1 / 15

Do Now: #1-8, p.346

Do Now: #1-8, p.346. Let. Check the graph first?. 1. Continuous for all real numbers. 2. . 3. H.A.: y = 0, y = 50. 4. In both the first and second derivatives, the denominator. will be a power of , which is never 0. Thus, the. domains of both are all real numbers.

jimbo
Download Presentation

Do Now: #1-8, p.346

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Do Now: #1-8, p.346 Let Check the graph first? 1. Continuous for all real numbers 2. 3. H.A.: y = 0, y = 50 4. In both the first and second derivatives, the denominator will be a power of , which is never 0. Thus, the domains of both are all real numbers.

  2. Do Now: #1-8, p.346 Let Check the graph first? 5. Graph f in [–30, 70] by [–10, 60]. f (x) has no zeros. 6. Graph the first derivative in [–30, 70] by [–0.5, 2]. Inc. interval: Dec. interval: None 7. Graph the second derivative in [–30, 70] by [–0.08, 0.08]. Conc. up: Conc. down: 8. Point of inflection:

  3. Logistic Growth Section 6.5a

  4. Review from last section… Many populations grow at a rate proportional to the size of the population. Thus, for some constant k, Notice that is constant, and is called the relative growth rate. Solution (from Sec. 6.4):

  5. Logistic Growth Models In reality, most populations are limited in growth. The maximum population (M) is the carrying capacity. The relative growth rate is proportional to 1 – (P/M), with positive proportionality constant k: or The solution to this logistic differential equation is called the logistic growth model. (What happens when P exceeds M???)

  6. A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential eq. with k = 0.1. (a) Draw and describe a slope field for the differential equation. Carrying capacity = M = 100 k = 0.1 Differential Equation: Use your calculator to get the slope field for this equation. (Window: [0, 150] by [0, 150])

  7. A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential eq. with k = 0.1. (b) Find a logistic growth model P(t) for the population and draw its graph. Differential Equation: Initial Condition: Rewrite Partial Fractions

  8. A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential eq. with k = 0.1. (b) Find a logistic growth model P(t) for the population and draw its graph. Rewrite Integrate Prop. of Logs

  9. A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential eq. with k = 0.1. (b) Find a logistic growth model P(t) for the population and draw its graph. Prop. of Logs Exponentiate Rewrite

  10. A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential eq. with k = 0.1. (b) Find a logistic growth model P(t) for the population and draw its graph. The Model: –c Let A = + e – Solve for P Graph this on top of our slope field! Initial Condition

  11. A national park is known to be capable of supporting no more than 100 grizzly bears. Ten bears are in the park at present. We model the population with a logistic differential eq. with k = 0.1. (c) When will the bear population reach 50? Solve: Note: As illustrated in this example, the solution to the general logistic differential equation is always

  12. More Practice Problems For the population described, (a) write a diff. eq. for the population, (b) find a formula for the population in terms of t, and (c) superimpose the graph of the population function on a slope field for the differential equation. 1. The relative growth rate of Flagstaff is 0.83% and its current population is 60,500. How does the graph look???

  13. More Practice Problems For the population described, (a) write a diff. eq. for the population, (b) find a formula for the population in terms of t, and (c) superimpose the graph of the population function on a slope field for the differential equation. 2. A population of birds follows logistic growth with k = 0.04, carrying capacity of 500, and initial population of 40. How does the graph look???

  14. More Practice Problems The number of students infected by measles in a certain school is given by the formula where t is the number of days after students are first exposed to an infected student. (a) Show that the function is a solution of a logistic differential equation. Identify k and the carrying capacity. This is a logistic growth model with k = 1 and M = 200.

  15. More Practice Problems The number of students infected by measles in a certain school is given by the formula where t is the number of days after students are first exposed to an infected student. (b) Estimate P(0). Explain its meaning in the context of the problem. Initially (t = 0), 1 student has the measles.

More Related