2. ESCUELA DE INGENIERÍA DE PETROLEOS 7. BIBLIOGRAFIA CONTENT 4. FUNCTIONS 3. MODELS 2. PROCESS 1. INTRODUCTION
3. ESCUELA DE INGENIERÍA DE PETROLEOS I. Introduction A mathematical model is defined as a description from the point of view of mathematics of a fact or phenomenon of the real world, the size of the population to physical phenomena such as speed, acceleration or density. The objective of the mathematical model is to understand fully the phenomenon and perhaps predict their behavior in the future.
4. ESCUELA DE INGENIERÍA DE PETROLEOS The process for developing a mathematical model is as follows: 1. Encounter a real world problem. 2. Formulate a mathematical model of the problem, identifying variables (dependent and independent) and establishing hypotheses simple enough to be treated mathematically. 3. Implement mathematical knowledge that has to reach mathematical conclusions. 4. Compare data obtained as predictions with real data. If the data are different, the process is restarted.
5. ESCUELA DE INGENIERÍA DE PETROLEOS It is worth mentioning that a mathematical model is not entirely accurate to real life problems, in fact, it is an idealization. There are a lot of functions that represent relationships observed in the real world, which will be discussed in the following paragraphs, both algebraically and graphically.
6. ESCUELA DE INGENIERÍA DE PETROLEOS It is said that a function is linear when its graph is a straight line and therefore has the form: y = f (x) = mx + b Where m represents the slope of the line and b the intercept (the point where the deceased to the axis line of the "y"). It is noteworthy that these functions grow at constant rate, and its domain and image are all real numbers.
7. ESCUELA DE INGENIERÍA DE PETROLEOS A function is polynomial if it has the form: P (x) = anxn + an-1xN-1 + ... ... + a2x2 + a0 A1X Where n is a negative integer and numbers a0, a1, a2, ... .. an, calls are constant coefficients of the polynomial. The domain of all polynomials are all real numbers (- ∞, ∞). .
8. ESCUELA DE INGENIERÍA DE PETROLEOS Polynomials of degree one are of the form: P (x) = mx + b, and are linear functions. Second-degree polynomials are called quadratic functions and have the form P (x) = axx + bx + c; its graph is a parabola. A third degree function is called cubic function, and has the form: P (x) = AX3 + BX2 + cx + d. Below are some graphs of polynomial functions
9. ESCUELA DE INGENIERÍA DE PETROLEOS A power function is called when the form: f (x) = xa, where a is constant. And several cases: The generic form of the graph depends on whether n is even or odd, if n is even, the graph of f is similar to the parabola y = x2, otherwise, the graph will look like the function y = x3. It is important to mention that whatever the case, when n increases, the graph becomes flatter near 0, and steeper where Ix I is less than or equal to 1.
10. ESCUELA DE INGENIERÍA DE PETROLEOS The two graphs above are examples of even functions: x2 and x6.
11. ESCUELA DE INGENIERÍA DE PETROLEOS The two graphs above are examples of even functions: x3 and x5.
12. ESCUELA DE INGENIERÍA DE PETROLEOS a = n, n is a positive integer The function f (x) = x1/n is a root function. As in the previous case, the graph depends on n, since if n is even the graphic will be similar to the square root and n is odd if its graph is similar to the cube root.
14. ESCUELA DE INGENIERÍA DE PETROLEOS a = -1 This type of function is called reciprocal function and its shape is f (x) = x -1 ó (x) = -1 / x. And the graphic corresponds to a hyperbola whose asymptotes are the coordinate axes.
15. ESCUELA DE INGENIERÍA DE PETROLEOS A rational function is called when a ratio or division of two polynomials. f (x) = P (x) / Q (x) Its domain is all values are not made to Q (x) = 0, since a division is indivisible from 0.
16. ESCUELA DE INGENIERÍA DE PETROLEOS For these functions, is convenient to use radian measure, it is important to mention that every function has a specific graphic. In the specific case of sine and cosine, its domain is (- ∞, ∞) and its image [-1, 1]. Here on the charts. .
17. ESCUELA DE INGENIERÍA DE PETROLEOS They are called exponential functions to those that have the form f (x) = ax, where the base is a positive constant. Its domain is (- ∞, ∞) and its image (0, ∞). It is important to note that if the base of the exponential function is greater than 1, the graph will be downward, and if the base is between 0 and 1 the graph is downward (but in the opposite quadrant).
19. ESCUELA DE INGENIERÍA DE PETROLEOS The functions which have the form f (x) = logax, where the base is a positive constant, it is important to mention that it is the inverse of the exponential functions, so its domain is (0, ∞) and its image (- ∞, ∞). Let's look at examples: As shown in the two previous graphs, as the base of the logarithm is greater, the graph of it sticks more to the axis Y.
20. ESCUELA DE INGENIERÍA DE PETROLEOS In reality, this classification covers all those functions that are not algebraic (ie those involving addition, subtraction, division and multiplication of variables). The transcendental functions are the trigonometric, logarithmic, exponential, and inverse trigonometric, among others.
21. ESCUELA DE INGENIERÍA DE PETROLEOS • Stewart, James. "Calculus, Early Transcendent." 4 ed. Tr. Andrew Sesti. Mexico, Ed Thomson, 2002. p. 1151
