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The Logistic Function Problem Set Math 140 This Logistic Function Problem Set will give you practice with a realistic logistic function. Purpose: To...

Logistic Problem need help

The Logistic Function Problem Set Math 140 Page 1 of 7 pages This Logistic Function Problem Set will give you practice with a realistic logistic function. It is an INDIVIDUAL assignment. Purpose: To extend your skill in using an extremely important generalized exponential function called the logistic function To develop, algebraically, the two forms (exponential and logarithmic) of the logistic function . To recognize valid applications of the logistic function in scenarios of constrained growth. You will want the use of Microsoft Mathematics (think of it as a cool graphing calculator) or other graphing tool (TI or Casio calculator or web-based applet, or Graphmatica), and the Equation Editor built into MS Word 2007 or 2010 or MathType. MS Excel (or equivalent) can also be used for graphing or other calculations. The primary submission must be composed in Microsoft Word or its equivalent. The rules are as follows: Submission via email is due on Day Seven of Module/Week 6 (2359 Eastern Time Zone). It must be in one of the following formats: *.docx/*.xlsx *.doc/*.xls *.pdf (if you convert your work into Adobe Acrobat format) * If you use Open Office.org be sure to convert your documents into one of the acceptable formats listed. Submission of .ods / *.oxs, suffixes for OpenOffice.org, will need to be approved by your instructor prior to beginning this problem set. The asterisk is where you enter “LastName FirstName Problem Set” (no special characters. Things like #, ? and * screw up your submission) Module 6 activities consists only of those related to taking the Test on Chapters 3 and 4, so there is time to produce a quality product on this Problem Set. This assignment is weighted 7.5% of the final grade.
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The Logistic Function Problem Set Math 140 Page 2 of 7 pages Math background. Pure exponential growth is not a real-world phenomenon (there are a finite number of atoms in the observable universe, and the exponential function is a continuous function), but it can be a fine model for short periods of time. There is an apocryphal story involving a king and a wise man; the wise man pleases the king, who then offers him the reward of his choice. The wise man asks for a chess board and asks the king to give him 1 grain of rice on the first square, 2 on the second, with successive doublings on successive squares. Ultimately, the king has to surrender the kingdom to the wise man because he had to give the wise man all the grain the king had. (and THEN some). Task 1: Calculate the number of grains assigned to the 64 th square—they won’t all fit on the board. For a little extra credit find how many metric tons of rice that is. To three decimal places, please. A BONUS to the enterprising individuals who calculate how many metric tons of rice are assigned to the entire chess board—all 64 squares. The cleverer the work, the higher the bonus. Even graphing an exponential function is far more difficult than textbooks lead you to believe. Using a blackboard with horizontal units 1 cm apart, graphing the function y = 2 x , the point (100 cm, 2 100 cm) is very difficult to actually plot because of how far “up” 2 100 cm is. It’s why some smart folks invented semi-log paper. Nonetheless, if the growth is slow enough—on the order of a couple percent per year, for example, an exponential model is simple and very good at tracking populations over a twenty year interval, but not much longer. Logistic growth described No living system exhibits exponential growth over any extended period of time simply due to the exhaustion of available resources to feed the “beast”. It is this resource limitation (among other limiting factors) that causes an eventual “leveling” of the population (or size) of the system . With this in mind, think about resource-constrained exponential growth , also known as logistic growth. Google or Bing a guy named Verhulst (he lived in the 1800’s) for background. It will be similar to the notes of this project. The basic theory. Once Newton and Leibnitz successfully showed calculus to be a precise tool for modeling the rates of change of physical systems (in Newton’s case, it was explaining the laws of planetary motion and the mathematics of gravity), RATE OF CHANGE Equations (referred to as Rate Equations for brevity) became The Way to describe a physical system.
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