Exponential Decay, Linear Approximation, sinh(x), MVT lecture notes - MA103 Week 8 Exponential Growth and Decay Approximation Techniques Hyperbolic

Exponential Decay, Linear Approximation, sinh(x), MVT lecture notes

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MA103 Week 8 Exponential Growth and Decay, Approximation Techniques, Hyperbolic Functions, MVT 1.ProportionalityWhen we say that a given quantity, sayy, is proportional to another quantity, sayt, the relationship isdenoted byyt.This means that the value ofywill be some constant multiple oftand the relationshipcan be represented by the equationy=k·t,kR.If a coordinate pair of values, (y, t),is given then thevalue ofkcan be determined, which then remains the same for any given application.In this case, ifk >0,then the quantityyincreases astincreases and ifk <0, thenydecreases astincreases.Ifyis inversely proportional tot, the relationship is represented byy1tor by the equationy=kt,kR,etc.2.Exponential Growth and Decay(Text: 3.8)Many practical applications involve a quantity which either grows or decays at a rate proportional to its size.In general, ify=f(t) gives the value of a quantity at timet, then the rate of change inywith respect totis given bydydt.Thus, from our first statement, we havedydt=kyfor some constantk.This is an exampleof a differential equation, and is called theLaw of Natural Growth[ ifk >0 ]or theLaw of Natural Decay[ ifk <0 ].We have already seen a function[ and it is, up to a constant multiple, the only such function! ]whosederivative is a multiple of itself – the exponential function.It can be shown that solutions for the abovedifferential equation are of the form:y=y0·ekt, wherey0denotes the initial value of the quantity[ i.e.,the value ofywhent= 0 ].Using given values in the application [ i.e., the size of the quantity at a specifictime ], the value ofkcan be determined to obtain the solution particular to that application, which can thenbe used to find the size of the quantity at timet.Examples of Exponential Growth/Decay that may be looked at during the lab:1)Malthusian Law of Population Growth:The rate of change of a populationP0(t) is proportional to the population sizeP(t) at timet.Specifically,P0(t)P(t)P0(t) =kP(t)Example: Suppose a population grows according to this model (i.e.P0(t) =kP(t)).IfP(0) = 5000,then (from above)P(t) =P(0)ekt= 5000ektNote: More information is required to findk(the relative growth rate of the population).

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