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AB Unit 6B

AB Unit 6B. 49. Steps for Euler’s Method. Approximation is based on tangent line y next = y old + 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

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AB Unit 6B

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  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

  8. Play Quizizz! https://quizizz.com/admin/quiz/start_new/5cab3db28af501001a7cd2de

  9. Free Response

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

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