Calculus – iGCSE Further Pure Mathematics Edexcel Revision

Definitions and Key Concepts of Calculus

Calculus is a branch of mathematics that deals with rates of change and the accumulation of quantities. There are two main types: differential calculus and integral calculus.
Differential calculus focuses on the concept of a derivative, which describes how a function changes as its input changes.
The derivative of a function represents the rate of change of the function at any given point, and it is found using a process known as differentiation.
Integral calculus involves the concept of an integral, which can be thought of as the area under a curve. It is the process of calculating the total accumulation of quantities, and it is found using a process known as integration.
The Fundamental Theorem of Calculus links differentiation and integration, showing that these two operations are essentially the reverse of each other.

Common rules for differentiation include the power rule, product rule, quotient rule, and chain rule. These allow calculation of the derivative of a wide variety of functions.
The power rule is used when differentiating a function of the form f(x) = x^n where n is any real number.
The product rule is useful when differentiating products of two or more functions, and the quotient rule is used for the quotient of two functions.
The chain rule is used when differentiating composite functions, which are functions composed of other functions.

Basic techniques for integration include direct integration, substitution, integration by parts, and the use of partial fractions. These help in calculating the integral of a variety of functions.
Direct integration is used when the integral is straightforward, based on standard forms.
The substitution method, also known as u-substitution, is useful when the integral is of a composite function.
Integration by parts is used when the integral is the product of two functions.
Partial fractions can simplify complex fractions to easier-to-integrate terms before performing integration.

Calculus plays a vital role in various fields of science and engineering. It can be used to calculate rates of change, areas under curves, and summations of infinite sequences.
The study of motion, known as kinematics, makes heavy use of calculus.
The optimisation problems, which involve finding the maximum or minimum values of functions, can be solved using the tools of calculus.
With calculus, it is possible to determine quantities such as the length of curves, volume of solids of revolution, and the surface area of solids.
Understanding calculus is also fundamental to the study of more advanced topics, like differential equations and vector calculus.