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

2. 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

3. 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
(4) = 4
0.5
6.25
(6.25) = 7.125 1
7.125

4. 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

5. 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

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

7. 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

8. Quizizz