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BASIC THEOREMS
DIFFERENTIAL CALCULUS 1. THEOREM ENDS. If f is a continuous function defined on the closed interval, there is (at least) a point at which f takes the greatest value, and there is (at least) a point where f takes the lowest value. This theorem states that every continuous function defined on a closed interval, you must have a point at which the curve takes its maximum value and a point at which the curve takes its value mínimo.Ver Figure 1.
2. Rolle's theorem. If a function f is continuous on derivatives, and f (a) = f (b) = 0, then there exists at least one number c, such that f '(c) = 0, ie, a point c where the tangent line is parallel to the x-axis. This theorem was enunciated and demonstrated by Michel Rolle.

Example: Given the function f given by y = 2x - 2 (x) ^ 3, determines the point c between such that f '(c) = 0
check: f (1) = f (- 1) = 0
f (1) =?
f (-1) =?
Find the derivative of f, Click f '(x) = 0,
for x = c, we have that c is about 1 over root three
c The two values \u200b\u200bare in the range (-1, 1). Figure 2 shows the two tangents
Mean Value Theorem. If a function f is continuous on derivatives, then there a number c in such that:
EXAMPLE. Consider the function. Determine the numbers c where the tangent line is parallel to the secant line through the points x = -1 x = 3
On the mean value theorem:
Then -1 is the slope of the line through the points - 1 and 3.
find f '(x), then
If there is, it must satisfy f' (c) = -1. Therefore
f '(c) = 2c-3 = -1. Then c = 1
Then in c = 1, the tangent line is parallel to the secant through the points -1 and 3. See chart.

plotting
REAL FUNCTION is a function f differentiable on, then:
CRITERIA The first derivative
SDE 1. If, for all x, then f is increasing. That is, if the slope of the tangent line is positive, then the function is increasing
2. If, for all x, then f is decreasing. That is, if the slope of the tangent line is negative, then the function is decreasing
3. If, for all x, then f is constant. That is, if the slope of the tangent line is zero, then the function does not grow or decrease
CRITICAL. Critical point is called the domain of a function f, those points where the derivative is zero or does not exist. That is, c is a critical point if f '(c) = 0. In general a critical point is that where change function of increasing to decreasing (0 or vice versa).
A curvature of a graph is called concave. The function is concave upward
CRITERIA FOR THE SECOND DERIVATIVE
f is a function whose second derivative exists in the interval, then:
1. If, for any x in, then the graph of f is concave upward. That is, if the derivative is positive, then the curve is concave upward
2. If, for any x in, then the graph of f is concave down. That is, if the derivative is negative, then the curve is concave down
3. If, for any x in, then the test does not decide. Ie can not be said about concavity. TURNING POINT

A point x = c is a turning point if, at that point. A turning point is that in which the graph of the function changes concavity.
A critical point may be a turning point, a maximum or minimum. MAXIMUM AND MINIMUM

say that a function y = f (x) has an absolute maximum at x = c, if f (c) is the largest value taken by the function, ie, if, for all x in the domain of f
say that a function y = f (x) has an absolute minimum at x = c, if f (c) is the largest value taken by the function, ie, if, for all x in the domain of f
is said a function y = f (x) has a relative maximum at x = c, if there is an interval containing c and such that for all x in the open interval.
is said that a function y = f (x) has a relative maximum at x = c, if there is an interval containing c and such that for all x in the open interval.
SEARCH CRITERIA FOR MAXIMUM AND MINIMUM
If c is a critical point of the function y = f (x), ie f '(c) = 0 and if the second derivative f''(x) exists, then:
1. If f has a relative minimum in c
2. If f has a relative maximum at c
FUNCTIONS CHART graph the function f (x) = x ^ 3 - 4x ^ 2 - 3x - 4
1.
Critical Yield function and equals zero drift, ie f '(x) = 0 Factoring the trinomial
get (3x +1) (x - 3) = 0 Do this!
each factor equal to zero: 3x +1 = 0 or, x - 3 = 0, then: x = -1 / 3, or, x = 3
For x = 3, then
For x = -1 / 3, f (-1 / 3) = Calculation! Therefore
critical points are: Write
2. Intervals where the function is increasing and decreasing
If f '(x)> 0. the function is increasing
Let (3x +1) (x - 3)> 0. Solving the above inequality gives the solution:
RESOLVE the inequality!
According to the criterion of the first derivative, say in which intervals the function is increasing and decreasing in the interval. Write them down!
3. Tipping Points
inflection points are where the second derivative is zero, ie f''(x) = 0
So f''(x) = 6x - 8, equal to zero: 6x - 8 = 0
Therefore the inflection point is?
4. Intervals where the function is concave up and concave down
If f''(x)> 0. function is concave upward. (Criterion of the second derivative)
Since f''(x) = 6x - 8. Do: 6x - 8> 0
Then the solution of the inequality is? Find it! Find
concave upward and downward. Write
intervals 5. Maximum and Minimum
Replaced the abscissa value of the critical points in the second derivative, to find the maximum and minimum (second derivative criteria), then:
For x = 3, f''(3) = 6 (3) - 8 = 18-8 = 10> 0, which is the maximum and minimum
6. GRAPH. The following is the graph obtained using the criteria of the derivative.

Creatine And Enlarged Heart

DNI-e

On Thursday 26 November I had the opportunity to participate in the III Congress electronic ID and digital identity organized by ASIMELEC . Specifically it was in the roundtable "The Digital Identity Accreditation and Data Protection ", which was invited the English Professional Association Privacy-APEP, of which I am Secretary.

Besides the general issues addressed in this event on the electronic signature and, specifically, the electronic identity card (see program ), the board addressed the implications that has the setup and operation of this tool the perspective of the right to protection of personal data of their owners. In particular, we discussed many aspects of great relevance in this regard, among which can be highlighted:
• the potential use of electronic signatures for identification of minors on the Internet and the difficulties to use for this purpose the DNI-e, both in the absence of an obligation on the part of children for that instrument and, in particular, because minors are not active functionality on the certificates;
• the higher incidence of use of electronic ID may result in the personal sphere of the owners to the extent that its use is required for actions in life "face" do not require reliable identification on especially given the increased technical capabilities to collect information that is not strictly necessary for the purpose that justifies the service concerned;
• the implications for the fundamental right of art. 18.4 of the Constitution posed by the processes of checking the revocation status of certificates, especially from the perspective of the actions undertaken by the authorities of validation
• and, above all, the controversy relating to personal data incorporated into electronic documents digitally sign when using the DNI-e, as the public information of the certificates become part of the document and, therefore, anyone who can get to know not only the number of owner NIF but also your birth date or even a summary of its digital fingerprint and other personal information.

main conclusion of the roundtable would highlight the importance of the DNI-e design and computer applications that will be used taking into account the requirements of the protection of personal data of their owners, in particular from the principle of data quality to address two of its manifestations: the compatibility of uses and, above all, the principle of finality.

Cervical Polyp Removal Time

data protection and electronic signatures in audio conferences

As announced few days ago, on Monday took place in Valencia a conference on the use of electronic signatures in government ... And are already available the audio of speeches and presentations that were used ! Thanks no doubt to the diligence, effort, dedication and expertise of the coordinator of the day, Lorenzo Cotino , once again ... going to say "strikes again", but honestly nothing that he can come and amaze me. Thank

Lorenzo, not only for the provision of materials but also and above all, for helping me not to forget the responsibility we have from the University in this kind of approach. Tomorrow