First Assessment

I will post here the exercises that you will have to give me as practice. This assessment will be done individually, there is another one individual an a group assessment. Recall that it counts the 30% of the total mark.

The first two exercises consists of proving two identites using mathematical induction. One is that posted as a voluntary exercise, that will be removed from that cathegory due to the little success it had:

1.-

And the other is the sum of the squares of the first n numbers:

2.-

3.- Solve the following optimization problem:

“Compute the dimensions of the isosceles triangle of maximum area that can be inscribed in a circle of radious 4″

4.- Give the points of discontinuity of each of the following functions:

5.- Let f(x) be the function defined as if and if . Discuss the continuity and the derivability of the function in .

6.- Use the rules of derivation to find the derivative of the function

7.- Analyze and represent the graph of the function

8.- Exercises 7, 8, 9, 10, 11, of lecture 4.

9.- Calculate

Hint: Factorize first the polynomial on the denominator.

10.- Calculate

I will go on editing this post with more exercises, so don’t forget to check it regularly. You can give me the solutions as the exercises appear or all together.

Note: That problems are taken or adapted from problems appeared in the extensive literature on the subject.

Pablo.

Questions

You can post me here the doubts you can have.

A common mistake

In fact this is not a mistake but it is something that most of you do and there is no reason to do in that way.

The problem is when you want to guess the zeros of a polynomial such that

                         

As it is in a factorized form, if and only if or .

The procedure I have seen in many ocasions is the following, you compute and then you solve it using the formula for a second order equation getting the values 3 and 6.

PLEASE don’t do things in that way. I would like you to come up with an example of this way of doing things in arquitecture.

Pablo.

Questions and problems.

First of all, I would like to remind you that the second practice is due to the day of the exam.

                                 I WON’T TAKE EXERCISES LATER

You can post here any problem or question you have.

Answering to a question of Manuel, the integral of ex 5 in the group assessment (I think that was the one you can have problems with) is similar to that one we did in class. It is solved by a change

                    

Pablo.

Second Assessment

1.- Exercises number 6 and 7 of lecture 5.

 

2.- Calculate the determinant of the matrix of the exercise 5 of lecture 6.

 

3.-  Find a candidate for the value of and prove using induction that your assumption is correct where:

        

4.- Solve, using Gauss method, the system of exercise number 1 of this link.

5.- Solve, using Crammer’s rule, the system of exercise number 1 of this other  link.

 6.- Solve the problems of Lecture 9.

The value of the exercises is 1 Point (1, 2 and 3) 2 points (4 and 5) and 3 points (number 6).

Note: That problems are taken or adapted from problems appeared in the extensive literature on the subject.

Pablo.

WARNING

JUST ONE OF THE WORKS YOU HAVE TO DO IS MEANT TO BE DONE IN GROUPS. AT THE FIRST SIGN OF EXERCISES DONE IN COMMON, THE MARK FOR EVERY STUDENT CHEATING WILL BE A ZERO.

Exercises

Hello

As I told you at the begining of the course, my idea for the module is the following. We are based in the new program for Higher Education where the student is no longer passive but has be the main character in its own education. This means that I assume that you have to work in your own (at least) as much as in the lectures. I also consider that you are mature enough to know the amount of work you need, I don’t see the point (at this age) to ask you to do 25 exercises of derivatives if you just need 10 and on the other way, if I ask 10 and you need 50, I think it is you the one who has to do the rest of the work.

Maybe I am making a mistake since I think that, in general, there is no much work done and you are not using much this blog (I will bet that there are people who still has not check the assessments). But it is a bigger mistake that you left mathematics for the last moment since the work you have done after the exam counts the same amount as the exam itself.

For those who need more exercises and don’t want to look for (or can’t find) them you have a web page with a lot of materials and exercises: http://www.sectormatematica.cl/educsuperior.htm

I hope that this will be useful.

Pablo.

Every continuous function is integrable

In the past lecture we saw this theorem without proof. I tried to prove a weaker version of this theorem:

Every continuous monotonous function is integrable.

Since I thought it could be done in few steps. My proof was incomplete and surfing the net I have found a simple (although not as simple as I thougth) of this fact where the continuity is not needed.

You can see the proof here: http://thales.cica.es/rd/Recursos/rd99/ed99-0244-01/integral2.html

The site is in Spanish, sorry for that but I think that it is easy to understand although it need some time of reflection. I don’t think it deserves to be seen in class but you will be wellcome to ask me any doubt you have (furthermore, it will be taken into account).

Pablo.

 

 

Lecture 4

You can ask questions about lecture 4 in this post. If we have time, I plan to devote the last lecture to solve problmes with the exercises of the assessments so It will be nice that you start working on them ASAP.

Lecture 5 is in reprography and I hope to send Lecture 6 in due course.

Pablo.

Group assessment

You have to do this practise in groups, I think that you can chose the composition between 3,4,4 or 3,3,3,2. The maximun number of people in a group is 4 and of course the minimum is 2, I think that two groups of 4 and one of 3 is better since a group of 2 is in disadvantage.

I post now the exercises of the first part of the module (up to integrals) and I recommend you that you do them as soon as possible since you have another individual practise and half of this one. 

1.- Prove using the definition that

Hint: Use that

2.- Draw the graph of the function latex x>0$. Use the graph representation this function and the definition of derivative to decide if the function for and is derivable at .

3.- Explain the geometrical interpretation of the operations with complex numbers (sum and multiplication of complex numbers, multiplication of a complex number times a real) and write several examples using the different representations of a complex number. Which one is better for each case?

4.- As we have seen, in a set, we can define operations giving the set an structure. These structures are called algebraic structures. List the principal algebraic structures with their properties and write examples. Prove that the set of complex numbers is a field (with the appropriate operations). 

5.- Calculate:

I will go on editing this post with more exercises, so don’t forget to check it regularly. You can give me the solutions as the exercises appear or all together.

6.- Prove that the space of polynomials of degree less or equal than n:

         

with the usual operation for the sum of polynomials and the multiplication by scalars, is a vector space.

7.- Find information (the equation, the history, problem that solve) about the following curves: cycloid, catenary and tautochrone.

8.-Find the area of a stained glass window limited from above for a parabola and from below for and arch of a circunference (see picture). Hint the area of a sector of a circle or radious r and of angle (in radians) is .

The value of the exercises is 1 point exept the number 7 and 8 that worth 2 points.

Note: That problems are taken or adapted from problems appeared in the extensive literature on the subject.

Pablo.