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Calculus II (MAT 146) Dr. Day Monday March 24, 2014

Calculus II (MAT 146) Dr. Day Monday March 24, 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)

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Calculus II (MAT 146) Dr. Day Monday March 24, 2014

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  1. Calculus II (MAT 146)Dr. Day Monday March 24, 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. What is a Solution to a Differential Equation? • A general solutionto a differential equation is a family of functions that satisfies a given differential equation. • A particular solutionto a differential equation (also called the solution to an initial-value problem) is a particular function that satisfies both a given differential equation and some specified ordered pair for the function. MAT 146

  3. DE Warm-Ups • For the differential equation here, what are the constant solutions? • For the following differential equation, determine whether any of the functions that follow are solutions. MAT 146

  4. More DE Questions (4) Solve this initial-value problem: MAT 146

  5. MAT 146

  6. MAT 146

  7. Solving Differential Equations Solve for y: y’ = −y2 MAT 146

  8. Separable Differential Equations MAT 146

  9. Separable Differential Equations Solve for y: y’ = 3xy MAT 146

  10. Separable Differential Equations MAT 146

  11. Separable Differential Equations MAT 146

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

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

  14. Slope Fields MAT 146

  15. Slope Fields MAT 146

  16. Euler’s Method MAT 146

  17. Separable DEs MAT 146

  18. 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

  19. 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

  20. Applications! MAT 146

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

  22. Applications! MAT 146

  23. Applications! MAT 146

  24. MAT 146

  25. Assignments WebAssignments • DE General and Special Solutions (9.1) • DE Solutions as Visual Representations: Slope Fields (9.2-(I)) • Numerical Approximations to DE: Euler’s Method (9.2-(II) • Analytical Solutions to DE: Separation of Variables (9.3) • DE Applications MAT 146

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