First Year:
Basic Differentiation
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
First year:
(I)
Basic functions
Polynomials
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 (
First year:
(I)
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 ).
(product rule)
Now, integrate on both sides:
u
u
' ( 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.
(I)
First order linear differential equation
A first order linear different
First year:
(I)
Complex numbers
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
i 1.
Note that since i 1 , we have i 2 1 .
Let z a ib .

T
First year:
(I)
Tangent, maximum and minimum
Tangents
The tangent to the graph of a function f at the point c, f (c) is a line such
that:

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.
2
n
Consider the partial sum S n 1 x x . x (called partial sum because it
only goes as
First year:
(I)
Integration and partial fractions
The method
1
.
( 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
1
i
First year:
(I)
Integration by substitution
The method
Remember that
f ( x)dx F ( x) , where
dF ( x)
f ( x) so, in a way, integration
dx
is the opposite of differentiation.
Integration by substitution is in this case the opposite of the chain rule for
di
First year:
Drawing graphs
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
First year:
(I)
Basic integration
Integration: areas
The symbol
b
f ( x)dx
a
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
First year:
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
3
like e x , cos(e x ) etc The chain rule tells you h