 1A tank's volume is modelled by V(t) = 40 + 6t − 0.5t², where V is in litres and t is time in minutes, for 0 ≤ t ≤ 12. (a) Find dV/dt. (b) Find the time at which the volume is a maximum. (c) State the maximum volume.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 2Marking analysis: A learner attempts the following task: “A tank's volume is modelled by V(t) = 40 + 6t − 0.5t², where V is in litres and t is time in minutes, for 0 ≤ t ≤ 12. (a) Find dV/dt. (b) Find the time at which the volume is a maximum. (c) State the maximum volume.” Their response addresses only this point: “Differentiates to obtain dV/dt = 6 − t.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 3The rate of change of the temperature T (°C) of a cooling drink is modelled by dT/dt = −0.12(T − 4), where t is in minutes and initially T = 20. (a) Use Euler's method with a step length of 1 minute to estimate T after 2 minutes, showing both steps. (b) State one way the estimate could be made more accurate without changing the model.✓ ReviewedPaper 2 styleMedium5 marksCalculator+
• 4Marking analysis: A learner attempts the following task: “The rate of change of the temperature T (°C) of a cooling drink is modelled by dT/dt = −0.12(T − 4), where t is in minutes and initially T = 20. (a) Use Euler's method with a step length of 1 minute to estimate T after 2 minutes, showing both steps. (b) State one way the estimate could be made more accurate without changing the model.” Their response addresses only this point: “States Euler's update rule Tₙ₊₁ = Tₙ + h·f(Tₙ).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium5 marksCalculator+
• 7The radius of a circular oil spill is increasing at a rate of 3 cm/s. Find the rate at which the area of the spill is increasing when the radius is 10 cm.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 8Marking analysis: A learner attempts the following task: “The radius of a circular oil spill is increasing at a rate of 3 cm/s. Find the rate at which the area of the spill is increasing when the radius is 10 cm.” Their response addresses only this point: “States A = πr² and differentiates with respect to time using the chain rule: dA/dt = 2πr·(dr/dt).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 9The curve x²y + y³ = 10 passes through the point (1, 2). Use implicit differentiation to find dy/dx at this point.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 10Marking analysis: A learner attempts the following task: “The curve x²y + y³ = 10 passes through the point (1, 2). Use implicit differentiation to find dy/dx at this point.” Their response addresses only this point: “Differentiates both sides with respect to x, using the product rule on x²y: 2xy + x²(dy/dx) + 3y²(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.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 11For the curve f(x) = x³ − 3x² − 9x + 5, find the x-coordinate of the point of inflection using the second derivative, and state the corresponding y-coordinate.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 12Marking analysis: A learner attempts the following task: “For the curve f(x) = x³ − 3x² − 9x + 5, find the x-coordinate of the point of inflection using the second derivative, and state the corresponding y-coordinate.” Their response addresses only this point: “Differentiates twice to obtain f''(x) = 6x − 6.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 13Given the differential equation dy/dx = x + y, with y(0) = 1, use Euler's method with a step length of 0.5 to estimate y(1), showing both steps.✓ ReviewedPaper 2 styleMedium5 marksCalculator+
• 14Marking analysis: A learner attempts the following task: “Given the differential equation dy/dx = x + y, with y(0) = 1, use Euler's method with a step length of 0.5 to estimate y(1), showing both steps.” Their response addresses only this point: “States Euler's update rule yₙ₊₁ = yₙ + h·f(xₙ, yₙ).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium5 marksCalculator+
• 15Solve the differential equation dy/dx = 2xy, given that y = 3 when x = 0, by separating variables. Hence find y when x = 1.✓ ReviewedPaper 2 styleMedium5 marksCalculator+
• 16Marking analysis: A learner attempts the following task: “Solve the differential equation dy/dx = 2xy, given that y = 3 when x = 0, by separating variables. Hence find y when x = 1.” Their response addresses only this point: “Separates variables: ∫(1/y) dy = ∫2x dx.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium5 marksCalculator+
• 17A particle's position vector is given by r(t) = (t², 3t) metres, where t is time in seconds. (a) Find the velocity vector v(t). (b) Find the speed of the particle at t = 2 seconds.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 18Marking analysis: A learner attempts the following task: “A particle's position vector is given by r(t) = (t², 3t) metres, where t is time in seconds. (a) Find the velocity vector v(t). (b) Find the speed of the particle at t = 2 seconds.” Their response addresses only this point: “Differentiates each component of r(t) to obtain v(t) = (2t, 3).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 21A spherical balloon is being inflated so that its volume increases at a constant rate of 100 cm³/s. Find the rate at which the radius is increasing when the radius is 5 cm.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 22Marking analysis: A learner attempts the following task: “A spherical balloon is being inflated so that its volume increases at a constant rate of 100 cm³/s. Find the rate at which the radius is increasing when the radius is 5 cm.” Their response addresses only this point: “States V = (4/3)πr³ and differentiates with respect to time: 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.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 23A population of bacteria grows according to dP/dt = 0.05P, with P(0) = 200. Find the general solution for P(t), and use it to find the population after 20 hours.✓ ReviewedPaper 1 styleMedium4 marksCalculator+
• 24Marking analysis: A learner attempts the following task: “A population of bacteria grows according to dP/dt = 0.05P, with P(0) = 200. Find the general solution for P(t), and use it to find the population after 20 hours.” Their response addresses only this point: “Separates variables and integrates to obtain ln(P) = 0.05t + c.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium4 marksCalculator+
• 25For the curve f(x) = x⁴ − 4x³, find all values of x where the curve has a point of inflection, using the second derivative.✓ ReviewedPaper 2 styleMedium5 marksCalculator+
• 26Marking analysis: A learner attempts the following task: “For the curve f(x) = x⁴ − 4x³, find all values of x where the curve has a point of inflection, using the second derivative.” Their response addresses only this point: “Differentiates twice to obtain f''(x) = 12x² − 24x.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium5 marksCalculator+
• 27Use the trapezoidal rule with 4 strips to estimate ∫₀⁴ x² dx, and compare your estimate with the exact value found by integration.✓ ReviewedPaper 2 styleMedium5 marksCalculator+
• 28Marking analysis: A learner attempts the following task: “Use the trapezoidal rule with 4 strips to estimate ∫₀⁴ x² dx, and compare your estimate with the exact value found by integration.” Their response addresses only this point: “Uses strip width h = 1 and evaluates f(x) = x² at x = 0, 1, 2, 3, 4 to obtain 0, 1, 4, 9, 16.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisMedium5 marksCalculator+