Mathematical Approximation Methods Guide

1. AB Unit 6B

2. 49. Steps for Euler’s Method • Approximation is based on tangent line • ynext= yold + dy/dx(△x) • Identify the step size, derivative, and initial value • Step size can be negative

3. Example Problem Given dy/dx = x + 2 and y is 3 when x is 0. Use Euler’s method with step size 1/2 Find the value of y when x is 1.

4. 50. Relationship Between Direct Variation and Exponential Decay • If problems states rate of change is proportional to amount. • dy/dx = ky • If problems states rate of change is inversely proportional to amount • dy/dx = k/y

5. Example Problem The rate of change of y is proportional to y. When t = 0, y = 2. When t = 2, y = 4. What is y when t = 3?

6. 51. Logistic Growth • This Logistic growth equation is presented ｙ＝M/(1+〖be〗^(-Mkt) ) ,dy/dt=ky(M-y) or dy/dt=ky(1-y/M) • Limit T approaches ∞, f(t) goes M. • Horizontal asymptote occur y=M • Point of maximum growth occur when y value get M/2

7. Example problem The number N(t)N(t)N, left parenthesis, t, right parenthesis of people who have adopted a certain fashion style after ttt months satisfies the logistic differential equation: dN/dt=N(0.1-N/700000) Initially, there were 3000 people who had adopted the style. What is the number of people who have adopted the style when it's growing the fastest? Solution • Find dN/dt=0 • Take average for N The answer is 35000

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10. Answers www.umath2.com/CalcBC/docs/chap6/BCEulerSlopeFieldDiffEq.pdf