School / IB / MATH AA SL / Calculus Question practice
Calculus About this practice Differentiation, stationary points and definite integration.
Mathematics AA: Standard Level Calculus
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1 Let g(x) = x³ − 6x² + 9x. Find the coordinates of all stationary points and classify each as a local maximum or local minimum. Paper 1 style Hard 6 marks No calculator + 2 Marking analysis: A learner attempts the following task: “Let g(x) = x³ − 6x² + 9x. Find the coordinates of all stationary points and classify each as a local maximum or local minimum.” Their response addresses only this point: “Differentiates to obtain g′(x) = 3x² − 12x + 9.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Hard 6 marks No calculator + 3 Evaluate the definite integral from 0 to 2 of (3x² + 2) dx. Interpret your answer as the signed area between the curve and the x-axis on this interval. Paper 1 style Easy 3 marks No calculator + 4 Marking analysis: A learner attempts the following task: “Evaluate the definite integral from 0 to 2 of (3x² + 2) dx. Interpret your answer as the signed area between the curve and the x-axis on this interval.” Their response addresses only this point: “Finds an antiderivative x³ + 2x.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks No calculator + 5 Find the equation of the tangent to the curve y = x² − 3x + 5 at the point where x = 2. Paper 1 style Medium 5 marks No calculator + 6 Marking analysis: A learner attempts the following task: “Find the equation of the tangent to the curve y = x² − 3x + 5 at the point where x = 2.” Their response addresses only this point: “Differentiates to obtain dy/dx = 2x − 3.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 7 Differentiate y = (3x + 1)⁴ using the chain rule. Hence find the gradient of the curve at x = 0. Paper 1 style Medium 4 marks No calculator + 8 Marking analysis: A learner attempts the following task: “Differentiate y = (3x + 1)⁴ using the chain rule. Hence find the gradient of the curve at x = 0.” Their response addresses only this point: “Applies the chain rule: dy/dx = 4(3x + 1)³ × 3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 9 Find the equation of the normal to the curve y = x³ at the point (1, 1). Paper 1 style Medium 5 marks No calculator + 10 Marking analysis: A learner attempts the following task: “Find the equation of the normal to the curve y = x³ at the point (1, 1).” Their response addresses only this point: “Differentiates to obtain dy/dx = 3x².” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 11 A particle moves along a line so that its displacement is s(t) = t³ − 6t² + 9t metres, t ≥ 0 seconds. Find the particle's velocity function and determine when the particle is at rest. Paper 1 style Easy 3 marks No calculator + 12 Marking analysis: A learner attempts the following task: “A particle moves along a line so that its displacement is s(t) = t³ − 6t² + 9t metres, t ≥ 0 seconds. Find the particle's velocity function and determine when the particle is at rest.” Their response addresses only this point: “Differentiates displacement to obtain v(t) = 3t² − 12t + 9.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks No calculator + 13 Find ∫(4x³ − 6x + 2) dx. Paper 1 style Easy 3 marks No calculator + 14 Marking analysis: A learner attempts the following task: “Find ∫(4x³ − 6x + 2) dx.” Their response addresses only this point: “Integrates each term using the power rule.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Easy 3 marks No calculator + 15 For the curve y = 12/x, x ≠ 0, find the equation of the tangent at x = 3. Paper 2 style Medium 4 marks Calculator + 16 Marking analysis: A learner attempts the following task: “For the curve y = 12/x, x ≠ 0, find the equation of the tangent at x = 3.” Their response addresses only this point: “Differentiates to obtain dy/dx = −12/x².” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 17 Determine the interval(s) over which the function h(x) = x³ − 3x is increasing. Paper 1 style Medium 5 marks No calculator + 18 Marking analysis: A learner attempts the following task: “Determine the interval(s) over which the function h(x) = x³ − 3x is increasing.” Their response addresses only this point: “Differentiates to obtain h′(x) = 3x² − 3.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 19 Water flows into a tank at a rate given by R(t) = 6t − t² litres per minute, for 0 ≤ t ≤ 6. Find the total volume of water that flows into the tank during this time. Paper 2 style Medium 4 marks Calculator + 20 Marking analysis: A learner attempts the following task: “Water flows into a tank at a rate given by R(t) = 6t − t² litres per minute, for 0 ≤ t ≤ 6. Find the total volume of water that flows into the tank during this time.” Their response addresses only this point: “Recognises that total volume is the definite integral of the rate function from 0 to 6.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 21 A farmer has 40 m of fencing to enclose a rectangular field, using an existing wall as one side (so fencing is only needed for the other three sides). Find the dimensions that maximise the enclosed area, and calculate this maximum area. Paper 2 style Medium 5 marks Calculator + 22 Marking analysis: A learner attempts the following task: “A farmer has 40 m of fencing to enclose a rectangular field, using an existing wall as one side (so fencing is only needed for the other three sides). Find the dimensions that maximise the enclosed area, and calculate this maximum area.” Their response addresses only this point: “Sets up the constraint x + 2y = 40, where x is the side parallel to the wall and y are the two perpendicular sides, and expresses x = 40 − 2y.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 23 A particle's displacement is given by s(t) = t³ − 9t² + 24t, for t ≥ 0 (metres, seconds). Find the particle's acceleration function, and determine the value of t at which the particle's acceleration is zero. Paper 2 style Medium 4 marks Calculator + 24 Marking analysis: A learner attempts the following task: “A particle's displacement is given by s(t) = t³ − 9t² + 24t, for t ≥ 0 (metres, seconds). Find the particle's acceleration function, and determine the value of t at which the particle's acceleration is zero.” Their response addresses only this point: “Differentiates displacement twice: v(t) = 3t² − 18t + 24, then a(t) = 6t − 18.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 25 Find the area of the region enclosed between the curves y = x² and y = 2x. Paper 2 style Medium 5 marks Calculator + 26 Marking analysis: A learner attempts the following task: “Find the area of the region enclosed between the curves y = x² and y = 2x.” Their response addresses only this point: “Finds the points of intersection by solving x² = 2x: x = 0 and x = 2.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + 27 For the curve y = x³ − 3x² + 2, find the coordinates of the point of inflection, using the second derivative. Paper 2 style Medium 4 marks Calculator + 28 Marking analysis: A learner attempts the following task: “For the curve y = x³ − 3x² + 2, find the coordinates of the point of inflection, using the second derivative.” Their response addresses only this point: “Finds y′ = 3x² − 6x and y″ = 6x − 6.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks Calculator + 29 The area of a circular oil spill is increasing at a rate of 8 m² s⁻¹. Find the rate at which the radius is increasing at the instant when the radius is 5 m. (A = πr²) Paper 2 style Medium 4 marks Calculator + 30 Marking analysis: A learner attempts the following task: “The area of a circular oil spill is increasing at a rate of 8 m² s⁻¹. Find the rate at which the radius is increasing at the instant when the radius is 5 m. (A = πr²)” Their response addresses only this point: “Differentiates A = πr² with respect to t, 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. Marking analysis Medium 4 marks Calculator + 31 For f(x) = x^2 e^(-x), with real x, find the stationary points and classify each using the sign of the derivative. For f(x) = x2 e(-x) , with real x, find the stationary points and classify each using the sign of the derivative. Paper 1 style Medium 5 marks No calculator + 32 Marking analysis: A learner attempts the following task: “For f(x) = x^2 e^(-x), with real x, find the stationary points and classify each using the sign of the derivative.” Their response addresses only this point: “f'(x) = e^(-x)(2x - x^2) = e^(-x)x(2 - x).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis: A learner attempts the following task: “For f(x) = x2 e(-x) , with real x, find the stationary points and classify each using the sign of the derivative.” Their response addresses only this point: “f'(x) = e(-x) (2x - x2 ) = e^(-x)x(2 - x).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 33 Find the intersections of y = 2x and y = x^2, then calculate the exact area enclosed between the curves. Find the intersections of y = 2x and y = x2 , then calculate the exact area enclosed between the curves. Paper 1 style Medium 4 marks No calculator + 34 Marking analysis: A learner attempts the following task: “Find the intersections of y = 2x and y = x^2, then calculate the exact area enclosed between the curves.” Their response addresses only this point: “x^2 = 2x gives x = 0 and x = 2, at (0,0) and (2,4).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis: A learner attempts the following task: “Find the intersections of y = 2x and y = x2 , then calculate the exact area enclosed between the curves.” Their response addresses only this point: “x2 = 2x gives x = 0 and x = 2, at (0,0) and (2,4).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 4 marks No calculator + 35 Find the tangent to y = ln x at x = e. Prove that ln x <= x/e for every x > 0, stating when equality holds. Paper 1 style Hard 5 marks No calculator + 36 Marking analysis: A learner attempts the following task: “Find the tangent to y = ln x at x = e. Prove that ln x <= x/e for every x > 0, stating when equality holds.” Their response addresses only this point: “At x = e, y = 1 and dy/dx = 1/e.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 37 f(x) = x^3 - 3kx, where k > 0. The vertical separation between its local maximum and local minimum is 32. Determine k. f(x) = x3 - 3kx, where k > 0. The vertical separation between its local maximum and local minimum is 32. Determine k. Paper 1 style Hard 5 marks No calculator + 38 Marking analysis: A learner attempts the following task: “f(x) = x^3 - 3kx, where k > 0. The vertical separation between its local maximum and local minimum is 32. Determine k.” Their response addresses only this point: “f'(x) = 3x^2 - 3k, so stationary inputs are +/-sqrt(k).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis: A learner attempts the following task: “f(x) = x3 - 3kx, where k > 0. The vertical separation between its local maximum and local minimum is 32. Determine k.” Their response addresses only this point: “f'(x) = 3x2 - 3k, so stationary inputs are +/-sqrt(k).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks No calculator + 39 A temperature is modelled by T(t) = 20 + 60e^(-0.15t), where t is minutes after cooling begins. Find when T first reaches 35 degrees, the initial rate of temperature change, and whether the model reaches 20 degrees at a finite time. Paper 2 style Hard 5 marks Calculator + 40 Marking analysis: A learner attempts the following task: “A temperature is modelled by T(t) = 20 + 60e^(-0.15t), where t is minutes after cooling begins. Find when T first reaches 35 degrees, the initial rate of temperature change, and whether the model reaches 20 degrees at a finite time.” Their response addresses only this point: “35 = 20 + 60e^(-0.15t) gives e^(-0.15t) = 1/4.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks. Marking analysis Medium 5 marks Calculator + Self-assessed Not marked yet
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