A-level Mathematics/AQA/MPC3

FunctionsEdit

Mappings and functionsEdit

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 functionEdit

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 functionEdit

The modulus of  , written  , is defined as

 

DifferentiationEdit

Chain ruleEdit

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 ruleEdit

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 ruleEdit

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 yEdit

In general,

 

Trigonometric functionsEdit

The functions cosec θ, sec θ and cot θEdit

 

 

 

Standard trigonometric identitiesEdit

 

 

 

Differentiation of sin x, cos x and tan xEdit

 

 

 

Integration of sin(kx) and cos(kx)Edit

In general,

 

 

Exponentials and logarithmsEdit

Differentiating exponentials and logarithmsEdit

In general,

 

 

Natural logarithmsEdit

If  , then

 

It follows from this result that

 

 

IntegrationEdit

Integration by partsEdit

 

Standard integralsEdit

 

 

Volumes of revolutionEdit

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 methodsEdit

Iterative methodsEdit

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

Numerical integrationEdit

Mid ordinate rule

 
 

Simpson's rule