Calculus — Free Maths AI SL Practice Questions

The population of a small island, $P$ (in thousands), is modelled by the differential equation $$\frac{dP}{dt} = 0.02P$$ where $t$ is the time in years after 1 January 2000. The general solution of $\frac{dP}{dt} = kP$ is $P = Ae^{kt}$, where $A$ and $k$ are constants.

A population of bacteria in a laboratory culture grows over time. The rate of change of the population, $P$ (in thousands), at time $t$ (in hours) is modelled by the differential equation $$\frac{dP}{dt} = kP(10 - P)$$ where $k$ is a positive constant.

A chemical processing plant uses a holding tank that initially contains $5000$ litres of a chemical solution. The solution contains $200\,\text{kg}$ of dissolved salt. A pipe feeds a new mixture into the tank at a rate of $40$ litres per minute; this incoming mixture contains $1.5\,\text{kg}$ of salt per litre. The well-stirred mixture leaves the tank at the same rate of $40$ litres per minute. Let $S(t)$ represent the amount of salt (in kg) in the tank after $t$ minutes.

A cup of coffee cools according to Newton's Law of Cooling. The temperature $T$ (in °C) of the coffee at time $t$ minutes satisfies the differential equation $$\frac{dT}{dt} = -0.1(T - 22)$$ where $22$°C is the room temperature.

A cylindrical water tank has a small hole at its base. The depth of water, $h$ metres, in the tank at time $t$ seconds satisfies the differential equation $$\frac{dh}{dt} = -0.4\sqrt{h}$$

Consider the function $$f(x) = \begin{cases} 2x + 1, & x < 2 \\ 5, & x = 2 \\ x^2 - 1, & x > 2 \end{cases}$$

A company models the cost, $C(x)$ dollars, of producing $x$ units of a product by the function $$C(x) = \frac{1000}{x} + 5x, \quad x > 0.$$

The population of a rare species of bird on an island is modelled by $P(t) = \dfrac{500t}{t+5}$, where $t$ is the time in years since monitoring began ($t \geq 0$).

A function $g$ is defined by $$g(x) = \begin{cases} 3x - 4, & x < 1 \\ 2, & x = 1 \\ 5 - x, & x > 1 \end{cases}$$

The function $f$ is defined by $f(x) = \dfrac{|x-1|}{x-1}$, $x \neq 1$.

A water tank is being filled. The volume of water in the tank, $V$ litres, after $t$ minutes is given by $V(t) = 2t^2 + 5t$, for $t \geq 0$.

A water tank is being filled. The volume of water, $V$ litres, in the tank at time $t$ minutes is given by $V(t) = 2t^2 + 10t$, for $0 \leq t \leq 10$.

A water tank at a desalination plant has a valve that controls the outflow. The volume of water in the tank, $V$ litres, at time $t$ minutes after the valve is opened, is modelled by $$V(t) = 500 + 240t - 6t^2, \quad 0 \leq t \leq 20.$$

The graph of $y = f(x)$ for $0 \leq x \leq 6$ is shown below, with points $A(1,\,2)$, $B(3,\,8)$, $C(5,\,4)$ marked. A tangent to the curve at $x = 3$ passes through $B(3,\,8)$ and the point $(0,\,2)$.

The graph of $y = f(x)$ for $0 \leq x \leq 4$ is shown below. The curve has a tangent drawn at point $P$ where $x = 1$. The tangent passes through the points $(1, 2)$ and $(3, 6)$.

The rate at which water flows into a reservoir is modelled by $f(t) = 12t^2 + 5$, where $f(t)$ is measured in thousands of litres per day and $t$ is the time in days from the start of monitoring.

The rate of change of the volume of water, $V$ (in litres), in a tank during a 10-minute period is given by $\dfrac{dV}{dt} = 12t - 3t^2$, where $t$ is the time in minutes, $0 \leq t \leq 10$.

A company manufactures a new type of solar panel. The rate of change of energy output, in kilowatt-hours per day, of a prototype panel is modelled by the function $$R(t) = 12\text{e}^{-0.2t} + 4\cos\left(\frac{\pi t}{12}\right)$$ where $t$ is the time in days since the panel was first activated, for $t \geq 0$. The total energy output $E(t)$, in kilowatt-hours, satisfies $E(t) = \int R(t)\,dt$ and $E(0) = 0$.

The marginal cost of producing $x$ units of a product is given by $C'(x) = 3x^2 - 10x + 20$ dollars per unit. The fixed costs are $500$ dollars (i.e. $C(0) = 500$).

The marginal cost (in dollars per unit) of producing $x$ units of a product is given by $C'(x) = 3x^2 - 12x + 15$.

The population of a colony of penguins on a remote island is modelled by the differential equation $$\frac{dP}{dt} = 0.02P(1000 - P)$$ where $P$ is the number of penguins at time $t$ years after 1 January 2020. Initially, there are $200$ penguins.

A tank initially contains $500$ litres of pure water. A salt solution of concentration $0.2\,\text{kg per litre}$ flows into the tank at a rate of $5\,\text{litres per minute}$. The well-mixed solution drains from the tank at the same rate of $5\,\text{litres per minute}$. Let $S(t)$ be the mass of salt (in kg) in the tank at time $t$ minutes.

The rate of decay of a radioactive substance is proportional to the amount present. Initially there are $80$ grams of the substance. After $12$ hours, $60$ grams remain. Let $N(t)$ be the amount in grams at time $t$ hours.

The velocity $v$ (in $\text{m s}^{-1}$) of a parachutist falling under gravity satisfies the differential equation $$\frac{dv}{dt} = 9.8 - 0.2v$$ where $t$ is time in seconds after the parachute opens. The initial velocity is $v(0) = 55\,\text{m s}^{-1}$.

A cup of coffee is initially at $90°\text{C}$ and is placed in a room maintained at $20°\text{C}$. Newton's law of cooling states that the rate of cooling of the coffee is proportional to the difference between its temperature and the room temperature. Let $T(t)$ be the temperature of the coffee in $°\text{C}$ at time $t$ minutes.

A particle moves along a straight line. Its displacement, $s$ metres, from a fixed point $O$ at time $t$ seconds is given by $$s(t) = \frac{t^2 - 1}{t - 1}, \quad t \geq 0,\ t \neq 1.$$

The graph below shows the function $f(x)$ defined for $-2 \leq x \leq 5$, consisting of a straight line segment from $(-2, 1)$ to $(2, 5)$, a horizontal line segment from $(2, 5)$ to $(3, 5)$, and a curve from $(3, 5)$ to $(5, 2)$.

A population of bacteria in a petri dish is modelled by $P(t) = \dfrac{5000t}{t+2}$, where $P$ is the number of bacteria and $t \geq 0$ is the time in hours since the start of the experiment.

A ball is thrown upwards. Its height $h$, in metres, at time $t$ seconds is modelled by $$h(t) = 1.5 + 14t - 4.9t^2, \quad t \geq 0.$$

The temperature of a chemical reaction, $T$ degrees Celsius, is modelled by the function $$T(x) = \frac{4x^2 - 3x + 2}{x - 1}, \quad x > 1$$ where $x$ is the time in minutes after the reaction begins.

The graph of the function $f(x) = -0.5x^2 + 3x + 1$ is shown below for $0 \leq x \leq 6$.

The graph shows the function $f(x) = \dfrac{1}{2}x^3 - 2x^2 + 1$ for $0 \leq x \leq 4$.

A water tank is being filled. The volume of water in the tank, $V$ litres, at time $t$ minutes is given by $$V(t) = 2t^3 - 15t^2 + 36t + 50, \quad 0 \leq t \leq 5.$$

The graph below shows the function $y = f(x)$ for $-3 \leq x \leq 4$. The tangent to the curve at point $A(1, 2)$ passes through the origin $(0, 0)$.

The volume of water, $V$ litres, in a storage tank during a 24-hour period is modelled by the function $$V(t) = 100 + 20t - t^2, \quad 0 \leq t \leq 24$$ where $t$ is the time in hours after midnight.

A company models the rate of change of profit, in thousands of dollars per year, for a new product using the function $P'(t) = 12t\,e^{-0.2t}$, where $t$ is the time in years since the product was launched.

The marginal cost $C'(x)$ of producing $x$ units of a product, in dollars per unit, is given by $$C'(x) = 3x^2 + 4x + 5$$ The fixed cost (the cost when $x = 0$) is USD 2000.

The rate at which water flows into a reservoir, measured in thousands of cubic metres per day, is modelled by the function $f(t) = 12 + 5\sin\!\left(\dfrac{\pi t}{6}\right)$, where $t$ is the time in days, $t \geq 0$.

A particle moves along a straight line. Its velocity $v\,\text{ms}^{-1}$ at time $t$ seconds is given by $$v(t) = 3t^2 - 12t + 9, \quad t \geq 0.$$ The particle starts at the origin.

A company designs a logo for a new brand. The logo is modelled by the region bounded by the curve $y = 6x^2 - x^3$ and the $x$-axis, for $0 \leq x \leq 6$.