Welcome to block three Calculus.
And now to practice with differentiation and integration.
The same content of the theory video,
and now we will see some examples about this.
First of all,
an example of the derivative of a function.
One of the most important examples is to find
maximum and minimum values of a function.
In this case, find the absolute
maximum and minimum value of this function,
a polynomial, in this domain,
a closed interval.
Since f is a continuous
function on a closed interval
then by Weierstrass theorem, f has maximum
and minimum in its a domain.
Our function is polynomial, so is in each point differentiable.
Then, the only critical numbers of f are in the points
that the derivative of f is equal to zero.
In this case,
x=0 and x=2
are critical numbers.
Notice that each of these critical numbers
lies in the interval open interval
of the domain.
Then, the values of f at these critical numbers
are f(0)=1 and f(2)=-3.
Then we compute the values of f
at the endpoints of the interval
-1/2 and 4.
Comparing these values, you can see
that the maximum value is
17 for x=4
and the absolute minimum value
is -3 for x=2.
Let's see another example
of the derivative of a function.
Differentiate this function h.
See that if we define
f and g these functions,
we can compose f and g
to get the function h.
And then the chain rule tell us
that the derivative of
h(x) is minus tangent of x
applying the chain rule.
Let's see
another example.
You know that is not the same
sin(x^2) than (sin x)^2.
If you calculate the derivative
of each of these function,
you can see that the derivative is not the same,
of course.
And to work with the other notation,
I put here a classic problem
by George F. Simmons that says:
If a car travels twice as fast as a bicycle
and the bicycle is four times
as fast as a walking man,
then the car travels eight times
as fast as the man.
Well, to prove this,
we can use the chain rule.
Let's z,
y and x
be the variable positions of the car,
the bicycle and a walking man, respectively.
The rate of change of relative positions of the car and the bicycle
is this one, two and four.
So, the rate of change of the
relative positions of the car and the walking man is the product.
So, effectively
the car travels eight times
as fast as the man.
Let's see now, an example
with several variables, in this case 2 variables, partial derivatives.
Here you have the wave equation
for a function u
of two variables.
This example ask to verify
that the function u(x,t)=sin(x-at)
satisfies the wave equation.
And what do we need to do?
We need to calculate these partial derivatives.
In this case second partial derivatives.
So, first we calculate the first partial derivative
of u with respect to x and t,
and then, we calculate the second partial derivative
See the results,
the function u
verifies the wave equation,
just as we wanted to prove.
And directional derivatives.
Here we have a real function of two variables,
x and y and f defined as
f(x,y)=xe^y
The problem is,
in what direction does f have the maximum rate of change
at the point (2,0)?
And what is the maximum rate of change?
The function f
increases faster in the direction of the gradient vector
and decreases faster
in the direction of minus gradient.
In this case, they ask for the maximum rate of change
so, we consider the gradient
of the function at the point as the direction.
We calculate the partial derivatives,
in this case with respect to x and respect to y, we calculate
and we evaluate at the point (2,0).
So (1,2) is the direction
what f takes the maximum rate
of change at this point (2,0).
And what's the maximum rate of change?
It's the norm of this vector,
we calculate the norm and results square of five.
And geometric interpretation,
in this case of a real function
with two variables.
We have a surface z=f(x,y)
and we suppose the function f
has continuous partial derivatives.
Then it exists the tangent plane to the surface at a point P
and is the equation you can see. As an example, find the tangent plane
to the elliptic paraboloid at the point (1,1,3).
So we need to calculate the partial derivatives,
in this case two, and we put these values in the equation
that we have seen. The result is the tangent plane
to the surface in this case of
the elliptic paraboloid.
You can plot it with different programs,
both, the surfaces and the tangent planes.
And to finish with these examples
of derivation and integration, let's see one of integration.
This problem says:
Find the area of the region bounded by the curves
y=sinx, y=cosx,
x=0 and
x=pi/2.
You can plot these functions and you can see that
we need to calculate
the red and blue regions.
We can calculate this
with the absolute value of the difference of these two functions.
Why the absolute value?
Because you can see that
in A_ 1 the upper function is the red one,
and in A_2 the upper function
is the blue one.
So we calculate, see that
we need to calculate the intersection
points of these two functions.
In this case, x=pi/4
and then we separate the integration
and we calculate knowing that
the primitives or these functions are what you see in the screen.
This is an example of calculating areas.
More examples in the references,
in the books, in the MOOCs you can plot with Geogebra,
you can plot and calculate with Wolfram Alpha
and Wikipedia also is good to know.
So enjoy with differentiation and integration and thank you!