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

2. 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.

3. Formula for Euler’s Method
slope
new y
old y

4. 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.

5. Answer Steps t y ∆y = −0.3y∆t 1 25 −0.3(25)(0.2) = −1.5 1 + 0.2 = 1.21 25
−0.3(25)(0.2) = −1.5
= 1.2
25 − 1.5 = 23.5
−0.3(23.5)(0.2) = −1.41 2
= 1.4
23.5 − 1.41 = 22.09
−0.3(22.09)(0.2) = −1.3254 3
= 1.6
22.09 − =

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

7. 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?

8. Solution

9. 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?

10. Solution

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

12. 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?

13. Solution

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

15. Solution