Answers to an Advanced Mathematics Test

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Question 1 A function defines a relation between two sets of quantities in such a way that each quantity in the first set, which is called the domain, is related to a certain quantity in the second set, which is called the range. Thus in the linear function y = x, the (simplest) 3rd degree polynomial y = x3, the exponential function y = ex, and the (natural) logarithmic function y = ln(x), the set of possible values of the quantity x are the domain, and the set of the corresponding values of y are the range. In the case of the quadratic function y = x2, for each y value in the range, there are two x values in the domain so the inverse function doesn’t exist unless more explicit conditions are stated. Similarly, in the case of a periodic function such as y = sin(x), there is an infinite set of values of x for a certain value of y, so in order to define the inverse function it is generally agreed that only the value of x between –π/2 and + π/2 is allowed. (Reference: Lecture notes, Answers to sample exam questions). Graph of y = ax + b (plot adopted from http://lasp.colorado.edu/~bagenal/MATH/math7.html) Graph of y = exp{x} (adopted from the same source) Graphs of periodic functions, e.g. a shifted sinusoidal function (sine or cosine, and a more complicated but still periodic function): (plot adopted from http://lasp.colorado.edu/~bagenal/MATH/math7.html) Question 2 Carbon dating is the process by which the age of ancient, and important, objects is estimated. The process is possible because of the existence of an unstable isotope of carbon, namely 14C, which is produced in the Earth’s atmosphere during nuclear reactions resulting from cosmic-ray bombardments. Since plants constantly ... ... middle of paper ... ...r than linear. However the same model (ideal gas model) can be improved by at least introducing a simple temperature variation with altitude of the linear form, T = T0 ̶ α y where T is the temperature at altitude y and T0 is temperature at the Earth surface (altitude zero). The same procedure as above then leads to the modified function ln⁡(p/p_0 )=mg/kα ln⁡(T_(0-α)/T_0 ). Coefficient α is called the lapse rate of temperature and has an average value of about 0.6 degrees C / 100 m (Young and Freedman, 2012, Chapter 18). This gives a better approximation to the P variations than the simple ideal gas and constant T result we discussed above. References H. Young and R. Freedman, University Physics, 13th Ed. Addison-Wesley, 2012. Poole and Poole, Carbon 14, University of California Press, 1999. P. Schmidt and F. Ayres, College Mathematics, 2010.

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