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This 5 page Class Notes was uploaded by Claudine Friesen on Monday September 28, 2015. The Class Notes belongs to MATH170 at University of Pennsylvania taught by E.So in Fall. Since its upload, it has received 53 views. For similar materials see /class/215393/math170-university-of-pennsylvania in Mathematics (M) at University of Pennsylvania.
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Date Created: 09/28/15
Lecture Notes for Math 170 Ideas in Mathematics Spring 2007 by Nathanael Leedom Ackerman January 19 2007 1 TALK SLOWLY AND WRITE NEATLY AND BIG 2 1 TALK SLOWLY AND WRITE NEATLY AND BIG 2 Review Recall from last time a discrete dynamical system is a way to model some behavior lt is done by nding a series of numbers 130131 related by a formula of the form Pn fltPn1 Further De nition 201 Let PnH We then say a state a is an equilibrium if a a and De nition 202 We say a is an unstable equilibrium of Pn fltPn1 if a a and for all su ioiently small 6 P0 a 6 implies limrH00 Pn y a 3 CHAOS 3 3 Chaos De nition 303 ln mathematics chaos is when a small change in initial conditions leads to a large change in outcome Describe the Butter y effect 0 Term coined by Edward Lorenz in 1963 o Phrase was One meteorologist remarked that if the theory were correct7 one flap of a seagull7s wings could change the course of weather forever77 Discuss errors and repeated calculations using a calcula tor l Population Formula l Give population formula PnH ltl gt explain about predators 3 CHAOS 4 Say we want to consider maps where we stay in the in terval 0 1 so we can consider it as a of the maximum possible population We want to discuss this over time One way is to simply write a computer program which spits out values How ever we want a pictorial representation of what is going 011 Functions 0 De ne what a function is 0 De ne it as a black box 0 Make connection to black box of probability 0 Range of a function Domain of a function l Graphing Functions l 3 CHAOS 5 0 De ne what a graph of a function 0 Go through an example Web Diagrams 0 Explain the relation of the graph to the dynamical system 0 Draw the 13 y line 0 Go step by step though the case of the exponential 0 Go step by step through a case where A g 1 so everything is below 13 y Observe that if A g 4 then everything Observe period points if A gt 33 Go through the isosolese triangle and cycles with a web
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