College Math Rational Functions The function is rational when it can be represented as the quotient of polynomials. Actually, all polynomials are rational functions. Standard rational function can be described as follows: f(x) = g(x)/h(x).
Both p and q are polynomials. G(x) is called the denominator, and p(x) is called the numerator. Asymptote for f(x) function represents a straight line that is approached but never reached by f(x) function. Lets examine three types of asymptotes: a) Vertical Asymptote. Vertical asymptotes exhibit vertical lines near which f(x) function becomes infinite.
In case the denominator of rational function has more factors of (x a) that the numerator, then the rational function f(x) will have a vertical asymptote at x = a (Rational Functions, n.p.).
b) Horizontal Asymptote. Horizontal asymptote exhibits line y = c, where the values of rational function f(x) get increasingly close to the number c as x gets large in either the positive of negative direction (Rational Functions, n.p.).
Rational function f(x) has horizontal asymptotes in case the degree of the numerator is the same as the degree of the denominator. We can determine the location of horizontal asymptotes if we look at the degrees of the numerator and denominator. For example, according to Chapter 3.5 Rational Functions and Asymptotes, – if n m, there is no horizontal asymptote.
Yet, if n = m + 1, there is an oblique or slant asymptote. c) Oblique Asymptote. Oblique asymptote represents and asymptote of the form y = a x + b with a non-zero. Rational function f(x) has oblique asymptotes in case the degree of the numerator is one more than the degree of the denominator. Ian Garbert in his article radioactive decay and exponential laws claims that the phenomenon of radioactive decay obeys an exponential law. Graphic asymptotes can illustrate the data generated through radiocarbon dating the half-life of fossil specimens. Actually, radio carbon dating is related to natural logarithms and exponential functions of the form: y = Cekx where C and k are specific constants. He reminds that Carbon 14 is a radioactive element that is found naturally. According to him, the ratio of C-14 to C-12 in the atmosphere’s carbon dioxide molecules is about 1.3?10-12, and this value is assumed constant for the main part of archaeological history since the formation of the earth’s atmosphere.
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If we know the level of activity of a sample, we can deduce how much C14 there is in the material at present. Further, when we know the ratio of C14 to C12 originally, we can find the time that has passed since carbon exchange ceased, that is, since the organic material died (n.p.).
Lets see it by the following example: lets suppose a sample (1g) is to be analyzed. The activity is measured at approx. 11.9 decays per minute. Lets denote the magnitude of the rate of nuclei C14 decay as R. R will be equal to the rate that beta particles are detected.
In such a way, dN dt = ?R The exponential law for the number of C14 nuclei present says that N = N0 / e?l t. It means that dN dt = ? l N0 / e?l t = ?l N, That tells us the R = IN (the activity at t = 0) is R(0) = lN0. Substituting will certainly give us an exponential relation in terms of the measured activity: R = R0 e?l t (Garbert n.p.).
So, the decay constant for Carbon-14 is l = 3.8394 ? 10-12 per second. This corresponds to a half life of 5,730 years. In such a way, we can calculate the number of C12 in 1g of C: 6.02 ?10?12 nuclei/mole 12.0 grams/mole ?1 gram = 5.0167 ?10?22 nuclei Lets use the living ratio of C14 to C12. We define that the original (t=0) number of C14 was (1.3 ?10?12) ?(5.0167 ?1022) = 6.5221 ?1010 Further lets rearrange the exponential activity law: t = ? ln(R/R0) l It gives us the value for t result: 6.0758 ?1010 ? 1 year 3.15?107 seconds , That means approx.
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1929 years (the approx. age for some of the Dead Sea Scrolls) (Garbert n.p.).
In such a way, we can see that radio carbon dating is related to natural logarithms and exponential functions.
Bibliography:
Garbert, I. Radioactive Decay and Exponential Laws. Retrieved May 29, 2006. http://plus.maths.org/issue14/features/garbett/ind ex.html Rational Functions. Retrieved May 29, 2006. http://oregonstate.edu/instruct/mth251/cq/FieldGui de/rational/lesson.html.
