Exploring Differential Equations and Power Series in Calculus II

1. Calculus II (MAT 146)Dr. Day Wednesday April 18, 2018 Differential Equations • What is a differential equation? (9.1) • Solving Differential Equations • Visual: Slope Fields (9.2) • Numerical: Euler’s Method (9.2) • Analytical: Separation of Variables (9.3) • Applications of Differential Equations Infinite Sequences & Series (Ch 11) • What is a sequence? A series? (11.1,11.2) • Determining Series Convergence • Divergence Test (11.2) • Integral Test (11.3) • Comparison Tests (11.4) • Alternating Series Test (11.5) • Ratio Test (11.6) • Nth-Root Test (11.6) • Power Series • Interval & Radius of Convergence • New Functions from Old • Taylor Series and Maclaurin Series Integration Applications • Area Between Curves (6.1) • Average Value of a Function (6.5) • Volumes of Solids (6.2, 6.3) • Created by Rotations • Created using Cross Sections • Arc Length of a Curve (8.1) • Probability (8.5) Methods of Integration • U-substitution (5.5) • Integration by Parts (7.1) • Trig Integrals (7.2) • Trig Substitution (7.3) • Partial-Fraction Decomposition (7.4) • Putting it All Together: Strategies! (7.5) • Improper Integrals (7.8)

2. Converge or Diverge?

5. Power Series • The sum of the series is a function with domain the set of all x values for which the series converges. • The function seems to be a polynomial, except it has an infinite number of terms.

6. Power Series: Example • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series:

7. Generalized Power Series • This is called: • a power series in (x – a), or • a power series centered at a, or • a power series about a.

8. Power Series Convergence • For what values of x does each series converge? • Determine the Radius of Convergence and theInterval of Convergencefor each power series.

9. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence.

10. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence.

11. Power Series Convergence • For what values of x does this series converge? • Use the Ratio Test to determine values of x that result in a convergent series.

12. Power Series Convergence • For what values of x does this series converge? • Use the Ratio Test to determine values of x that result in a convergent series.

13. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence.

14. Power Series Convergence • For what values of x does this series converge? • Determine its Radius of Convergence and its Interval of Convergence.

15. Geometric Power Series • If we let cn = 1 for all n, we get a familiar series: • This geometric series has common ratio x and we know the series converges for |x| < 1. • We also know the sum of this series:

16. Geometric Power Series

17. Geometric Power Series

18. Geometric Power Series

19. Why StudySequences and Seriesin Calc II? Taylor Polynomials applet • Infinite Process Yet Finite Outcome . . . How Can That Be? • Transition to Proof • Re-Expression!

20. 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:

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

25. Create a 3rd-degree polynomial approximator for

26. Beyond Geometric Series Connections:Taylor Series How can we describe the cnso a power series can represent OTHER functions? ANY functions? Now we go way back to the ideas that motivated this chapter’s investigations and connections: Polynomial Approximators!

27. Taylor Series Demo #1 Taylor Series Demo #2 Taylor Series Demo #3 Taylor Series Demo #4

28. Taylor Series Look at characteristics of the function in question and connect those to the cn. Example: f(x) = ex, centered around a = 0.

29. Taylor Series Example: f(x) = ex, centered around a = 0. And…how far from a = 0 can we stray and still find this re-expression useful?

30. General Form: Coefficients cn

31. Examples: Determining the cn f(x) = cos(x), centered around a = 0.

32. Examples: Determining the cn f(x) = sin(x), centered around a = 0.

33. Examples: Determining the cn f(x) = ln(1-x), centered around a = 0.