FERMAT LIMIT: WHAT IT CONSISTS OF AND EXERCISES SOLVED - MATHS - 2024

The Fermat limit is a numerical method used to obtain the value of the slope of a line, which is tangent to a function at a certain 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.

What is the Fermat limit?

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 limits of the type K between zero (K / 0). Where through different factoring techniques these indeterminacies can be broken.

The most commonly used operating techniques are:

-Difference of squares (a ² - b ²) = (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 (ax ² + 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) needed to factor. For this, it is carefully observed which elements are repeated in each factor of the expression.

Application of the Fermat limit for maximums and minimums

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 tops or floors of 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 cubic parable

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 point of inflection 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.

Maximus and minimous

The study of maximums and minimums of a function was a challenge for classical mathematics, where an unambiguous and practical method was needed to define them.

Fermat created a method based on the operation of small differential values, which after factoring processes, are eliminated, giving way to the maximum and minimum value sought.

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.

Method

In his method, Fermat uses the literal symbolism of Vieta, which consisted of the exclusive use of capital letters: vowels, for unknowns, and consonants for known quantities.

For the case of radical values, Fermat implemented a particular process, which later would be used in the factorizations of the limits of indeterminacy infinite between 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 sought value of the critical point becomes clearable.

History

The Fermat limit is in fact one of the least renowned contributions in the mathematician's long list. His studies went from prime numbers to basically creating the basis for calculation.

In turn, Fermat was known for his eccentricities with respect to 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 either loved or hated working with him.

His last theorem was the main responsible for his worldwide fame, where he stated that a generalization of the Pythagorean theorem for any degree "n" 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, mathematicians Andrew Wiles and Richard Taylor put an end to Fermat's anxiety, proving that he was right through a valid proof of his last theorem.

Exercises

Exercise 1

Define the slope of the tangent line to the curve f (x) = x ² 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

Exercise 2

Define the critical point of the expression f (x) = x ² + 4x using the Fermat limit

A strategic grouping of elements is carried out, seeking to group the pairs XX ₀

The least squares are developed

Observe the common factor XX _{0 and extract}

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 X ₀

2 X ₀ + 4 = 0

X ₀ = -4/2 = -2

To get the missing coordinate it is only necessary to evaluate the point in the original function

F (-2) = (-2) ² + 4 (-2) = 4 - 8 = - 4

The critical point is P (-2, -4).

References

1. Real Analysis. A Historical Approach Sauhl Stahl, John Wiley & Sons, Aug 5. 1999.
2. The Mathematical Career of Pierre de Fermat, 1601-1665: Second Edition. Michael Sean Mahoney. Princeton University Press, Jun 5. 2018
3. From Fermat to Minkowski: Lectures on the Theory of Numbers and Its Historical Development. W. Scharlau, H. Opolka, Springer Science & Business Media, 1985
4. Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. Harold M. Edwards. Springer Science & Business Media, Jan 14 2000
5. Fermat Days 85: Mathematics for Optimization. J.-B. Hiriart-Urruty Elsevier, Jan 1. 1986