The Fermat limit is a numerical method used to get the value of the slope of a line, which is tangent to a function at a given point in its domain. It is also used to obtain critical points of a function. Its expression is defined as:
It is obvious that Fermat did not know the fundamentals of derivation, however it was his studies that prompted a group of mathematicians to inquire about tangent lines and their applications in calculus..
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It consists of an approach of 2 points, which in previous conditions form a secant line to the function with intersection in pairs of values.
By approaching the variable to the value "a", the pair of points is forced to meet. In this way the previously secant line becomes tangent to the point (a; f (a)).
The value of the quotient (x - a), when evaluated at point “a”, yields an indeterminacy of K-type limits between zero (K / 0). Where through different factoring techniques these indeterminacies can be broken.
The most commonly used operating techniques are:
-Difference of squares (atwo - btwo ) = (a + b) (a - b); The existence of the element (a-b) implies in most of the cases the factor that simplifies the expression (x-a) in the quotient of the Fermat limit.
- Completion of squares (axtwo + bx); After completing squares, a Newton binomial is obtained, where one of its 2 factors is simplified with the expression (x - a), breaking the indeterminacy.
- Conjugate (a + b) / (a + b); Multiplying and dividing the expression by the conjugate of some factor can be of great help to break the indeterminacy.
- Common factor; In many cases the result of operating the numerator of the Fermat limit f (x) - f (a) hides the factor (x - a) necessary to factor. For this, it is carefully observed which elements are repeated in each factor of the expression.
Even though the Fermat limit does not differentiate between maximums and minimums, since it can only identify the critical points according to its definition, it is commonly used in the calculation of caps or floors of the functions in the plane..
A basic knowledge of the graphical theory of functions in conjunction with this theorem may be sufficient to establish maximum and minimum values between functions. In fact the inflection points can be defined by means of the mean value theorem in addition to Fermat's theorem.
The most significant paradox for Fermat came from studying the cubic parabola. Because his attention was directed to the tangent lines of a function for a given point, he ran into the problem of defining said tangent line at the inflection point in the function.
It seemed impossible to determine the tangent line to a point. Thus begins the inquiry that would give rise to the differential calculus. Defined later by important exponents of mathematics.
The study of maximum and minimum of a function was a challenge for classical mathematics, where an unambiguous and practical method was needed to define these.
Fermat created a method based on the operation of small differential values, which after factoring processes, are eliminated, giving way to the value sought as a maximum and minimum.
This variable will have to be evaluated in the original expression to determine the coordinate of said point, which together with analytical criteria will be defined as the maximum or minimum of the expression.
In his method, Fermat uses the literal symbolism of Vieta, which consisted in the exclusive use of capital letters: vowels, for unknowns, and consonants for known quantities.
In the case of radical values, Fermat implemented a particular process, which would later be used in the factorizations of the limits of indeterminacy infinity among infinity.
This process consists of dividing each expression by the value of the differential used. In the case of Fermat, he used the letter E, where after dividing by the highest power of E, the value sought for the critical point becomes clearable..
The Fermat limit is in fact one of the least renowned contributions in the mathematician's long list. His studies went from the prime numbers, to basically create the bases for the calculation.
In turn, Fermat was known for his eccentricities regarding his hypotheses. It was common for him to leave a kind of challenge to the other mathematicians of the time, when he already had the solution or proof.
He had a great variety of disputes and alliances with different mathematicians of the time, who loved or hated working with him.
His last theorem was the main responsible for his world fame, where he stated that a generalization of the Pythagoras theorem for any grade "n", it was impossible. He claimed to have a valid proof of it, but died before making it public.
This demonstration had to wait approximately 350 years. In 1995 the mathematicians Andrew Wiles and Richard Taylor, put an end to the anxiety left by Fermat, showing that he was right through a valid proof of his last theorem.
Define the slope of the tangent line to the curve f (x) = xtwo at the point (4, 16)
Substituting in the expression of the Fermat limit we have:
The factors (x - 4) are simplified
When evaluating you have
M = 4 + 4 = 8
Define the critical point of the expression f (x) = xtwo + 4x using the Fermat limit
A strategic grouping of elements is carried out, seeking to group the X-X pairs0
The least squares are developed
The common factor X-X is observed0 and is extracted
The expression can now be simplified and the indeterminacy broken
At the minimum points it is known that the slope of the tangent line is equal to zero. In this way we can equalize the expression found to zero and solve for the value X0
2 X0 + 4 = 0
X0 = -4/2 = -2
To get the missing coordinate it is only necessary to evaluate the point in the original function
F (-2) = (-2)two + 4 (-2) = 4 - 8 = - 4
The critical point is P (-2, -4).
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