Ewww-ler’s Method, Exponentials, & Logistics

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Presentation transcript:

Ewww-ler’s Method, Exponentials, & Logistics Izmir, Karie, Melody

Euler’s Method step size Euler’s method is used to approximate y-values utilizing differential equations... step size dy/dx new y value old y value

given dy/dx=x2+y+2 approximate f(1) using step size of 0 given dy/dx=x2+y+2 approximate f(1) using step size of 0.5 given f(0)=2 x y dy/dx yold + △x (dy/dxold) =ynew 2 4 2 + 0.5 (4) = 4 0.5 6.25 4 + 0.5 (6.25) = 7.125 1 7.125

dp/dt=kP integrate P(t)=Ce^(kt) Exponentials Population growth rate: dp/dt=kP integrate P(t)=Ce^(kt) C is the initial population and k is the growth/ decay factor differential equation for population growth/decay

The table below shows the rabbit population on a certain island where t is the number of years beginning with the year 2000. Determine the relative growth rate B. Estimate the population in 2010 Exponentials year population 2000 1500 2001 1577 2002 1658 2003 1743 2004 1832 2005 1926 P(t) = C(initial value) e ^ kt 1577 = 1500 e ^ k(1) k = ln(1577/1500) k = 0.05 P(t) = 1500 e ^ 0.05t P(10) = 2473

Logi(stic)s POI: greatest growth potential M : Karie-ing capacity P : Population of Karie, NC K : a constant as random as Karie

Let f be a function such that all points (x,y) on the graph of f satisfy the differential equation: dy/dx = 2y (3-y) (a) Find the carrying capacity. (b) Find the y value when the function has the greatest growth rate. L = 3 Y = 3/2 = 1.5

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