Calculus
In this topic we will look at
Rates of Change- Tangents and Normals
- Chain Rule
- Product and Quotient Rule
- Local Maximum and Minimum Points and Points of Inflexion
- Applications and Optimisation
- Implicit Differentiation
- Related Rates of Change
- L'Hôpital's Rule
- Maclaurin Series
- Indefinite Integration
- Definite Integration
- Areas between Graphs
- Volumes of Revolution
- Kinematics
- Integration by Substitution
- Integration by Parts
- Differential Equations
On each page, you will find
- Video tutorials explaining key concepts.
- Concise revision notes providing you with reminders of all the key content you need
- Onscreen quizzes giving you instant feedback
- Exam style questions with hints, detailed solutions and explanation
Introducing Derivatives
Differentiation allows us to determine rates of change. When we differentiate a function \(f\), we obtain its derivative function \(f'\). The value \(f'(x)\) gives the instantaneous rate of...
Graphs and Derivatives
On this page, we will look at graphs and derivatives. We will practise sketching derivative functions and carefully consider stationary points (local maxima, local minima and stationary points of inflexion),...
Chain Rule
The chain rule is used to differentiate composite functions. The rule itself is quite simple, but it is important to recognise when it is needed and to become fluent in applying it. This can save time...
Product and Quotient Rule
The product rule is a formula that we can use to differentiate the product of two (or more) functions. The quotient rule is for the quotient of two functions (one function divided by another). The rules...
Equation of Tangent and Normal
On this page, we look at how to find the equation of a tangent and the equation of a normal to a curve. A tangent to a curve at a point is the straight line through that point whose gradient is equal...
Equation of Tangent and Normal
On this page, we look at how to find the equation of a tangent and the equation of a normal to a curve. A tangent to a curve at a point is the straight line through that point whose gradient is equal...
Optimisation
Optimisation questions involve finding the best solution to a problem. Usually, that is the maximum or minimum value of a function. Questions on this topic require you to take information from a practical...
Implicit Differentiation HL
This page is all about implicit equations and how we find their gradients. Implicit equations are equations which are not written explicitly. Examination questions typically ask you to find the equation...
Related Rates of Change HL
This page is about finding related rates of change, otherwise known as connected rates of change. In questions on this topic, you will be given the rate of change of one quantity and be required to find...
L'Hôpital's Rule HL
You will probably have already considered limits in work on the sum of a geometric series, rational functions and asymptotes and differentiation from first principles. On this page, we will take the...
Maclaurin Series HL
A Maclaurin series is a power series that enables us to find a polynomial that approximates a function that cannot be written algebraically. It is what calculators and computers use to work with transcendental...
Standard Integrals SL
In this page, we will look at some of the standard integration techniques that you need to know about involving x^n, \sin x, \cos x, \dfrac{1}{x}, e^x and the composite of these with...
Standard Integrals HL
In this page, we will look at some of the standard integration techniques that you need to know about involving x^n, \sin x, \cos x, \dfrac{1}{x}, e^x, a^x, \dfrac{1}{a^2+x^2}, \dfrac{1}{\sqrt{a^2-x^2}} and...
Definite Integration
Integration is sometimes called antidifferentiation, as it is the opposite process of differentiation. When we find an indefinite integral, we find a function with an arbitrary constant, C. Whereas, when...
Area between Graphs SL
On this page, we'll look at how we can use integration to find areas. This is a really common question in examinations. The technique is not too hard, but there are just a couple of pitfalls to avoid...
Area between Graphs HL
On this page, we'll look at how we can use integration to find areas. This is a really common question in examinations. The technique is not too hard, but there are just a couple of pitfalls to avoid...
Volume of Revolution HL
On this page, we will learn how to find the volume generated by rotating a region about the x-axis and the y-axis. The formulae for these are not difficult to use. The difficulty often comes with applying...
Integration by Substitution SL
Integration by substitution or U-substitution is a method that will help you integrate many different functions. By changing the variable of the integrand, we can make an apparently difficult problem...
Integration by Substitution HL
Integration by substitution or U-substitution is a method that will help you integrate many different functions. By changing the variable of the integrand, we can make an apparently difficult problem...
Integration by Parts
Integration by parts is a method of integration that we use to integrate the product (usually!) of two functions. The aim is to change this product into another one that is easier to integrate. Although...
Differential Equations HL - Separable Variables
If you want to describe the world around you, be it the forces acting on a body, the growth of a virus or the temperature of the coffee in your cup, you will be dealing with differential equations. On...
Differential Equations HL - Integrating Factor
On this page, we will look at another type of differential equation: first-order linear differential equations of the form \displaystyle \dfrac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x), which can be solved using an integrating factor. There are many applications...
Differential Equations HL - Homogeneous
When a first-order differential equation is not immediately separable or linear, it may still be possible to solve it using a substitution. A first-order homogeneous differential equation can be written...
Navigating the pages
Each of the topic pages is designed in the same way:
Had an explanation in class that you didn't quite understand? The short, high quality videos in this section will help you understand all the key concepts from each topic. Some are designed to explain concepts from the beginning and others are focussed on very specific skills and typical exam questions. All videos come with a set of notes that you can keep.
This section is all about getting practice. Before you try any exam-style questions, you will need a quick check to see if you have understood all the concepts from the page. These onscreen quizzes allow you to get just that. You can get instant feedback on your answers including advice about how to tackle the question and worked solutions.
This is the section where you can practise questions just like you the ones that you will get in the exam. If you need some help getting started, there are hints. Each question has a full solution with explanation
Each of the questions is marked by difficulty level
Easy
Medium
Hard