AB Unit 6B

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

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.

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

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?

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

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