Parabola
Definition 5. The locus of points for which the distance to a fixed point equals the distance to a given straight line (a directrix) is called a parabola.
L
et
us draw the perpendicular through a fixed point to the given straight
line and take it for the x-axis.
From the middle point of the segment joining the focus to the given
straight line we draw a perpendicular and take it for the y-axis.
p М(х;у)
r
0
x
To derive the equation of the parabola, we take an arbitrary point М(х;у) on it and write down the characteristic feature of a parabola as a mathematical formula.
The distance from the focus to the directrex is called the parameter of the parabola and denoted by p. Let us find the distance from the point М(х;у) to the focus:
,
and 
=
.
By definition, these distances are equal:
.
Let us transform this, relation by squaring both sides:
;
.
We obtain
.
This is the classical equation of a parabola.
The parabola passes through the origin (0;0), because it satisfies equation (20).
Suppose that the parameter is a positive number р>0; then, since у2>0, we have x>0, and the parabola is contained in the right half-plane. If p<0, then x<0, and the parabola is contained in the left half-plane
y
у
p>0 p<0
0 x 0 х
M0(x0,y0)
C
onsider
the equation of a parabola in the “school” form
.
Let us analyze this equation by analogy with (20):
if p>0,
then y>0,
and the branches of the parabola are directed upward; if p<0,
then y<0,
and the branches of the parabola are directed downward.
p
>0
y
p<0
y
0 x
0 x
The eccentricity of the parabola, that is, the ratio of the focal radius to the distance from a point to the directrix, equals 1, i.e.,
.
Lecture 4 Function. The function limit. Fundamental theorems on limits. Infinitely small and infinitely large quantities
1. Functions
Definition
1.
If to each value of an independent variable x
one value of a variable y
is
assigned then у
is called a single-valued function of the argument х
and is denoted by
.
Definition 2. The set of values of х for which the value of a function is defined is called the domain (of definition) of this function.
Definition 3. The set of values of the argument у corresponding to the values х from the domain is called the range of the function.
The domain of a function may be of one of the following types: (1) a closed interval [a;b]; (2) an open interval ]a;b[ , or (a;b); (3) a half-open interval [a;b[ or ]a;b].
Definition 4. The locus of the points with coordinates (х;у) satisfying an equation у=f(x) is called the graph the of function .
Definition
5.
A
function
is even if
.
A function is odd if
.
Definition
6. A
function
is
said to be increasing if larger values of the function correspond to
larger values of the argument, i.e.,
implies
.
Definition
7. A
function
is
said to be decreasing if smaller values of the function correspond to
larger values of the argument, i.e.,
implies
.
Elementary functions. The following functions are considered elementary: y=xn (a power function); у=ах (an exponential); у=logаx, у=lgx (a logarithmic function); y=cosx, у=sinx, у=tanx, у=cotx, (trigonometric functions); y=arccosx, у=arcsinx, у=arctanx, у=arccotx (antitrigonometric functions). These functions are studied at school in detail, and do not dwell on them.
Definition
8. A
function of the form
,
where
,
is called a composite function and denoted by
;
it is also
the composition of the functions
f and
.
Definition 9. A function is said to be periodic with period Т if f(x+T)=f(x) for all х from the domain.
Definition 10. A function is said to be implicit if its equation is given in the form F(x;y)=0, i.e., у is not explicitly expressed in terms of х.