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.Paper 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.Marking analysisMedium4 marksNo calculator+
• 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.Paper 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.Marking analysisMedium5 marksNo calculator+
• 5A city's population of a rare bird species, P(t) hundreds, t years after monitoring begins, is modelled by the logistic differential equation dP/dt = 0.3P(1 − P/8), with P(0) = 1. (a) Show that P = 8 is an equilibrium solution and interpret it in context. (b) By separating variables, show that the general solution can be written P(t) = 8 / (1 + 7e^(−0.3t)). (c) Find the population predicted after 10 years. (d) Find the time at which the population is growing fastest, and state the population at that time. (e) Comment on one limitation of using this model to predict the population after 50 years.Paper 3 style (extended)Hard8 marksCalculator+
• 6Marking analysis: A learner attempts the following task: “A city's population of a rare bird species, P(t) hundreds, t years after monitoring begins, is modelled by the logistic differential equation dP/dt = 0.3P(1 − P/8), with P(0) = 1. (a) Show that P = 8 is an equilibrium solution and interpret it in context. (b) By separating variables, show that the general solution can be written P(t) = 8 / (1 + 7e^(−0.3t)). (c) Find the population predicted after 10 years. (d) Find the time at which the population is growing fastest, and state the population at that time. (e) Comment on one limitation of using this model to predict the population after 50 years.” Their response addresses only this point: “Sets dP/dt = 0 and solves 0.3P(1 − P/8) = 0 to obtain P = 0 or P = 8, identifying P = 8 as the non-trivial equilibrium.” Evaluate the response against the complete 8-mark task. Identify what earns credit and state every additional requirement needed for full marks.Marking analysisHard8 marksNo calculator+