Taylor‘s Method

1. Taylor‘s Method 수학과 2004023371 이창혁 수학과 2008019455 김종현 수학과 2008057903 조예원

2. Main Theorem Taylor’s Theorem

3. Subject for Inquiry • Problem 1 • Compute the Taylor polynomials of degree 4 for the solutions to the given initial value problems. Use these Taylor polynomials to approximate the solution at x = 1. • (i) dy/dx = x – 2y; y(0) = 1. (ii) dy/dx = y(2 – y); y(0) = 4. • Problem 2 Compare the use of Euler’s method with that of Taylor series to approximate the solution Ф(x) to the initial value problem dy/dx + y = cos x – sin x; y(0) = 0.

4. Result of Research (1/12) • (i) dy/dx = x – 2y; y(0) = 1

5. Result of Research (2/12) • Graphs

6. Result of Research (3/12) • Codes

7. Result of Research (4/12) • (ii) dy/dx = y(2 – y); y(0) = 4.

8. Result of Research (5/12) • Graphs

9. Result of Research (6/12) • Codes

10. Result of Research (7/12) • dy/dx + y = cos x – sin x; y(0) = 0.

11. Result of Research (8/12) • X = 1 • X = 3

12. Result of Research (9/12) • Codes

13. Result of Research (10/12) • Results

14. Result of Research (11/12) • [ ] : error

15. Result of Research (12/12) • [ ] : error

16. Conclusion & Discussion • Taylor Method is approaching the exact value • closer than Euler Method. • Other approaching method… •  Improved Euler Method •  RungeKutta Method

17. Thank You!