1. NUMERICAL DIFFERENTIATION The derivative of f (x) at x 0 is: An approximation to this is: for small values of h. Forward Difference Formula

2. Find an approximate value for 0.10.58778670.6418539 0.5406720 0.010.58778670.5933268 0.5540100 0.0010.58778670.5883421 0.5554000 The exact value of

3. Assume that a function goes through three points: Lagrange Interpolating Polynomial

6. If the points are equally spaced, i.e.,

7. Three-point formula:

8. If the points are equally spaced with x 0 in the middle:

9. Another Three-point formula:

10. Alternate approach (Error estimate) Take Taylor series expansion of f(x+h) about x:.............. (1)

11. Forward Difference Formula

12. ................. (2)

13. .............. (1)................. (2) 2 X Eqn. (1) – Eqn. (2)

15. Three-point Formula

16. Take Taylor series expansion of f(x+h) about x: Take Taylor series expansion of f(x-h) about x: The Second Three-point Formula Subtract one expression from another

18. Second Three-point Formula

19. Forward Difference Formula Summary of Errors Error term

20. First Three-point Formula Summary of Errors continued Error term

21. Second Three-point Formula Error term Summary of Errors continued

22. Example: Find the approximate value of 1.9 12.703199 2.0 14.778112 2.1 17.148957 2.2 19.855030 with

23. Using the Forward Difference formula:

24. Using the 1 st Three-point formula:

25. Using the 2 nd Three-point formula: The exact value of

26. Comparison of the results with h = 0.1 FormulaError Forward Difference23.7084501.541282 1st Three-point22.0323100.134858 2nd Three-point22.2287900.061622 The exact value ofis 22.167168

27. Second-order Derivative Add these two equations.

29. NUMERICAL INTEGRATION area under the curve f(x) between In many cases a mathematical expression for f(x) is unknown and in some cases even if f(x) is known its complex form makes it difficult to perform the integration.

32. Area of the trapezoid The length of the two parallel sides of the trapezoid are: f(a) and f(b) The height is b-a

33. Simpson’s Rule:

40. Composite Numerical Integration

41. Riemann Sum The area under the curve is subdivided into n subintervals. Each subinterval is treated as a rectangle. The area of all subintervals are added to determine the area under the curve. There are several variations of Riemann sum as applied to composite integration.

42. In Left Riemann sum, the left- side sample of the function is used as the height of the individual rectangle.

43. In Right Riemann sum, the right-side sample of the function is used as the height of the individual rectangle.

44. In the Midpoint Rule, the sample at the middle of the subinterval is used as the height of the individual rectangle.

45. Composite Trapezoidal Rule: Divide the interval into n subintervals and apply Trapezoidal Rule in each subinterval. where

46. by dividing the interval into 20 subintervals. Find

48. Composite Simpson’s Rule: Divide the interval into n subintervals and apply Simpson’s Rule on each consecutive pair of subinterval. Note that n must be even.

49. where by dividing the interval into 20 subintervals. Find