Mathematics: Applications & Interpretation HL
Advanced calculus and modelling — Topics 5 and 1
- 1.
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.
[4 marks] - 2.
Marking 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.
[4 marks] - 3.
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.
[5 marks] - 4.
Marking 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.
[5 marks] - 5.
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.
[8 marks] - 6.
Marking 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.
[8 marks] - 7.
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.
[4 marks] - 8.
Marking 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.
[4 marks] - 9.
The curve x²y + y³ = 10 passes through the point (1, 2). Use implicit differentiation to find dy/dx at this point.
[4 marks] - 10.
Marking 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.
[4 marks] - 11.
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.
[4 marks] - 12.
Marking 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.
[4 marks] - 13.
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.
[5 marks] - 14.
Marking 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.
[5 marks] - 15.
Solve the differential equation dy/dx = 2xy, given that y = 3 when x = 0, by separating variables. Hence find y when x = 1.
[5 marks] - 16.
Marking 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.
[5 marks] - 17.
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.
[4 marks] - 18.
Marking 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.
[4 marks] - 19.
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.
[6 marks] - 20.
Marking 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.
[6 marks] - 21.
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.
[4 marks] - 22.
Marking 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.
[4 marks] - 23.
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.
[4 marks] - 24.
Marking 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.
[4 marks] - 25.
For the curve f(x) = x⁴ − 4x³, find all values of x where the curve has a point of inflection, using the second derivative.
[5 marks] - 26.
Marking 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.
[5 marks] - 27.
Use the trapezoidal rule with 4 strips to estimate ∫₀⁴ x² dx, and compare your estimate with the exact value found by integration.
[5 marks] - 28.
Marking 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.
[5 marks] - 29.
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.
[6 marks] - 30.
Marking 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.
[6 marks] - 31.
A slope field for a differential equation shows that wherever a solution curve crosses the line y = x, its gradient is zero, and that the gradient is positive below this line and negative above it. (a) Suggest a differential equation dy/dx = f(x, y) consistent with this description. (b) Describe the long-term behaviour of a solution curve that starts below the line y = x.
[2 marks] · no calculator - 32.
Marking analysis: A learner attempts the following task: “A slope field for a differential equation shows that wherever a solution curve crosses the line y = x, its gradient is zero, and that the gradient is positive below this line and negative above it. (a) Suggest a differential equation dy/dx = f(x, y) consistent with this description. (b) Describe the long-term behaviour of a solution curve that starts below the line y = x.” Their response addresses only this point: “Suggests a valid equation such as dy/dx = x − y, which is zero when y = x, positive when y < x, and negative when y > x.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator