1. Euler, Exponential, and Logistic Functions

2. Classwork & Homework Only if you stay on task!!! FRQ 11 & 13
Rest of the FRQs
All Multiple choices
Only if you stay on task!!!

3. Card 49: Steps for Euler’s Method
1.Create Chart in terms of “x”, “y”, and “dy/dx”
2.Use your initial x-value and your step size to fill out all of the x-values in the chart
3.Using the initial x-value and initial y-value, solve for dy/dx
4.Use equation ynew= yold + (Δx)dy/dx

4. Chart x y dy/dx 1 2 2(2)+(1)=5 1.5 2+(0.5)5=4.5 ynew= yold+(Δx)dy/dx
dy/dx=2y+x
F(1)=2
Step-size (Δx) =0.5
Find f(1.5) x y
dy/dx 1 2
2(2)+(1)=5
1.5
2+(0.5)5=4.5

5. Direct Variation Formula: y = k*x
Card 50: Relationship Between Direct Variation and Exponential Growth/Decay
Direct Variation Formula: y = k*x
“y” varies directly with “x”; as “x” increases, so does “y”
Inverse Variation Formula: k = x*y
“y” varies inversely with “x”; as “x” increases, “y” decreases and the opposite applies

6. Card 50: Relationship Between Direct Variation and Exponential Growth/Decay

7. Exponential Growth and Decay Formula: dy/dt = k*y or y = C*ekt
Card 50: Relationship Between Direct Variation and Exponential Growth/Decay
Exponential Growth and Decay Formula:
dy/dt = k*y or y = C*ekt
When asked to find the “general solution” use the equation to the right
C is initial value (constant)
k is proportionality constant
k>0 : exponential growth; k<0 : exponential decay

9. Card 50: Relationship Between Direct Variation and Exponential Growth/Decay

11. Card 51: Logistic Growth “M” represents the carrying capacity
A logistic growth model is a more realistic model which accounts for limiting factors and where growth rate is proportional to the amount present (P)

12. Card 51: Logistic Growth

13. Card 51: Logistic Growth

14. The “S” curve or “S” function’s proper name is the Sigmoid Function