Calculus methods

Practice

Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.

Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.

Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.

Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.

What actually comes up

Three questions every year, each opening with a short differentiation/integration calculation before building to an optimisation or rates-of-change problem wrapped in a real scenario. Being given the DERIVATIVE and asked to recover the original function — via integration, not further differentiation — appears in some form almost every year.

Across 4 years

The pattern — what to prepare

The 2025 paper ↗ Marking schedule ↗

Question One

Several short calculus tasks: a cubic function, a tangent, a gradient function to integrate, and a catch-up problem.

• A cruise ship moves at a constant 5 m s⁻¹, already 200 m from the dock
• A smaller boat accelerates at 0.5 m s⁻² from the dock

• a A function f is given. Use calculus to find the gradient of the function at the point where x = −½. Achieved
• b For a given cubic function, find the equation of the tangent to the curve at x = 2. Merit
• c The gradient function f'(x) = 2x² + 2x − 24 is given. Find the equation of the original function f(x), given the y-value of the local minimum is −46. Excellence
• d The smaller boat accelerates at 0.5 m s⁻² to catch the cruise ship, already 200 m ahead and moving at a constant 5 m s⁻¹. Find the distance from the dock at which the boat catches up. Excellence

Question Two

A sketch-the-derivative task, a function to recover from its gradient, and a caffeine-concentration model monitored over 180 minutes.

• f'(x) = 2x³ − 6x² + 2x + 4, passing through (2, 5)
• Caffeine concentration modelled by a cubic C(t), monitored over 180 minutes

• a Given the graph of f(x), sketch the graph of f'(x). Achieved
• b The graph of f(x) passes through (2, 5) and f'(x) = 2x³ − 6x² + 2x + 4. Find the equation of f. Merit
• c Show that the rate of change of caffeine concentration is −16 μg/L/min after 60 minutes, then use calculus to justify when the coffee was given. Excellence

Question Three

A gradient-equals-3 problem, an expanding triangle, a lidless container design, and a quartic with an unknown constant.

• Triangle: base always 4× the perpendicular height
• Container: max 4.32 m² material, width = 2 × height, maximise volume
• f(x) = ax³ + bx² + ax + 2, intersects x-axis at x = 1, stationary point at x = −1

• a For f(x) = 2x² − 7x − 20, find the coordinate of the point where the gradient equals 3. Achieved
• b Find the rate of change of a triangle's area with respect to its height, given the base is always 4 times the height, when the area equals 98 cm². Merit
• c Using calculus, find the maximum volume of a lidless rectangular container made from at most 4.32 m² of material, with width twice the height, and prove it is a maximum. Merit
• d A function f(x) = ax³ + bx² + ax + 2 intersects the x-axis at x = 1 and has a stationary point at x = −1. Using calculus, determine the regions where f(x) is decreasing. Excellence

The 2024 paper ↗ Marking schedule ↗

Question One

A cubic function and its gradient, a curve's local minimum, and a cylindrical cup designed to minimise surface area.

• f(x) = 4x³ − 6x² + 5x + 2
• f'(x) = 12 − 4x + 6x², passing through (2, 36)
• y = 3x³ − 9x² − 27x + 4
• Cup volume: 500 mL, V = πr²h, SA = 2πr² + 2πrh

• a Use calculus to find the gradient of f(x) = 4x³ − 6x² + 5x + 2 at x = 2. Achieved
• b f'(x) = 12 − 4x + 6x², and f(x) passes through (2, 36). Find the equation of f. Merit
• c For y = 3x³ − 9x² − 27x + 4, find the x-coordinate of the local minimum and explain how you know it is a minimum. Excellence
• d A cylindrical cup must hold 500 mL. Calculate the dimensions that minimise its surface area (including a lid). Excellence

Question Two

A sketch-the-derivative task, a tangent, a bacteria population model, and a trapezium inscribed under a parabola.

• f(x) = 3 − 12x + 3x³, tangent at (−2, 3)
• P(t) = t³ − 60t² + 768t + 40 960
• y = 8x − x², trapezium with parallel sides at x and 2x, 0 ≤ x ≤ 4

• a Sketch the gradient function y = f'(x) from the given graph of y = f(x). Achieved
• b Find the equation of the tangent to f(x) = 3 − 12x + 3x³ at (−2, 3). Merit
• c For the bacteria population P(t) = t³ − 60t² + 768t + 40 960, find the rate of decrease on day 10, then the change in population from when nutrients ran out to when more were added. Excellence
• d A trapezium sits under y = 8x − x² with parallel sides at x and 2x. Use calculus to find the maximum area. Excellence

Question Three

A gradient-equals-3 problem, a truck accelerating to top speed, a gradient-function graph to integrate and sketch, and a minimisation proof.

• h(x) = 0.25x² − 2x + 4
• Truck: accelerates at 0.64 m s⁻² to a top speed of 25 m s⁻¹
• y = k − x, minimise x² + 2y²

• a For h(x) = 0.25x² − 2x + 4, find the coordinate where the gradient equals 3. Achieved
• b A truck accelerates at 0.64 m s⁻² from rest to a top speed of 25 m s⁻¹. Use calculus to find the distance travelled. Merit
• c From a graph of the gradient function f'(x), find the equation of f(x) passing through (1, 0), then draw f(x) labelling intercepts and stationary points. Excellence
• d If y = k − x, use calculus to show the minimum value of x² + 2y² is 2k²/3. Excellence

The 2023 paper ↗ Marking schedule ↗

Question One

A quadratic gradient problem, a cubic recovered from its gradient function, an increasing-region question, and a running-track infield to maximise.

• f(x) = 4x² − 12x
• f'(x) = 4x³ − 6x² − 4x, passing through (2, −5)
• f(x) = x⁴/4 − 2x³/3 − 12x² + 10
• 400 m running track: two straights + two semicircles, maximise the infield rectangle

• a For f(x) = 4x² − 12x, find the point on the curve where the gradient equals 4. Achieved
• b Find the function whose gradient function is f'(x) = 4x³ − 6x² − 4x, passing through (2, −5). Merit
• c Use calculus to find the values of x for which f(x) = x⁴/4 − 2x³/3 − 12x² + 10 is increasing. Excellence
• d A 400 m running track has two straights and two semicircular ends. Find the dimensions of the rectangular infield that maximise its area, and use calculus to show this is a maximum. Excellence

Question Two

A sketch-the-derivative task, a tangent-line proof, and a braking-car problem with old vs new brake pads.

• f(x) = 3x³ − x + 4; line y = 8x + 10
• Car: 27.78 m s⁻¹ (100 km/hr), decelerating at 2.5 m s⁻²
• Older brakes: 2.1 m s⁻² deceleration, same ≈154 m stopping distance

• a Sketch the graph of the gradient function y = f'(x) from the given graph of y = f(x). Merit
• b Use calculus to show that y = 8x + 10 is a tangent to f(x) = 3x³ − x + 4. Merit
• c A car at 27.78 m s⁻¹ decelerates at 2.5 m s⁻² when the brakes are applied. Find the time to stop, and show it travels approximately 154 m. Then find the highest speed an older car (2.1 m s⁻² deceleration) could travel and still stop within the same 154 m. Excellence

Question Three

A gradient-matching problem, a social-media follower-growth cubic, and a tangent that also crosses the curve elsewhere.

• y = 2x³ − 9x² + 12x + 3, point (1, 8)
• F = 5t³ − 840t² + 42 180t, 0 ≤ t ≤ 90
• f(x) = x³ − px² + 21x − 7, tangent at x = 4 also crosses at (0, −7)

• a Find the gradient of y = 2x³ − 9x² + 12x + 3 at (1, 8), then find another point on the curve with the same gradient. Merit
• b A social-media account's followers are modelled by F = 5t³ − 840t² + 42 180t. Find the rate of increase on day 5, the maximum followers in the first 90 days, and how many days the account loses followers, justifying with calculus. Excellence
• c The curve f(x) = x³ − px² + 21x − 7 has a tangent at x = 4 that also crosses the curve at (0, −7). Find the value of p. Excellence

The 2022 paper ↗ Marking schedule ↗

Question One

A quartic gradient, a cubic recovered from its gradient function, a decreasing-region question, and a stylised "H" logo made of two semicircles and two straight lines with a fixed perimeter.

• f(x) = 2x⁴ + 4x³ − 20x² − 5
• f'(x) = 4 − 6x + 2x², passing through (3, 4)
• f(x) = 2x³/3 + 3x²/2 − 20x − 3
• Logo: two semicircles (radius r) + two straight sides, perimeter ≤ 80 cm, maximise enclosed area

• a Use calculus to find the gradient of f(x) = 2x⁴ + 4x³ − 20x² − 5 at x = 3. Achieved
• b f'(x) = 4 − 6x + 2x², and f(x) passes through (3, 4). Find the equation of f. Merit
• c Use calculus to find the values of x for which f(x) = 2x³/3 + 3x²/2 − 20x − 3 is decreasing. Merit
• d The tangent to f(x) = px − qx² at (2, −10) has a gradient of −6. Find the values of the constants p and q. Merit
• e(i) A logo shape (two semicircles and two straight sides) has a total perimeter of at most 80 cm. Find the maximum enclosed area. Excellence
• e(ii) Use calculus methods to show that this area is a maximum. Excellence

Question Two

A sketch-the-original-function task from a given gradient graph, a cricket ball's height, and spectator numbers at the match.

• h = 22.5t − 4.9t² + 1
• P = 100t² − 2t⁴ + 750 (0 ≤ t ≤ 7.5)
• A second match: P = kt² − 2t⁴ + 750, spectators growing fastest at t = 4

• a Given the graph of a gradient function f'(x), sketch a possible graph of the original function f(x). Achieved
• b A cricket ball's height is h = 22.5t − 4.9t² + 1. Using calculus, find the maximum height the ball reaches. Merit
• c(i) Spectator numbers follow P = 100t² − 2t⁴ + 750. Show that the rate of change of spectators is −528 people per hour when t = 6. Achieved
• c(ii) What is the meaning of this value for the rate of change? Merit
• c(iii) At another match, P = kt² − 2t⁴ + 750, and spectator numbers grow fastest when t = 4. Find the value of k, explaining your reasoning with correct mathematical statements. Excellence

Question Three

A tangent to a quadratic, sketching gradient functions from three given graphs (including one with a sharp corner), and two 3-D wireframe shapes made from a shared length of wire.

• y = 2x(x − 3), tangent at x = 1
• Three graphs y = f(x), g(x), h(x) to differentiate graphically; h(x) has a sharp corner
• Tetrahedron surface area 3a², octahedron surface area 2√3b², total wire 180 cm: 6a + 12b = 180

• a Find the equation of the tangent to y = 2x(x − 3) at x = 1. Achieved
• b(i) Given the graphs of y = f(x), y = g(x) and y = h(x) (the last with a sharp corner at x = 0), sketch their respective gradient functions. Merit
• b(ii) Given the graph of y = p(x), sketch the graph of its gradient function y = p'(x). Merit
• b(iii) Is it true to say the gradient of p(x) is zero when x = 0? Justify your response using the graphs and/or mathematical reasoning. Excellence
• c A tetrahedron (edge length a, surface area 3a²) and an octahedron (edge length b, surface area 2√3b²) are built from a total of 180 cm of wire (6a + 12b = 180 cm), so the total surface area of the two shapes is minimised. Determine a and b, and use calculus to justify this is a minimum. Excellence

Grades above are read from each year's own Achievement/Merit/Excellence evidence statement, condensed to the actual task — the schedules also grade holistically per question on a 0–8 scale, not a fixed mark per bullet.

Past exams & resources

⏱ Timed opens the paper and starts a countdown matching the exam length. Official NZQA PDFs. If a link 404s, that year’s paper may sit under a slightly different code: browse the NZQA subject page.