Welcome to block 3 Calculus.
My name's Rosa Estela and I am going to present this video.
The contents of this video is
derivatives and integration.
First of all, we want to define the concept of differentiability
the derivatives of basic functions
and geometric interpretation of derivative.
We will talk about partial derivatives
and vector gradient and descent directions
at the point for a differentiable function.
Also, we'll talk about the chain rule.
About integration,
we will define the concept of primitive,
we will see the primitives of basic functions
the definite integral,
the geometric interpretation of integration,
and the Fundamental Theorem of Calculus.
Learning objectives.
Why do we work with this concept?
Because differentiation is used to study
the small change of a quantity with respect
to unit change of another.
In Calculus, derivatives are incredibly important
because they allow individuals to study
how functions change over time.
In other words, derivatives provide information
about the direction a function is moving at any given point.
The derivative and integral
permeate all aspects
of modeling nature in the physical sciences.
Differentiation is used to find
the slope of a function at a point.
Integration is used to find the area
under the curve of a function.
So in physics
and engineering, differentiation helps us
understand motion and change.
By differentiating displacement
with respect to time, we obtain velocity and acceleration,
and this knowledge is crucial in designing
vehicles, predicting the behavior of objects in motion,
and developing control systems from robotics. For example,
the velocity of a car can be calculated by taking the derivative
of its position with respect to time.
Let's start with the concept of differentiability.
Let be f a real function.
We say that f is differentiable
in x=x_0 if exists
this limit that you can see on the screen
and we note like that.
A real function f
is differentiable in its domain,
if it's differentiable in each point
of the domain.
I already know that you have studied
the derivatives of basic functions,
but here you have a reminder.
If f is a real function,
then these are the
derivatives of the basic functions.
You can see.
Here you have trigonometric functions,
also the basic functions
and their derivatives.
What's the geometric
interpretation of a real function f?
As you can see in the graph,
the derivative of the real function f in a point
x equal to a is the slope
of the tangent line in the point x=a
and is the result of this limit.
If we have several variables,
for example, this function f
that represents a surface, then
if function f is differentiable
there is a tangent plane
at the point we are studying,
as you can see in the figure.
So, partial derivatives.
In mathematics, a partial derivative
of a function of several variables is its derivative
with respect to one of those variables,
with the others heald constant.
This definition, as in one dimension,
is given by the calculation of a limit.
In this case, the limit is this one.
This is the partial derivative of f
at a point a with respect to the i-th variable.
An important definition
is the gradient vector.
Let be f a function of n variables,
the gradient of f is denoted like this
and is the vector with components
all the partial derivatives of f.
For example, let be this function f
The gradient of f is,
in this case, a vector of three components
because we have three variables.
See that the first parcial respect
x is y^2,
the partial
respect y is 2xy,
and the partial respect z is 2.
So the gradient is this vector
of three components.
How about descent directions
at a point for a differentiable function.
If we are working on several variables
with a differentiable function,
then a nonzero vector v is a descent direction
if and only if, this product of vectors is negative.
For example, minus gradient
is a descent direction at any x.
The chain rule.
Let be now two real functions
f and g, differentiables
and f(I) included in J,
then we can talk about the composition
of these functions,
and you have the notation here.
We have also expressed the chain rule in the Leibniz notation
used most in physics and engineering.
If a variable z depends on the variable y,
which itself depends on the variable x,
that is, y and z are dependent variables,
then z depends on x as well,
via the intermediate variable y.
In this case, the chain rule is expressed
as you can see in the screen.
Let's see now integration.
First of all, the concept of primitive.
F is a primitive of a real function f
if F is continuous in the interval [a,b]
and differentiable in the open interval (a,b)
with the derivative of F is f
and denoted by this symbol. In Calculus,
is the same, an antiderivative,
inverse derivative, primitive function, primitive integral or indefinite
integral. Is the same concept.
Here we have primitives
of basic functions. Note that is the antiderivative.
So you can see that these is related
with the differentiation of functions.
You have here
some basic functions
and their primitives.
Also
as a reminder.
And here
you have important
methodologies to calculate
the primitive of a function.
Lineality.
The primitive of a sum of functions is the sum of the primitives.
And if we multiply a function by a scalar,
the scalar comes out of the integral.
Integration by parts
and important, change of variables.
They are methodologies to calculate
the primitive of a function.
Definite integrals
A definite integral
is an integral
with upper and lower limits
If x is restricted to lie on the real line,
the definite integral is known as a Riemann integral
and the representation graphically
is the area under the curve of the function.
This is plotted with Geogebra, but you can plot
with another program if you want.
Let's talk about
Fundamental theorem of Calculus.
This theorem relate derivatives
and integrals with one another.
The first theorem
fundamental of Calculus say that for f a real-valued
continuous function on an open interval I,
and a any number in I,
if F is defined by the integral
then the derivative of capital F
is the function f
at each number in the domain I.
The second fundamental
theorem of Calculus also known as Barrow's rule
says that, if f is a real-valued continuous function on the closed
interval [a,b] and F is
the indefinite integral of f on [a,b],
then we can calculate this number, not a function now,
this is a number, like the difference
of evaluating F in b and a.
So you can work
with this references here.
Two books. The first is a really
nice book in English
and you can work with these concepts.
You have also two MOOCs El Lenguaje de la IngenierĂ­a in Spanish
and MathCom T.I.M.E in English.
And then to plot, I recommend you Geogebra, and to calculate
WolframAlpha ,for example.
And with Wikipedia you can always
work with any concept.
Enjoy differentiation and integration!