Calculus: 36 IB Maths AI SL practice questions

Consider the function $f(x) = x^3 - 2x^2 + 5x - 7$.

(a) Find $f(1)$.[2]

(b) Find $f'(x)$.[2]

(c) Find the equation of the line that is tangent to the curve $y = f(x)$ at $x = 1$.[2]

Consider the curve $y = 5x^3 - 3x$.

(a) Find $\frac{dy}{dx}$.[2]

The curve has a tangent at the point $P(-1, -2)$.

(b) Find the gradient of this tangent at point $P$.[2]

(c) Find the equation of this tangent. Give your answer in the form $y = mx + c$.[2]

Consider the curve $y = x^3 + kx$.

(a) Write down $\frac{dy}{dx}$.[1]

The curve has a local minimum at the point where $x = 2$.

(b) Find the value of $k$.[3]

(c) Find the value of $y$ at this local minimum.[2]

When the brakes of a car are fully applied the car will continue to travel some distance before it completely stops. This stopping distance, $d$, in metres is directly proportional to the square of the speed of the car, $v$, in kilometres per hour ($\text{km}\,\text{h}^{-1}$).

When a car is travelling at a speed of $50 \text{ km}\,\text{h}^{-1}$ it will travel $12.3 \text{ m}$ after the brakes are fully applied before it completely stops.

(a) Determine an equation for $d$ in terms of $v$.[2]

The police can use this equation to estimate if cars are exceeding the speed limit.

A car is found to have travelled $33 \text{ m}$, after fully applying its brakes, before it completely stopped.

(b) Use your equation from part (a) to estimate the speed at which this car was travelling before the brakes were applied.[2]

(c) After the brakes have been fully applied, identify one other variable besides speed that could affect stopping distance.[1]

Consider the curve $y = 5x^2 - \frac{3}{x^3}$.

(a) Find $\frac{dy}{dx}$.[3]

(b) Write down the gradient of the curve at $x = 1$.[1]

(c) Hence, find the equation of the normal to the curve at $x = 1$.[3]

Mongolia is one of the coldest countries in the world, with the air temperature in winter as low as $-40°C$. It can feel even colder if a wind is blowing. This effect is called wind chill.

On a day when the air temperature is $-40°C$, the approximate wind chill index, $W$, can be calculated by the equation

$$W = -34.1 - 7.33 \ln(v),$$

where $v$ is the wind speed, in kilometres per hour (km h$^{-1}$).

(a) Find the approximate wind chill index on a day when the air temperature is $-40°C$ and the wind speed is measured as 13 km h$^{-1}$.[2]

Due to errors in the measuring device, the percentage error in the approximate wind chill index calculated in part (a) could be as high as 6%.

(b) Predict the maximum and the minimum wind chill index for this day.[3]

Line $L_1$ is tangent to the graph of a function $f(x)$ at the point P$(3, -1)$. Line $L_2$ is given by the equation $y = -\frac{1}{2}x - \frac{5}{2}$ and is perpendicular to $L_1$.

(a) Write down the gradient of $L_1$.[1]

(b) Find the equation of $L_1$ in the form $y = mx + c$.[2]

(c) Show that $L_2$ is not the line that is normal to $f(x)$ at point P.[2]

(a) Expand the expression $x(2x^3 - 1)$[2]

(b) Differentiate $f(x) = x(2x^3 - 1)$[2]

(c) Find the $x$-coordinate of the local minimum of the curve $y = f(x)$[2]

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]

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]

The number of words that a child understands is modelled by the equation

$$y = -167 + 6264\log_{10} x, \text{ for } x \in \mathbb{R}^+,$$

where $x$ is the number of years since the child was born, and $y$ is the number of words the child understands.

(a) Use the model to predict the number of words that a child understands when $x = 10$.[2]

To be fluent in a language, a child must understand at least 5000 words.

(b) Use the model to predict how long it takes for a child to become fluent. Give your answer correct to four significant figures.[2]

(c) Determine if the model is realistic for a child on their first birthday. Give a reason to support your decision.[2]

Consider the function $f(x) = \frac{x^4}{4}$.

(a) Find $f'(x)$.[1]

(b) Find the gradient of the graph of $f$ at $x = -\frac{1}{2}$.[2]

(c) Find the $x$-coordinate of the point at which the normal to the graph of $f$ has gradient $-\frac{1}{8}$.[3]

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 two limitations of using $h(t)$ to model the height of the basketball.[2]

Consider the function $f(x) = ax^2 + c$.

(a) Find $f'(x)$.[1]

Point $A(-2, 5)$ lies on the graph of $y = f(x)$. The gradient of the tangent to this graph at $A$ is $-6$.

(b) Find the value of $a$.[3]

(c) Find the value of $c$.[2]

Let $f(x) = x^4$.

(a) Write down $f'(x)$.[1]

Point P $(2, 16)$ lies on the graph of $f$.

(b) Find the gradient of the tangent to the graph of $y = f(x)$ at P.[2]

(c) Find the equation of the normal to the graph at P. Give your answer in the form $ax + by + d = 0$, where $a$, $b$ and $d$ are integers.[3]