Mathematics: Applications & Interpretation SL
Introductory calculus — Topic 5
- 1.
Find the derivative of f(x) = 3x⁴ − 2x² + 5x, and hence find f'(2).
[3 marks] - 2.
Marking analysis: A learner attempts the following task: “Find the derivative of f(x) = 3x⁴ − 2x² + 5x, and hence find f'(2).” Their response addresses only this point: “Applies the power rule to each term to obtain f'(x) = 12x³ − 4x + 5.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 3.
The curve y = x² − 3x + 1 passes through the point where x = 4. Find the equation of the tangent to the curve at this point.
[4 marks] - 4.
Marking analysis: A learner attempts the following task: “The curve y = x² − 3x + 1 passes through the point where x = 4. Find the equation of the tangent to the curve at this point.” Their response addresses only this point: “Calculates y(4) = 4² − 3(4) + 1 = 5, giving the point (4, 5).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 5.
Find the stationary points of f(x) = x³ − 6x² + 9x + 2, and use the second derivative to determine whether each is a local maximum or a local minimum.
[6 marks] - 6.
Marking analysis: A learner attempts the following task: “Find the stationary points of f(x) = x³ − 6x² + 9x + 2, and use the second derivative to determine whether each is a local maximum or a local minimum.” Their response addresses only this point: “Differentiates to obtain f'(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.
[6 marks] - 7.
A company's profit, in thousands of dollars, from selling x units is modelled by P(x) = −2x² + 80x − 200. Find the number of units that maximises profit, and find the maximum profit.
[4 marks] - 8.
Marking analysis: A learner attempts the following task: “A company's profit, in thousands of dollars, from selling x units is modelled by P(x) = −2x² + 80x − 200. Find the number of units that maximises profit, and find the maximum profit.” Their response addresses only this point: “Differentiates to obtain P'(x) = −4x + 80.” 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 volume of water in a tank, in litres, is modelled by V(t) = 100 − 5t², where t is time in minutes. Find the rate of change of volume at t = 3 minutes, and interpret the sign of your answer.
[3 marks] - 10.
Marking analysis: A learner attempts the following task: “The volume of water in a tank, in litres, is modelled by V(t) = 100 − 5t², where t is time in minutes. Find the rate of change of volume at t = 3 minutes, and interpret the sign of your answer.” Their response addresses only this point: “Differentiates to obtain V'(t) = −10t.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 11.
Use the chain rule to differentiate f(x) = (3x + 1)⁴, and find f'(1).
[3 marks] - 12.
Marking analysis: A learner attempts the following task: “Use the chain rule to differentiate f(x) = (3x + 1)⁴, and find f'(1).” Their response addresses only this point: “Applies the chain rule: f'(x) = 4(3x + 1)³ × 3 = 12(3x + 1)³.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 13.
Use the product rule to differentiate f(x) = x²(2x + 3), and find f'(2).
[4 marks] - 14.
Marking analysis: A learner attempts the following task: “Use the product rule to differentiate f(x) = x²(2x + 3), and find f'(2).” Their response addresses only this point: “Applies the product rule with u = x², v = (2x + 3): f'(x) = 2x(2x + 3) + x²(2).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 15.
Use the quotient rule to differentiate f(x) = (2x + 1)/(x − 3), and find f'(5).
[4 marks] - 16.
Marking analysis: A learner attempts the following task: “Use the quotient rule to differentiate f(x) = (2x + 1)/(x − 3), and find f'(5).” Their response addresses only this point: “Applies the quotient rule with u = 2x + 1, v = x − 3: f'(x) = [2(x−3) − (2x+1)(1)]/(x−3)².” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 17.
Evaluate the definite integral ∫₀³ (3x² − 4x + 2) dx.
[4 marks] - 18.
Marking analysis: A learner attempts the following task: “Evaluate the definite integral ∫₀³ (3x² − 4x + 2) dx.” Their response addresses only this point: “Finds the antiderivative x³ − 2x² + 2x.” 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.
Find the area enclosed between the curve y = 4 − x² and the x-axis, between x = −2 and x = 2.
[4 marks] - 20.
Marking analysis: A learner attempts the following task: “Find the area enclosed between the curve y = 4 − x² and the x-axis, between x = −2 and x = 2.” Their response addresses only this point: “Recognises the area is given by ∫₋₂² (4 − x²) dx.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 21.
A particle moves with velocity v(t) = 6t² − 4t m/s. Find the displacement of the particle between t = 1 and t = 3 seconds.
[4 marks] - 22.
Marking analysis: A learner attempts the following task: “A particle moves with velocity v(t) = 6t² − 4t m/s. Find the displacement of the particle between t = 1 and t = 3 seconds.” Their response addresses only this point: “Recognises that displacement is the definite integral of velocity: ∫₁³ (6t² − 4t) 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.
The displacement of a particle is given by s(t) = t³ − 6t² + 9t. Find the velocity and acceleration of the particle at t = 2 seconds.
[4 marks] - 24.
Marking analysis: A learner attempts the following task: “The displacement of a particle is given by s(t) = t³ − 6t² + 9t. Find the velocity and acceleration of the particle at t = 2 seconds.” Their response addresses only this point: “Differentiates once to obtain velocity v(t) = 3t² − 12t + 9.” 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.
The cost, in dollars, of producing x units is modelled by C(x) = 0.01x³ − 0.6x² + 13x + 100. Find the marginal cost when x = 10, and interpret its meaning.
[4 marks] - 26.
Marking analysis: A learner attempts the following task: “The cost, in dollars, of producing x units is modelled by C(x) = 0.01x³ − 0.6x² + 13x + 100. Find the marginal cost when x = 10, and interpret its meaning.” Their response addresses only this point: “Differentiates to obtain C'(x) = 0.03x² − 1.2x + 13.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 27.
A farmer has 200 m of fencing to enclose a rectangular field. Let the width be x metres. (a) Express the area A of the field in terms of x. (b) Find the value of x that maximises the area, and find the maximum area.
[5 marks] - 28.
Marking analysis: A learner attempts the following task: “A farmer has 200 m of fencing to enclose a rectangular field. Let the width be x metres. (a) Express the area A of the field in terms of x. (b) Find the value of x that maximises the area, and find the maximum area.” Their response addresses only this point: “Uses the perimeter condition 2x + 2(length) = 200 to express the length as 100 − x.” 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.
Find the inflection points of the curve f(x) = x⁴ − 8x², using the second derivative.
[5 marks] - 30.
Marking analysis: A learner attempts the following task: “Find the inflection points of the curve f(x) = x⁴ − 8x², using the second derivative.” Their response addresses only this point: “Differentiates twice to obtain f''(x) = 12x² − 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] - 31.
Find the average value of the function f(x) = x² + 1 over the interval [0, 4].
[4 marks] - 32.
Marking analysis: A learner attempts the following task: “Find the average value of the function f(x) = x² + 1 over the interval [0, 4].” Their response addresses only this point: “Uses the average value formula: (1/(4−0)) ∫₀⁴ (x² + 1) dx.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks]