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AP Calculus BC Tuesday , 04 March 2014

AP Calculus BC Tuesday , 04 March 2014. OBJECTIVE TSW (1) investigate the logistics curve. ASSIGNMENT DUE Sec. 8.4: p. 549 (5-16 all, 21-41 eoo , 43, 45, 53, 54 ) W ire basket ASSIGNMENT DUE TOMORROW/THURSDAY Sec. 8.5: p. 559 (1, 5-15 odd) REMINDERS

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AP Calculus BC Tuesday , 04 March 2014

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  1. AP Calculus BCTuesday, 04 March 2014 • OBJECTIVETSW (1) investigate the logistics curve. • ASSIGNMENT DUE • Sec. 8.4: p. 549 (5-16 all, 21-41 eoo, 43, 45, 53, 54) • Wire basket • ASSIGNMENT DUE TOMORROW/THURSDAY • Sec. 8.5: p. 559 (1, 5-15 odd) • REMINDERS • PI Day: Friday, 14 March 2014

  2. The Logistics of the Logistic Curve

  3. Here is an exponential growth problem: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 HOURS, how many organisms will be produced from a single cell? Solution: 10 hours = 30 20-minute periods There will be 1 ∙ 2^30 = 1,073,741,824 (1.073 billion) bacteria after 10 hours.

  4. A problem that seems just as reasonable: Under favorable conditions, a single cell of the bacterium Escherichia coli divides into two about every 20 minutes. If the same rate of division is maintained for 10 DAYS, how many organisms will be produced from a single cell? Solution: 10 days = 720 20-minute periods There will be 1 ∙ 2^720 ≈ 5.5 ∙ 10^216 (much more than 1.073 billion) bacteria after 10 days.

  5. Makes sense… …until you consider that there are probably FEWER THAN 10^80 ATOMS IN THE ENTIRE UNIVERSE! Real world Bizarro world

  6. Why didn’t they tell us the truth? Most of those classical “exponential growth” problems should have been “logistic growth” problems! Exponential Logistic

  7. The Logistics Curve

  8. The Logistics Curve We have seen exponential growth and decay differential equations of the form whose general solution is

  9. The Logistics Curve We have seen exponential growth and decay differential equations of the form whose general solution is

  10. The Logistics Curve Logistics curve problems combine these: where L = carrying capacity andP = the infected population.

  11. The Logistics Curve Before solving this, let's rewrite the equation:

  12. The Logistics Curve Now integrate: Use partial fractions to rewrite the integral on the left:

  13. The Logistics Curve Rewrite the integral as

  14. The Logistics Curve Since L is positive and L – P is positive, we can omit the absolute value bars:

  15. The Logistics Curve

  16. L P0 The Logistics Curve Some important points to remember: The rate of change (slope) of the curve is most rapid at L/2 (inflection point). The curve starts concave up and finishes concave down.

  17. The Logistics Curve L= carrying capacity y0 = initial population k = constant of proportionality t = time NOTES: The Logistics Curve(Handout) Formulas: or 1) a) Label when units are given. b) c) d) Solve for t (using equation solver): on the 2nd day Differential equation Both are included in an initial-value problem !!! e) Particular solution

  18. The Logistics Curve L= carrying capacity y0 = initial population k = constant of proportionality t = time NOTES: The Logistics Curve(Handout) Formulas: or 2) a) b) c) d) in the 2nd month e)

  19. The Logistics Curve L= carrying capacity y0 = initial population k = constant of proportionality t = time NOTES: The Logistics Curve(Handout) Formulas: or 3) a) b) c) d) y = 75 guppies

  20. The Logistics Curve • WS The Logistics Curve • Due on Monday, 10 March 2014. • ASSIGNMENT DUE TOMORROW/THURSDAY • Sec. 8.5: p. 559 (1, 5-15 odd) • QUIZ: Sec. 8.3 – 8.5 on Monday, 10 March 2014. • TEST: Sec. 8.1 – 8.5, 8.7, 8.8 on Wednesday/Thursday, 13 March 2014.

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