Calculus

An environmental engineer measures the depth of a straight river channel at equal intervals across its width of 12 m. The depths, d metres, at horizontal distance x metres from one bank are shown below.

x (m): 0, 2, 4, 6, 8, 10, 12
d (m): 0, 1.4, 2.3, 2.8, 2.5, 1.6, 0

A small bakery sells x loaves of speciality bread per day. The marginal profit, in dollars per loaf, is modelled by
\(P'(x) = 48 - 0.6x\).
When no loaves are sold the bakery makes a loss of $200 per day due to fixed costs, so $P(0) = -200$.

A car pulls away from a set of traffic lights. Its distance, \(s\) metres, from the lights \(t\) seconds after it starts moving is modelled by
\(s(t) = 0.4t^2 + 3t\), for \(0 \le t \le 10\).

A tree seedling is planted in a nursery. Its height, h cm, t weeks after planting is modelled by

\(\frac{dh}{dt} = 2.4 - 0.15t, \quad 0 \le t \le 16.\)

When it is planted, the seedling is 5 cm tall.

A section of a cycle path seen from above is modelled by the curve
\(y = 0.03x^2 - 1.2x + 15\), for \(0 \le x \le 40\),
where x and y are measured in metres from a fixed point.

A small bakery sells x loaves of sourdough per day. Its daily profit, P dollars, is modelled by
\(P(x) = -0.4x^2 + 36x - 250\), for \(0 \le x \le 90\).

A farmer uses 120 m of fencing to make a rectangular pen for goats. One side of the pen is formed by a long straight stone wall, so it needs no fencing. The two sides perpendicular to the wall each have length x metres.

A small bakery sells x loaves of speciality bread per day. The daily profit, P dollars, is modelled by
\(P(x) = -0.4x^2 + 36x - 300\), for \(0 \le x \le 90\).

An engineer measures the depth of a river at equal intervals across its width. The river is 12 m wide. The depth, d metres, is recorded at a horizontal distance x metres from one bank.

x (m): 0, 2, 4, 6, 8, 10, 12
d (m): 0, 1.4, 2.1, 2.6, 2.3, 1.5, 0

A small company makes handmade chairs. The marginal cost of producing the xth chair is modelled by
\(C'(x) = 0.03x^2 - 1.2x + 25\) US dollars per chair.
The company has fixed costs of 400 US dollars, so that \(C(0) = 400\).

Water flows into a storage tank. The rate of flow, in litres per minute, is modelled by
\(r(t) = 40 + 6t - 0.5t^2\), for \(0 \le t \le 12\),
where t is the time in minutes after the inflow begins.

An engineer models the vertical profile of a section of a mountain road by
\(y = 0.001x^3 - 0.06x^2 + 1.5x\), for \(0 \le x \le 50\),
where \(x\) is the horizontal distance in metres from the start of the section and \(y\) is the height in metres above the start.

A farmer builds a rectangular chicken run against the long straight wall of a barn. The wall forms one side of the run. The other three sides use 80 m of fencing. The two sides perpendicular to the wall each have length \(x\) metres.

The profile of a small hill is modelled by
\(y = 12 + 0.3x^2 - 0.01x^3\), for \(0 \le x \le 31\),
where x is the horizontal distance in metres from a marker post and y is the height of the hill in metres. A straight support cable is fixed to the hill at the point where x = 30. The cable lies along the tangent to the hill at this point and is extended down the slope until it reaches ground level, y = 0.

The profile of a hill is modelled by
\(y = -0.002x^3 + 0.12x^2\), for \(0 \le x \le 50\),
where \(x\) is the horizontal distance in metres from the foot of the hill and \(y\) is the height in metres.

A farmer owns a field with one straight boundary, of length 80 m, along a road. The other boundary is curved. The perpendicular distance, d metres, from the road to the curved boundary is measured at points x metres along the road, as shown below.

x (m): 0, 20, 40, 60, 80
d (m): 32, 41, 45, 38, 27

The farmer uses the trapezoidal rule with the five measurements to estimate the area of the field.

A ball is thrown upwards from a balcony. Its height above the ground, h metres, t seconds after it is thrown is modelled by
\(h(t) = 1.5 + 14t - 4.9t^2\), for \(0 \le t \le 2.9\).

A small bakery sells handmade sourdough loaves at a weekend market. The weekly profit, \(P\) dollars, from selling \(x\) loaves is modelled by
\(P(x) = -0.4x^2 + 36x - 250\), for \(0 \le x \le 90\).

A hydrologist measures the depth of water across a straight river channel that is 12 m wide. Depths are recorded every 2 m from one bank to the other.

Distance from left bank (m): 0, 2, 4, 6, 8, 10, 12
Depth (m): 0, 1.2, 1.9, 2.4, 2.1, 1.5, 0

The cross-section of the river is modelled as the region between the water surface and the river bed.

A small bakery sells \(x\) celebration cakes per week. The weekly profit, in dollars, is modelled by
\(P(x) = -0.04x^2 + 6x - 80\), for \(0 \le x \le 140\).

A farmer builds a rectangular sheep pen against a long straight stone wall. The wall forms one side of the pen, and 84 m of fencing is used for the other three sides. The two sides perpendicular to the wall each have length \(x\) metres.

During the dry season the volume of water, \(V\) m³, in a small reservoir is modelled by
\(V(t) = 5000 - 12t^2 + 0.1t^3\), for \(0 \le t \le 20\),
where \(t\) is the number of days after the start of the dry season.

The depth of water, d metres, in a reservoir t hours after a gate is opened is modelled by
d(t) = 0.02t³ − 0.3t² + 1.5t + 2,   0 ≤ t ≤ 10.

The height, h metres above sea level, of a scenic road at a horizontal distance x km from its start is modelled by
h(x) = −x³ + 6x² − 5x + 40,   0 ≤ x ≤ 5.

A farmer wants to estimate the area of an irregular field. A straight road runs along one side. The width of the field, measured at right angles to the road, is recorded at 10 m intervals along the road.
Distance along road (m): 0, 10, 20, 30, 40
Width of field (m): 0, 14, 20, 17, 0

A robot moves along a straight rail. Its displacement from a fixed point O is s metres at time t seconds, where
s(t) = t³ − 9t² + 24t,   0 ≤ t ≤ 6.

In a video game, a player earns a bonus score B points, t seconds after activating a power-up, modelled by B(t) = 24t − 1.5t², for 0 ≤ t ≤ 16.

The main cable of a footbridge hangs in a curve modelled by y = 0.02x² − 0.6x + 6, where x is the horizontal distance in metres from the left tower and y is the height of the cable in metres above the water.

Water flows into a treatment plant at a rate Q thousand litres per hour. The rate is recorded every two hours, as shown.

t (hours): 0, 2, 4, 6, 8
Q (thousand litres per hour): 12, 16, 20, 19, 14

A nurse programs an infusion pump so that the drug is delivered at a rate R(t) = 0.6t² + 2 mL per minute, for 0 ≤ t ≤ 4, where t is in minutes.

A light aircraft climbs after take-off. Its height above the ground is modelled by h(t) = 900 + 240t − 6t² metres, where t is the time in minutes and 0 ≤ t ≤ 30.

A farmer models the profile of a sloping terrace by y = 0.01x³ − 0.15x + 3, where x is the horizontal distance in metres from the lower edge and y is the height in metres.

The amplitude A of a note played on a flute, in arbitrary units, is measured every 0.1 s after the note starts.

A farmer irrigates a field using a sprinkler. The rate at which water is delivered is modelled by r(t) = 15 − 0.03t² litres per minute, for 0 ≤ t ≤ 20, where t is the time in minutes after the sprinkler is switched on.

During an outbreak at a school, the total number of pupils who have caught an infection is modelled by \(N(t) = 240 + 36t - 1.5t^2\), where \(t\) is the time in days since monitoring began, for \(0 \le t \le 10\).

The power output, \(P\) kW, of a small wind turbine is modelled by \(P = 0.05v^3\), where \(v\) m s⁻¹ is the wind speed, for \(0 \le v \le 12\).

A solar panel array records its power output, \(P\) kW, every two hours after 08:00, as shown.

Time after 08:00 (hours): 0, 2, 4, 6, 8
Power \(P\) (kW): 0.4, 2.9, 4.6, 5.1, 3.2

A hydroelectric intake channel has a cross-section whose depth is measured at 2 m intervals across its 10 m width.

Distance from left bank (m): 0, 2, 4, 6, 8, 10
Depth (m): 0, 0.9, 1.6, 1.8, 1.1, 0

A wind farm's cumulative energy output is modelled by E(t) = 0.5t³ − 6t² + 30t, where E is in MWh and t is the time in hours after midnight, 0 ≤ t ≤ 10.

A wind farm maintenance compound is to be a rectangle fenced on three sides, the fourth side being the straight wall of a storage building. There is 48 m of fencing. The sides perpendicular to the wall each have length x metres.

The power output P (kW) of a wind turbine was recorded every two hours during a 12-hour period.

A wind farm lies on a ridge whose height is modelled by h(x) = 4x − 0.25x², 0 ≤ x ≤ 10, where x is the horizontal distance and h is the height, both in units of 100 m. A turbine stands at the point where x = 3.

The total number of books issued by a city library t hours after it opens is modelled by N(t) = 4t² − 0.1t³, for 0 ≤ t ≤ 10.

A wind farm records its power output P (in MW) once an hour for four hours:
Time t (hours): 0, 1, 2, 3, 4
Power P (MW): 18.0, 24.5, 31.0, 27.5, 21.0

A swimming club ropes off a rectangular practice zone in a lake. The shore forms one side of the rectangle, so rope is needed for the other three sides only. The club has 120 m of rope. The sides perpendicular to the shore each have length x metres.

The profile of a water slide is modelled by y = 0.04x² − 0.8x + 5, for 0 ≤ x ≤ 20, where x is the horizontal distance in metres from the top and y is the height in metres above the pool surface.

A grey seal makes a dive from a research boat. Its depth below the surface is modelled by D(t) = 0.04t³ − 1.2t² + 9t, where D is in metres and t is the time in seconds after the dive begins, for 0 ≤ t ≤ 15.

The roof of a market canopy has a cross-section modelled by h(x) = −0.04x² + 0.9x + 3 for 0 ≤ x ≤ 20, where h is the height above the ground in metres and x is the horizontal distance in metres from the left-hand wall.

A tidal channel in a marine reserve is 8 m wide. The depth of water is measured at equal intervals across the channel.
Distance from bank (m): 0, 2, 4, 6, 8
Depth (m): 0, 1.2, 1.9, 1.5, 0

The temperature of seawater at a monitoring station varies with depth. It is modelled by T(d) = 20 − 0.05d + 0.00005d², where T is in °C and d is the depth in metres, 0 ≤ d ≤ 300.

A landing module descends towards a moon. Its height above the surface is modelled by h(t) = 0.05t² − 14t + 1000, where h is in metres and t is the time in seconds after the parachute opens, for 0 ≤ t ≤ 120.

The main cable of a footbridge is modelled by the curve y = 0.01x² − 0.8x + 20 for 0 ≤ x ≤ 80, where x is the horizontal distance in metres from the left tower and y is the height of the cable above the deck in metres.

During a coffee roast the electrical power P kW supplied to the roaster is measured every 2 minutes.
t (minutes): 0, 2, 4, 6, 8, 10
P (kW): 4.0, 5.2, 6.1, 6.6, 6.8, 6.9
The graph of P against t is increasing, and its gradient decreases throughout the 10 minutes.

The number of people, N, in a town who are infected with a virus is modelled by N(t) = 40 + 18t − 0.6t², for 0 ≤ t ≤ 25, where t is the time in days since the first case was recorded.

A shop sells a desk lamp at a price of €x. The daily profit, €P, is modelled by P(x) = −2x² + 56x − 150, for 5 ≤ x ≤ 25.

The height, h metres, of a mountain road above sea level is modelled by h(x) = x³ − 6x² + 9x + 20, for 0 ≤ x ≤ 4, where x is the horizontal distance in kilometres from the start of the road.

The cumulative energy, E MWh, generated by a solar farm t hours after a test began satisfies dE/dt = 3t² − 18t + 30. After 2 hours the farm had generated 60 MWh in total.

The height, H km, of an ash plume above a volcano vent is modelled by H(t) = −0.05t² + 1.2t, for 0 ≤ t ≤ 24, where t is the time in minutes after the start of the eruption.

A theatre sells tickets at €p each. The number of tickets sold for a performance is modelled by N = 600 − 20p, for 0 < p < 30.

The fuel cost, €f per nautical mile, for a cargo ship travelling at a steady speed of v knots is modelled by f(v) = 2744/v + 0.5v², for 5 ≤ v ≤ 25.

The depth, h cm, of ash on a village roof t hours after an eruption begins satisfies dh/dt = 0.12t² − 1.2t + 3.6. After 5 hours the depth of ash is 14 cm.

The concentration of a drug in a patient's blood is modelled by C(t) = −0.8t² + 6.4t, where C is in mg/L and t is the time in hours after an injection, for 0 ≤ t ≤ 8.

Water flows into a storage tank at a rate of r(t) = 0.6t² + 2 m³ per hour, where t is the time in hours, for 0 ≤ t ≤ 6.

A gardener uses 40 m of edging to mark three sides of a rectangular irrigation bed. The fourth side is a straight wall that needs no edging. The sides perpendicular to the wall each have length x m.

In an online game, the total experience points (XP) needed to reach level n is modelled by X(n) = 2n² + 30n, for 1 ≤ n ≤ 30. Treat n as a continuous variable where needed.

The density D of krill, in individuals per m³, at a monitoring station is modelled by
D(t) = 0.5t³ − 6t² + 24t + 10,   0 ≤ t ≤ 10,
where t is the time in hours after sunset.

The roof of a small pavilion has a cross-section modelled by the curve
y = 6 − 0.04x²,   −12 ≤ x ≤ 12,
where x and y are measured in metres, with the origin at floor level directly below the highest point.

Delivery vans arrive at a distribution depot at a rate that is recorded every two hours during an eight-hour shift.

Seawater flows into a tidal pool at a rate of r(t) = 6 + 0.9t² litres per minute, where t is the time in minutes after the tide starts to rise, 0 ≤ t ≤ 5.

The depth d metres below ground level of a drainage channel at an archaeological dig is modelled by \(d = 0.25x^2 - 2x + 6\), where x is the horizontal distance in metres from the start of the channel, for 0 ≤ x ≤ 12.

The side of an excavation trench is modelled by the curve \(y = 0.25x^2 - 3x + 10\), where x and y are in metres. The point P(2, 5) lies on the curve.

A cooling plate in a data centre has temperature T °C at time t minutes after it is switched off, given by \(T = 25 + \frac{144}{t}\) for t ≥ 2.

The cross-section of an ancient river channel is measured at 2 m intervals across its 12 m width. The table shows the depth d of water at each position.

Distance from bank (m): 0, 2, 4, 6, 8, 10, 12
Depth (m): 0, 1.1, 2.3, 3.0, 2.2, 1.4, 0

A robotic arm moves its gripper along a rail. The distance, x centimetres, of the gripper from the start of the rail at time t seconds is modelled by x(t) = 0.5t³ − 3t² + 8t, for 0 ≤ t ≤ 6.

The number of fish, N thousand, in a coastal fishery t months after a new quota is introduced is modelled by N(t) = −2t² + 24t + 150, for 0 ≤ t ≤ 10.

A designer models the curve of a neckline by y = 0.5x² − 3x + 6, where x and y are measured in centimetres from a fixed origin. The point A on the curve has x-coordinate 4.

The angle, θ degrees, turned by a robot's elbow joint after t seconds is modelled by θ(t) = 2t³ − 9t² + 12t, for 0 ≤ t ≤ 3. The joint momentarily stops turning when dθ/dt = 0.

A climber leaves a mountain hut and ascends a snow slope. Her height, h metres, above the valley floor is modelled by
h = 150 + 72t − 4.5t², for 0 ≤ t ≤ 6,
where t is the time in hours since she left the hut.

A mobile network records the data rate of a video stream. During the first 10 seconds the rate is modelled by
r(t) = 12 + 3t − 0.2t², for 0 ≤ t ≤ 10,
where r is in megabits per second (Mbit s⁻¹) and t is in seconds.

A food scientist studies a fermentation process. The mass of product obtained from one batch is modelled by
y(T) = −0.8T² + 52T − 700, for 20 ≤ T ≤ 45,
where y is in grams and T is the temperature of the batch in °C.

A food manufacturer designs a closed cylindrical tin of volume 500 cm³. The tin has base radius r cm. The total outer surface area of the tin is given by
A(r) = 2πr² + 1000r⁻¹, for r > 0,
where A is in cm². The manufacturer wants to use as little metal as possible.

Ash falls steadily on a village near a volcano. The depth of ash on a flat roof, d mm, is modelled by d(t) = 0.05t³ − 0.9t² + 5.4t, where t is the time in hours after the ash fall begins, for 0 ≤ t ≤ 10.

The pressure, P MPa, of gas in a magma chamber is modelled by P = 2400/V, where V km³ is the volume of the chamber, for V ≥ 1.

A scenery panel has a curved top edge. Its height, h m, at a horizontal distance x m from the left edge is modelled by h(x) = 4 + 0.5x − 0.05x², for 0 ≤ x ≤ 6.

A road climbs a hill. The height of the road, h metres above sea level, at a distance x km along the climb is modelled by h(x) = 60 + 15x² − x³, for 0 ≤ x ≤ 8. The gradient of the road is the rate of change of height with distance.

After a cooling failure the temperature of a server room, T °C, is modelled by T(t) = 22 + 0.8t + 0.05t², where t is the time in minutes since the failure, 0 ≤ t ≤ 20.

Coolant is pumped through a data centre rack at a rate of r(t) litres per minute, where r(t) = 0.6t² + 2 and t is the time in minutes, 0 ≤ t ≤ 5.

The concentration of dust in a burial chamber, c mg m⁻³, at a distance x metres from the chamber entrance is modelled by c(x) = 12 + 90/x, for x ≥ 1.

A space probe is moving away from Earth. Its distance from Earth is modelled by d(t) = 2t³ − 15t² + 60t, where d is measured in thousands of kilometres and t is the time in days after a course correction, 0 ≤ t ≤ 6.

A coffee roaster burns gas at a rate of r(t) = 0.3t² + 2 litres per minute, where t is the time in minutes after the start of a roast, 0 ≤ t ≤ 6.

The depth of a river is measured at points across its width. The table shows the depth y metres at distance x metres from the left bank.

x (m): 0, 4, 8, 12, 16, 20
y (m): 0, 1.2, 2.6, 3.1, 2.4, 0