Class Note for MATH 250A with Professor Lega at UA
Class Note for MATH 250A with Professor Lega at UA
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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 15 views.
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Date Created: 02/06/15
o Newton39s law for an object moving in one dimension dv Calculus and Differential Equations I F quotW quot7E7 MATH 250 A where F is the sum of forces applied along the positive X direction In is the mass of the object X is the position of its Modeling with differential equations center of mass and v dXdt is its velocity o If the only force is gravity then F 7mg if X points upward in the vertical direction In this case we have dvi dt g7 which is solved by direct integration Madam wnh differential equations Calculus and D renual Equationsl Madam wnh d mal equations Calculus and al Equationsl Objects in motion continued Mixture problems l the Pquotesence Of graVlty and fI39lCtloni We typically haVe 9 These problems typically involve a fluid of volume Vt in o F 7mg 7 c v c gt 0 if the object is moving slowly which a substance is dissolved The goal is to find the amount F 7mg CV2I C gt OI if M gtgt 1IV lt 0 At or the concentration Ct AtVt of the substance in the fluid a It is possible to have other types of friction forces especially in the case 01 SOIid friction 0 The general way of addressing such a problem is to write a balance equation for the amount At of the substance in the o If we restrict ourselves to the above examples we have H d dv uI o 7g7 v which is linearin v dA dt m dt input rate 7 output rate 0 7g v2 which is separable dt quot7 0 Example 5 page 207 Take a ZOO gallon container filled o Fora springmass system we have F 71X X0 k gt OI with pure water Add a salt concentration with 3 pounds of d2X I I I salt per gallon at a rate of 4 gallons per minute At the same Then m TMX T XO Wthh 395 a second order lmear time drain the container at a rate of 5 gallons per minute equatlon Find the amount of salt in the container as a function of time Madam wnh differential equations Calculus and Differential Equationsl Madam wnh differential equations Calculus and ON nual Equationsl Cooling and heating 0 Newton s law of cooling and heating says that the rate of change of the temperature T of an object is a linear function of the difference between T and the ambient temperature To 17urany dt kgt0 o This equation can be solved as a linear equation or as a separable equation to find Tt To Hexp7kt where H is an arbitrary constant a As expected T 7 To as t 7 00 Calculus and D Modeling with differential equations rential Equations Compounding interest o If money in a bank account is compounded continuously at a rate of r percents per year then in the absence of deposits or withdrawals we have dM 7 r dt T 100 7 where M is the account balance and t is time measured in years a The above equation describes the exponential growth of M 0 After one year the amount of money in the account is given W M1 expr100 M0 o The annual interest rate is therefore larger than rlOO since APY expr10071 Calculus and Modeling mm d mai equations al Equationsl Population ynamics o If N is the population density of a region then one can write dN bN 7 dN l immigration 7 emigration dt assuming that resources are not limited 9 In the above equation b is the birth rate and d is the death rate of the population The growth rate r of the population is given by r b 7 d o If immigration and emigration are given functions oft then the above equation is linear in N pulation dynamics continued o If a population is growing exponentially at rate r gt 0 we can define its doubling time T ln2 T r a Note the analogy with the half life of a substance decaying exponentially at rate r lt 0 7 ln2 ln2 r lrl T12 o If resources are limited one can expect that r will depend on N With r a7 N a gt 0 gt 0 and in the absence of immigration or emigration we have logistic growth de 2 gaaN7 N Modeling with differential equations Calculus and Differential Equations Modeling With differential equations Calculus and Du ntial Equationsl Chemical reactions k1 o Fora chemical reaction of the form A B C7 the law k2 of mass action says that dlCl 7 a kiiAi B e kle where X is the concentration of chemical X and k1 and k2 are the forward and backward rate constants respectively 0 For an autocatalytic reaction of chemical X one may have d X k18X 7 2X27 where a IQ and k2 are constants This is again the logistic equation Modeling with differential equations Calculus and Differential Equations
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