Euler’s Method, Logistic, and Exponential Growth Shlok Yeolekar, Anish Bondre, Tyler Lee

Euler’s Method A process that allows a numerical way to approximate a particular solution This method allows a point on a curve to be mathematically found even if the curve is not know.

Formula for Euler’s Method slope new y old y

Example With a step size of ∆t = 0.2, compute three steps of Euler’s method to approximate the solution of y’ = −0.3y starting with y = 25 for t = 1.

Answer Steps t y ∆y = −0.3y∆t 1 25 −0.3(25)(0.2) = −1.5 1 + 0.2 = 1.2 1 25 −0.3(25)(0.2) = −1.5 1 + 0.2 = 1.2 25 − 1.5 = 23.5 −0.3(23.5)(0.2) = −1.41 2 1.2 + 0.2 = 1.4 23.5 − 1.41 = 22.09 −0.3(22.09)(0.2) = −1.3254 3 1.4 + 0.2 = 1.6 22.09 − 1.3254 = 20.7646

Logistic Functions Calculus formula: Pre-calculus formula: M = carrying capacity K = growth constant A = constant Initial Value:

Example #1 Ten grizzly bears are introduced to a national park ten years ago. There are 23 bears in the park at the time. The park can support a maximum of 100 bears. Assuming logistic growth, when will population reach 50? 75? 100?

Solution

Example #2 Growth rate of population P of bears in a newly established wildlife preserve is modeled by t = years. What is population when it is growing fastest?

Solution

Exponential Growth/Decay Growth: Proportional rate of change, Decay: Inversely proportional rate of change,

Example #1 Rate of change of y is proportional to y. When t = 0, y = 2, When t = 2, y = 4. What is u when t = 3?

Solution

Example #2 Rate of change of P is proportional to P. When t = 0, P = 5000 and when t = 1, P = 4750. Find P when t = 5.

Solution