A-level Mathematics/AQA/MPC3

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Functions[edit]

Mappings and functions[edit]

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]

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 ,

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;

Differentiate

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

Let

Now we can rewrite the original function,

We can now differentiate each part;

and

Now applying the rule above;

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:

Or in Lagrange notation:

If ,

then

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