A-level Mathematics/AQA/MPC3

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Functions[edit | edit source]

Mappings and functions[edit | edit source]

We think of a function as an operation that takes one number and transforms it into another number. A mapping is a more general type of function. It is simply a way to relate a number in one set, to a number in another set. Let us look at three different types of mappings:

  • one-to-one - this mapping gives one unique output for each input.
  • many-to-one - this type of mapping will produce the same output for more than one value of .
  • one-to-many - this mapping produces more than one output for each input.

Only the first two of these mappings are functions. An example of a mapping which is not a function is

Domain and range of a function[edit | edit source]

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 | edit source]

The modulus of , written , is defined as

Differentiation[edit | edit source]

Chain rule[edit | edit source]

The chain rule states that:

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

As you can see from above, the first step is to notice that we have a function that we can break down into two, each of which we know how to differentiate. Also, the function is of the form . The process is then to assign a variable to the inner function, usually , and use the rule above;


We can see that this is of the correct form, and we know how to differentiate each bit.


Now we can rewrite the original function,

We can now differentiate each part;


Now applying the rule above;

Product rule[edit | edit source]

The product rule states that:

If , where and are both functions of , then

An alternative way of writing the product rule is:

Or in Lagrange notation:

If ,


Quotient rule[edit | edit source]

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 | edit source]

In general,

Trigonometric functions[edit | edit source]

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

Standard trigonometric identities[edit | edit source]

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

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

In general,

Exponentials and logarithms[edit | edit source]

Differentiating exponentials and logarithms[edit | edit source]

In general,

Natural logarithms[edit | edit source]

If , then

It follows from this result that

Integration[edit | edit source]

Integration by parts[edit | edit source]

Standard integrals[edit | edit source]

Volumes of revolution[edit | edit source]

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 | edit source]

Iterative methods[edit | edit source]

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

Numerical integration[edit | edit source]

Mid ordinate rule

Simpson's rule