Solving integral functional equation using integration and properties of definite integral.Olympiad

Tchebychev polynomials few few elements definitions and properties Tn(x)cos(n arcos(x))
As the cosine of n r cosine of x, We will find the first few of paranos. We will find the polynomial order, zero of order. One order two and three. We can also find the recursive relationship that We have between n and N plus 1 and N minus 1. These are like recursive relation can help us find any polynomial that we need. We will come later and we're going to prove that we can find the nodes and the approximation by polynomials of chubby chief. This is a very great relationship that we can have them. And we can see that we can use them in interpolation and find in polynomials that are easy to follow and to master.
In using integration bypart. Sometimes We don't know which Function to choose to integrate first. We know that the integration bypass, let us choose between two functions, F and T, we have to choose. Which one is the first derivative? And which one is the second derivative And how to combine them to get to change the function and make it easier to compute using integration by a part? We can let f equal F. Prime and G is the integration of G And combine in the integration and differentiation. We can get the result that we need. We can use substitution as well or we can use a change of variable. This is this work sometime And we can use the sign and the cosine, or maybe the tangent to combine both, or all these to find the function that we need By the end. Doing a change of variable or an integration bypass or any kind of substitution will help us find the Primitive that we look for the fundamental theorem of calculus that has find the derivative. And from there, we see if the function is increasing or decreasing. Sometimes we are interested in Computing, the second derivative. And we want to see if the function is concave up or concave down, and that's what we need to do.
Today, we're gonna work some problems on the floor function. We're going to Define what we mean by the floor function. We're also going to solve some equations that have to deal with the floor function. We're going to see the properties of the floor function, and we're going to see some equations contained in the floor function. Also, we get a solve some integral equations, those equations that have integrals on them. We're gonna solve integral functional equations and we can see what we can do when we have this kind of equations. We also going to see an apply the calculus to find the area. Lighted. By some street bulb. Okay, this case can help us see that calculus can help us find the area, how the light extend, how how much the area covered by the light? We want to compute that we got to do a lot of simplification of this model. This model is linear. Therefore zooming a lot of simplification but we just want to see that calculus can help us solve many problems. So the key idea here is that using the definite integral to find the area that the light has covered. This is a simple application of calculus using definite integral and rules and the properties of integral, we're gonna see our see more application in the next few days.
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