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.

• 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.✓ ReviewedPaper 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.✓ ReviewedMarking analysisHard8 marksCalculator+
• 19A cylindrical can must have a volume of 500 cm³. (a) Express the surface area S of the can in terms of its radius r only. (b) Find the radius and height that minimise the surface area, and find the minimum surface area.✓ ReviewedPaper 2 styleHard6 marksCalculator+
• 20Marking analysis: A learner attempts the following task: “A cylindrical can must have a volume of 500 cm³. (a) Express the surface area S of the can in terms of its radius r only. (b) Find the radius and height that minimise the surface area, and find the minimum surface area.” Their response addresses only this point: “Uses V = πr²h = 500 to express h = 500/(πr²).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisHard6 marksCalculator+
• 29A particle moves along a line with acceleration a(t) = 6t − 4 m/s², where t is in seconds. Given that the initial velocity is v(0) = 2 m/s and the initial displacement is s(0) = 0, find (a) the velocity function v(t), (b) the displacement function s(t), and (c) the displacement at t = 3 seconds.✓ ReviewedPaper 2 styleHard6 marksCalculator+
• 30Marking analysis: A learner attempts the following task: “A particle moves along a line with acceleration a(t) = 6t − 4 m/s², where t is in seconds. Given that the initial velocity is v(0) = 2 m/s and the initial displacement is s(0) = 0, find (a) the velocity function v(t), (b) the displacement function s(t), and (c) the displacement at t = 3 seconds.” Their response addresses only this point: “Integrates acceleration to obtain v(t) = 3t² − 4t + c.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.✓ ReviewedMarking analysisHard6 marksCalculator+