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Calculus - IB Questionbank | RevisionDojo

Practice Calculus with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ .

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

HLPaper 1

By using the substitution $u = \sec x$ or otherwise, find an expression for $\int_0^{\frac{\pi}{3}} \sec^n x \tan x \, dx$ in terms of $n$ , where $n$ is a non-zero real number.

[6]

Question 3

SLPaper 1

The function $j$ is defined for all $x \in \mathbb{R}$ . The line with equation $y = 6x - 1$ is the tangent to the graph of $j$ at $x = 4$ .

Write down the value of $j'(4)$ .

[1]

Find $j(4)$ .

[1]

The function $k$ is defined for all $x \in \mathbb{R}$ where $k(x) = x^2 - 3x$ and $m(x) = j(k(x))$ . Find $m(4)$ .

[3]

Hence find the equation of the tangent to the graph of $m$ at $x = 4$ .

[5]

Question 4

SL & HLPaper 1

A car accelerates from rest with an acceleration given by $a(t) = 4t - 2$ , where $a$ is in meters per second squared and $t$ is in seconds.

Find the velocity of the car as a function of time.

[4]

Calculate the displacement of the car in the first 5 seconds.

[5]

Question 5

HLPaper 1

Evaluate the limit of the function $g(x) = \frac{\ln(x)}{x - 1}$ as $x$ approaches 1.

Use L'Hôpital's Rule to find the limit of $g(x)$ as $x$ approaches 1.

[3]

Question 6

SLPaper 1

Given the derivative of a function is $h'(x)=e^x-2$ , determine the intervals where the original function $h(x)$ is increasing or decreasing.

[3]

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ .

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

HLPaper 1

By using the substitution $u = \sec x$ or otherwise, find an expression for $\int_0^{\frac{\pi}{3}} \sec^n x \tan x \, dx$ in terms of $n$ , where $n$ is a non-zero real number.

[6]

Question 3

SLPaper 1

The function $j$ is defined for all $x \in \mathbb{R}$ . The line with equation $y = 6x - 1$ is the tangent to the graph of $j$ at $x = 4$ .

Write down the value of $j'(4)$ .

[1]

Find $j(4)$ .

[1]

The function $k$ is defined for all $x \in \mathbb{R}$ where $k(x) = x^2 - 3x$ and $m(x) = j(k(x))$ . Find $m(4)$ .

[3]

Hence find the equation of the tangent to the graph of $m$ at $x = 4$ .

[5]

Question 4

SL & HLPaper 1

A car accelerates from rest with an acceleration given by $a(t) = 4t - 2$ , where $a$ is in meters per second squared and $t$ is in seconds.

Find the velocity of the car as a function of time.

[4]

Calculate the displacement of the car in the first 5 seconds.

[5]

Question 5

HLPaper 1

Evaluate the limit of the function $g(x) = \frac{\ln(x)}{x - 1}$ as $x$ approaches 1.

Use L'Hôpital's Rule to find the limit of $g(x)$ as $x$ approaches 1.

[3]

Question 6

SLPaper 1

Given the derivative of a function is $h'(x)=e^x-2$ , determine the intervals where the original function $h(x)$ is increasing or decreasing.

[3]

Paper

Level

Question 1

HLPaper 1

Consider the function $g(x) = e^x \cos(x)$ .

Find the derivative of the function $g(x) = e^x \cos(x)$ .

[2]

Determine the equation of the tangent line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Find the equation of the normal line to the curve $g(x) = e^x \cos(x)$ at the point where $x = 0$ .

[2]

Question 2

HLPaper 1

By using the substitution $u = \sec x$ or otherwise, find an expression for $\int_0^{\frac{\pi}{3}} \sec^n x \tan x \, dx$ in terms of $n$ , where $n$ is a non-zero real number.

[6]

Question 3

SLPaper 1

The function $j$ is defined for all $x \in \mathbb{R}$ . The line with equation $y = 6x - 1$ is the tangent to the graph of $j$ at $x = 4$ .

Write down the value of $j'(4)$ .

[1]

Find $j(4)$ .

[1]

The function $k$ is defined for all $x \in \mathbb{R}$ where $k(x) = x^2 - 3x$ and $m(x) = j(k(x))$ . Find $m(4)$ .

[3]

Hence find the equation of the tangent to the graph of $m$ at $x = 4$ .

[5]

Question 4

SL & HLPaper 1

A car accelerates from rest with an acceleration given by $a(t) = 4t - 2$ , where $a$ is in meters per second squared and $t$ is in seconds.

Find the velocity of the car as a function of time.

[4]

Calculate the displacement of the car in the first 5 seconds.

[5]

Question 5

HLPaper 1

Evaluate the limit of the function $g(x) = \frac{\ln(x)}{x - 1}$ as $x$ approaches 1.

Use L'Hôpital's Rule to find the limit of $g(x)$ as $x$ approaches 1.

[3]

Question 6

SLPaper 1

Given the derivative of a function is $h'(x)=e^x-2$ , determine the intervals where the original function $h(x)$ is increasing or decreasing.

[3]

Calculus Questionbank