Recall that a nite sequence has the form fyngN n=0 and an innite sequence has the form fyng1 n=0. A dierence equation of order m has the form (1) yn+m = F(n; yn; yn+1; : : : ; yn+m

An ordinary dierential equation of order k has the form (1) F  x; y; dy dx ; d2y dx2 ; : : : ; dky dxk  = 0; where F is a function of k+2 variables. We will usually write ODE as an abreviation for ordinary dierential equation. A function y(x) is a solution of the ODE (1) in the interval a < x < b if it satises the equation (1) for all x in this interval. We will usually be interested in solving an initial value problem. For an ODE of order k, this has the form Reca...

These notes introduce the notion of derivatives for functions of several variables. It is worthwhile to rst recall the derivative of a real-valued function of a single variable. Let f : R ! R, and xed x0 2 R. The classical denition of f having a derivative f0(x0) at x0 is that the limit Notice that any linear map from R to itself is just multiplication by some constant a. This is the important conceptual leap we must make: the derivative of a function at a point is not a number, or a ve...

In these notes one can find almost everything important about trigonometry: definitions and concepts, formulas and exercises; solved(to understand the methodology) and unsolved (to practice by yourself).

In this document, the quadratic formula (for roots of x) and x- and y-values of the turning point is derived from a general quadratic equation.