A-level Mathematics/AQA/MPC3

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Functions[edit]

Domain and range of a function[edit]

In general:

  • is called the image of .
  • The set of permitted values is called the domain of the function
  • The set of all images is called the range of the function

Modulus function[edit]

The modulus of , written , is defined as

Differentiation[edit]

Chain rule[edit]

The chain rule states that:

If is a function of , and is a function of ,

Product rule[edit]

The product rule states that:

If , where and are both functions of , then

An alternative way of writing the product rule is:

Quotient rule[edit]

The quotient rule states that:

If , where and are functions of , then

An alternative way of writing the quotient rule is:

x as a function of y[edit]

In general,

Trigonometric functions[edit]

The functions cosec θ, sec θ and cot θ[edit]



Standard trigonometric identities[edit]



Differentiation of sin x, cos x and tan x[edit]



Integration of sin(kx) and cos(kx)[edit]

In general,


Exponentials and logarithms[edit]

Differentiating exponentials and logarithms[edit]

In general,


Natural logarithms[edit]

If , then


It follows from this result that


Integration[edit]

Integration by parts[edit]

Standard integrals[edit]


Volumes of revolution[edit]

The volume of the solid formed when the area under the curve , between and , is rotated through 360° about the -axis is given by:


The volume of the solid formed when the area under the curve , between and , is rotated through 360° about the -axis is given by:

Numerical methods[edit]

Iterative methods[edit]

An iterative method is a process that is repeated to produce a sequence of approximations to the required solution.

Numerical integration[edit]

Mid ordinate rule



Simpson's rule