1 / 26

Calculus II (MAT 146) Dr. Day Wednesday March 26, 2014

Calculus II (MAT 146) Dr. Day Wednesday March 26, 2014. Solutions of Differential Equations (9.1) Slope Fields: Graphical Solutions to DEs (9.2 - I) Euler’s Method: Numerical Solutions to DEs (9.2 - II)

devona
Download Presentation

Calculus II (MAT 146) Dr. Day Wednesday March 26, 2014

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. Calculus II (MAT 146)Dr. Day Wednesday March 26, 2014 • Solutions of Differential Equations (9.1) • Slope Fields: Graphical Solutions to DEs (9.2 - I) • Euler’s Method: Numerical Solutions to DEs (9.2 - II) • Separation of Variables: One Method for Determining an Exact (Analytical) Solution to a DE (9.3) • Applications of DEs MAT 146

  2. Separable Differential Equations MAT 146

  3. Applications! Rate of change of a population P, with respect to time t, is proportional to the population itself. MAT 146

  4. Rate of change of the population is proportional to the population itself. Slope Fields Euler’s Method Separable DEs MAT 146

  5. Slope Fields MAT 146

  6. Slope Fields MAT 146

  7. Euler’s Method MAT 146

  8. Separable DEs MAT 146

  9. Applications! The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (A) with respect to time (t) is proportional to the amount present (A). MAT 146

  10. Exponential Decay The radioactive isotope Carbon-14 exhibits exponential decay. That is, the rate of change of the amount present (A) with respect to time (t) is proportional to the amount present (A). Carbon-14 has a half-life of 5730 years Write and solve a differential equation to determine the function A(t) to represent the amount, A, of carbon-14 present, with respect to time (t in years), if we know that 300 grams were present initially. Use A(t) to determine the amount present after 250 years. MAT 146

  11. Applications! Mixtures A tank contains 1000 L of brine with 15 kg of dissolved salt. Pure water enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? After 20 minutes? MAT 146

  12. Applications! Mixtures A tank contains 2000 L of brine with 30 kg of dissolved salt. A solution enters the tank at a rate of 20 L/min with 0.25 kg of salt per L . The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after t minutes? After 60 minutes? MAT 146

  13. Applications! MAT 146

  14. Applications! Known information: A= 20° C and (0,95) and (20,70) MAT 146

  15. Applications! MAT 146

  16. Applications! MAT 146

  17. MAT 146

  18. MAT 146

  19. Polynomial Approximators Our goal is to generate polynomial functions that can be used to approximate other functions near particular values of x. The polynomial we seek is of the following form: MAT 146

  20. MAT 146

  21. Polynomial Approximators Goal: Generate polynomial functionsto approximate other functions near particular values of x. Create a third-degree polynomial approximator for MAT 146

  22. Create a 3rd-degree polynomial approximator for MAT 146

  23. What is a Sequence? MAT 146

  24. List the first five terms of each sequence. MAT 146

  25. Sequence Characteristics Convergence/Divergence: As we look at more and more terms in the sequence, do those terms have a limit? Increasing/Decreasing: Are the terms of the sequence growing larger, growing smaller, or neither? A sequence that is strictly increasing or strictly decreasing is called a monotonic sequence. Boundedness: Are there values we can stipulate that describe the upper or lower limits of the sequence? MAT 146

  26. Assignments WebAssignments • DE Applications • 11.1: Sequences • 11.2 Series MAT 146

More Related