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.- $1+2+\dots +n= \frac{n(n+1)}{2}$

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

2.- $1^2+2^2+\dots +n^2= \frac{2n^3+3n^2+n}{6}$

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:

$f(x)=\frac{x}{(x-2)(x-4)}$

$f(x)=\sqrt{(x-3)(x-6)}$

$f(x)=x^2 \sin (1/x)$

$f(x)=\frac{1}{1+2\sin x}$

5.- Let f(x) be the function defined as $2x-3$ if $0\leq x \leq 2$ and $x^2-3$ if $2<x\leq 4$ . Discuss the continuity and the derivability of the function in $0\leq x \leq 4$ .

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

$f(x)=ln(3x^2+\sqrt{x}\cdot \frac{1}{cos(x^2+1)})$

7.- Analyze and represent the graph of the function

$f(x)=\frac{x^3}{x^2-1}$

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

9.- Calculate

$\int \frac{4x-2}{x^3-x^2-2x}dx$

Hint: Factorize first the polynomial on the denominator.

$\int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx$

10.- Calculate

$\int \frac{2x}{\sqrt[3]{6-5x^2}}dx$

$\int \frac{ln(ln x)}{x}dx$

$\int \frac{\sin(2x)}{\sqrt{\cos(2x)}}dx$

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.

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IE Mathematical Foundations in Architecture

First Assessment

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