Calculus: 53 IB Maths AI HL practice questions

Professor Vinculum investigated the migration season of the Bulbul bird from their natural wetlands to a warmer climate.

He found that during the migration season their population, $P$ could be modelled by $P = 1350 + 400(1.25)^{-t}$, $t \geq 0$, where $t$ is the number of days since the start of the migration season.

(a) Find the population of the Bulbul birds,[3]

(i) at the start of the migration season.

(ii) in the wetlands after 5 days.

(b) Calculate the time taken for the population to decrease below 1400.[2]

(c) According to this model, find the smallest possible population of Bulbul birds during the migration season.[1]

A climate scientist is modelling an ice sheet as a rectangle.

She believes that the width $(x \text{ km})$ is increasing at a constant rate of $10$ km per year and the length $(y \text{ km})$ is decreasing at a constant rate of $5$ km per year.

The time, $t$, is measured in years, and the area, $A$, is measured in $\text{km}^2$.

When $t = 0$ then $x = 75$ and $y = 40$.

(a) Find $\frac{dA}{dt}$ when $t = 0$.[3]

(b) State, with justification, whether the area of the ice sheet is increasing or decreasing when $t = 0$.[1]

(c) Find the change in the area of the ice sheet between $t = 0$ and $t = 1$.[3]

A meteorologist models the height of a hot air balloon launched from the ground. The model assumes the balloon travels vertically upwards and travels 450 metres in the first minute.

Due to the decrease in temperature as the balloon rises, the balloon will continually slow down. The model suggests that each minute the balloon will travel only 82% of the distance travelled in the previous minute.

(a) Find how high the balloon will travel in the first 10 minutes after it is launched.[3]

(b) The balloon is required to reach a height of at least 2520 metres. Determine whether it will reach this height.[2]

(c) Suggest a limitation of the given model.[1]

The pH scale is a measure of the acidity of a solution. Its value is given by the formula $\text{pH} = -\log_{10}[\text{H}^+]$, where $[\text{H}^+]$ is the concentration of hydrogen ions in the solution (measured in moles per litre).

(a) Calculate the pH value if the concentration of hydrogen ions is 0.0003.[2]

The pH of milk is 6.6.

(b) Calculate the concentration of hydrogen ions in milk.[2]

The strength of an acid is measured by its concentration of hydrogen ions.

A lemon has a pH value of 2 and a tomato has a pH value of 4.5.

(c) Calculate how many times stronger the acid in a lemon is when compared to the acid in a tomato.[3]

A spherical balloon is being inflated such that its volume is increasing at a rate of $15 \text{ cm}^3 \text{ s}^{-1}$.

(a) Find the radius of the balloon when its volume is $288\pi \text{ cm}^3$.[2]

(b) Hence or otherwise, find the rate of change of the radius at this instant.[3]

A particle, A, moves so that its velocity ($v \, \text{ms}^{-1}$) at time $t$ is given by $v = 2 \sin t$, $t \geq 0$.

The kinetic energy ($E$) of the particle A is measured in joules (J) and is given by $E = 5v^2$.

(a) Write down an expression for $E$ as a function of time.[1]

(b) Hence find $\frac{dE}{dt}$.[2]

(c) Hence or otherwise find the first time at which the kinetic energy is changing at a rate of $5 \, \text{Js}^{-1}$.[2]

A player throws a basketball. The height of the basketball is modelled by

$$h(t) = -4.75t^2 + 8.75t + 1.5, t \geq 0,$$

where $h$ is the height of the basketball above the ground, in metres, and $t$ is the time, in seconds, after it was thrown.

(a) Find how long it takes for the basketball to reach its maximum height.[2]

(b) Assuming that no player catches the basketball, find how long it would take for the basketball to hit the ground.[2]

Another player catches the basketball when it is at a height of 1.2 metres.

(c) Find the value of $t$ when this player catches the basketball.[2]

(d) Write down one limitation of using $h(t)$ to model the height of the basketball.[1]

Consider the differential equation $\frac{dy}{dx} = xy - 1$, given that $y = 2$ when $x = 1$.

Use Euler's method with step size $0.1$ to find the approximate value of $y$ when $x = 1.5$.[4]

A vet wants to find a relationship between the age in days of a breed of puppy ($d$) and its weight in kg ($w$).

To do this he collects a large quantity of data and plots two graphs.

He finds the regression line for each graph. His results are summarized in the table.

Based on these results, find the best of the two possible relationships between $w$ and $d$.

Express your relationship in the form $w = f(d)$ where $f$ is a simplified expression.

Justify your choice of expression.[6]

The height of a plant, $h$ cm, depends on the time, $t$ days, after it is planted.

For the first 15 days, the rate of change of the height of the plant is modelled as

$$\frac{\mathrm{d}h}{\mathrm{d}t} = 1.2 - 0.1t, \text{ where } 0 \leq t \leq 15.$$

(a) (i) Find the value of $\frac{\mathrm{d}h}{\mathrm{d}t}$ at $t = 3$.

(ii) Interpret the meaning of your answer to part (a)(i) in context.[2]

(b) Use $\frac{\mathrm{d}h}{\mathrm{d}t}$ to find the value of $t$ when the plant reaches its maximum height.[2]

Two days after it is planted, the height of the plant is 6.5 cm.

(c) Find an expression for $h$ in terms of $t$, for $0 \leq t \leq 15$.[5]

(d) Hence, find the maximum height of the plant.[2]

A second plant was planted at the same time as the first plant. Its height, $H$ cm, is modelled by $H = 2(1.2)^t$, where $0 \leq t \leq 15$.

(e) Write down the initial height of the second plant.[1]

Each day, the height of the second plant increases by $p\%$.

(f) Find the value of $p$.[2]

(g) Find the value of $t$ when the two plants have the same height.[2]

Rea is investigating how the number of healthy skin cells changes following infection by a virus. At the beginning of the investigation, she inserts 27000 virus particles into a sample of 1350 healthy skin cells.

Let $h$ be the number of healthy skin cells at time $t$, where $t$ is the number of hours after the investigation has begun. Let $v$ be the number of virus particles, in thousands, at time $t$.

Rea finds that the rate of change, $\frac{dh}{dt}$, is proportional to $\frac{h}{v}$.

(a) Given that $\frac{dh}{dt} = -5$ at time $t = 0$, show that $\frac{dh}{dt} = -\frac{h}{10v}$.[3]

Rea models the number of virus particles, in thousands, after $t$ hours as $v = 27 + 0.3t$.

(b) Find an expression for the number of healthy skin cells at time $t$.[7]

Rea asks her colleague Artem to attempt the same investigation. Artem's models differ from those of Rea.

Artem models the number of virus particles, in thousands, after $t$ hours as $v = 27 + 0.28t$ and finds that $h = 4380(27 + 0.28t)^{-0.36}$.

(c) Using Artem's models, predict how many hours it will take for the number of virus particles to be at least 100 times the number of healthy skin cells.[4]

A particle moves such that its velocity, $v$ metres per second, at time $t$ seconds, is given by $v = t \sin(t^2)$.

(a) Find an expression for the acceleration of the particle.[2]

(b) Hence, or otherwise, find its greatest acceleration for $0 \le t \le 8$.[2]

The particle starts at the origin.

(c) Find an expression for the displacement of the particle.[3]

(d) Hence show that the particle never has a negative displacement.[2]

A duck is sitting in a duck pond at point $A(7, 4, 0)$ relative to an origin $O$, where lengths are measured in metres and time, $t$, is measured in seconds. It takes off and flies in a straight line with vector equation

$$\mathbf{d} = \begin{pmatrix} 7 \\ 4 \\ 0 \end{pmatrix} + t \begin{pmatrix} 6 \\ 6 \\ 3 \end{pmatrix}$$

(a) Find the speed of the duck through the air (in $\text{m s}^{-1}$).[2]

A hawk hovering at position vector $\begin{pmatrix} -38 \\ 134 \\ 315 \end{pmatrix}$ relative to $O$, sees the duck take off and immediately dives from its position with constant velocity vector $\begin{pmatrix} 15 \\ -20 \\ -60 \end{pmatrix}$ to intercept the duck.

(b) Write down the vector equation for $\mathbf{h}$, that models the flight of the hawk.[1]

(c) Find the position vector at which the hawk intercepts the duck.[4]

A ship S is travelling with a constant velocity, $\mathbf{v}$, measured in kilometres per hour, where $$\mathbf{v}=\begin{pmatrix}-12\\15\end{pmatrix}$$

At time $t=0$ the ship is at a point $A(300, 100)$ relative to an origin $O$, where distances are measured in kilometres.

(a) Find the position vector $\overrightarrow{OS}$ of the ship at time $t$ hours.[1]

A lighthouse is located at a point $(129, 283)$.

(b) Find the value of $t$ when the ship will be closest to the lighthouse.[6]

An alarm will sound if the ship travels within 20 kilometres of the lighthouse.

(c) State whether the alarm will sound. Give a reason for your answer.[2]

(a) Find the indefinite integral $\int xe^{-x^2} dx$[4]

(b) Hence find the area bounded by the $x$-axis, the curve $y = xe^{-x^2}$ and the line $x = k$. Give your answer in terms of $k$.[3]