Welcome to block 3 Calculus.
My name is Rosa Estela,
and I'm going to present this video
with examples working with functions.
First example, we'll work
with asymptote of the graph of a
real function f.
Remember that the asymptote
of the graph of a function is called a straight line
that the graph of that function continually approaches.
There are three types of asymptote: vertical asymptote,
horizontal asymptote and oblique asymptote.
Vertical asymptote is when
x= constant , horizontal when
y= constant
and oblique y is a line
not parallel tor perpendicular to the axes.
For example,
see the first example this function f.
(All of them are real functions)
f(x)=1/(x-1)
if x tends to 1
the function f tends to infinity.
So this function has vertical asymptote the line x equals one.
The other example
f(x)=1/(x^2+1)
when x tends to infinity,
the function goes to zero
as you can see in the graph.
So the line
y=0 is an horizontal asymptote.
Another example,
the function
f(x)=x^3/(x^2+1)
This function
graphycally represented
in black
has the blue line
as an oblique asymptote.
The line y=x
holds the properties of asymptote.
Now this example is to see the graph of the function.
You can go to Geogebra
this is remember,
a parameterized curve. You put x
as a function of t, and y as a function of t,
and then giving values
you have the curve that represent that function.
This is an example of
parameterized curve.
You can also represent surfaces
giving functions.
For example, the ellipsoid
or the hyperbolic paraboloid
which is the second equation.
You can play with Geogebra
there are a lot of options.
This is another graphic of surfaces
it's similar than the previous cases
but now
we are using WolframAlpha.
You can play. Here also are represented
the intersection with planes.
Another example,
we have a pyramid
and if we ntersect by a plane, we have a triangle.
This is the intersection about surfaces.
Now,
another example
working with WolframAlpha.
You have here as you can see two surfaces
given by the functions h and g
If you intersect,
in this case, you have a curve
painted in blue color.
You're seeing the intersection of these two surfaces.
Let's see here
an example of inverse of a function,
because you could be wrong and confused
with the inverse of the function f(x)
The inverse is not f(x)=1/f(x). It's not true.
Here you can see the function
f(x)=1/x
the domain is all the real numbers except zero
and the inverse of the function is the same function.
And remember the symmetry in the graphic
with the line
y=x.
Well,
example of continuity.
Here you have an example of continuity and this function is continue
in every point of the domain that you can plot.
See the graphic of the function.
And here you have an example of discontinuity
because you can see in the point x=-2
you have a jump.
And in the point x=4
see that the function when you approach from the right,
the function goes to infinity.
So this is a discontinuity,
in this case,
essential discontinuity.
You can work with this example
a little more.
References
you have books and you have MOOCs
with theory and practice.
There are two MOOCs here, one
in Spanish, El Lenguaje de la IngenierĂ­a, and another in English,
The MathCom T.I.M.E.
in the web page of the moocs UPC
and also you can work with Geogebra
or WolframAlpha.
If you need some specific definition
also you can go to Wikipedia, you know.
So enjoy with functions!
Thank you