One of the aims of the first years maths courses is to bring all students within a
group to the same level. In first years calculus in particular, differentiation and
integration are central notions.
Below you will find a
A polynomial f is a function of the form
f ( x) a n x n . a1 x a 0 ,
where a n ,., a 0 are constants. The polynomial f is said to be of degree n if
a n 0 .
We will only consider polynomials of degree 2.
Let f (
Integration by parts
Formula and example
It is relatively easy to derive the formula:
u ( x)v( x) ' u ' ( x)v( x) u ( x)v' ( x)
u ' ( x)v( x) u ( x)v( x) ' u ( x)v ' ( x ).
Now, integrate on both sides:
' ( x)v( x)dx
First year: Introduction to differential equations
We will only talk about first and second order homogeneous and nonhomogeneous differential equations with constant coefficients.
First order linear differential equation
A first order linear different
Definition, addition and multiplication
A complex number is a number of the form a ib , where a and b are real
numbers and where i is an imaginary number such that
Note that since i 1 , we have i 2 1 .
Let z a ib .
Tangent, maximum and minimum
The tangent to the graph of a function f at the point c, f (c) is a line such
its slope is equal to f ' (c).
it passes through the point c, f (c) .
The equation of the tangent to the graph of a
First year: Introduction to power series
We limit ourselves to the study of one particular power series, although youll
see it is one of the most useful basic examples.
Consider the partial sum S n 1 x x . x (called partial sum because it
only goes as
Integration and partial fractions
( x 2)( x 3) 2
Note that neither substitution nor integration by parts is likely to help, here.
This method allows you to integrate functions of the form
However, it is possible to split
Integration by substitution
f ( x)dx F ( x) , where
dF ( x)
f ( x) so, in a way, integration
is the opposite of differentiation.
Integration by substitution is in this case the opposite of the chain rule for
In order to draw graphs, you first have to be able to evaluate a function f (x)
at a given point x .
For example, evaluate the function f ( x) 2( x 2 1) at the point x 3 :
f (3) 2(32 1) 20.
Now evaluate the function f at the poi
f ( x)dx
means area of the set situated between the graph of f , the x axis, the line
of equation x a and the line of equation x b .
Its easier to understand this notion through an exampl
Differentiation: chain rule
So far we have been differentiating simple functions, like x 3 , e x or cos(x ) but
most of the functions you are likely to encounter will be composite functions
like e x , cos(e x ) etc The chain rule tells you h