Numerical Differentiation and Integration

University:
California State University, Northridge
Course:
MATH 331 | Mathematical Explorations
Academic year:

2023
Views:

328

Pages:

32
Author:

Yung D

When 5 [½ (4%+Ay-1)-+(y-3) dy dy ()=[(-2)+...] 2=70 Derivative ying Bressel's formala Maxima For and minimal of a tabulated function maxima of mintona dyo Hence canating the right hand side of (1) to zero and retaining only upto third diftrencar we obtain 0% + 24-14%+ 2 34-64+2 83% = 0 6 (4)(40)²+(42%-24%) + (4%-x4²+4%)=0 Substituting the values of 4%, 4²o, o trom the différence table, wo solve this quadratic foru. Then the cabresponding Values of rare given by x = lothh at which y is maximum (01) minimum. Problems: ① Find &'(3) and (3) for one following data: Slution! since we require of (3) and of"(3) we se Newton's forward formula. Difference Table Ay 3.0 -14 -10.032 3.968 y 0-768 3-2 4-736 34-5-296 0-304 -0-464 2.048 36 5.04 -0266 1-882 -3.072 -5-12 [3-968-(0-768) +5 (-0.464)-$(2.048)+(-5-12) 38 1.928 -1-488 6-672 0.4 4.0 7-328 14 By Newton's forward formula Idy = (dy Here h=0.2 1-1584 2=70 [3-968-0-384-0-1547-0-5124-024] =[1.8933] = 9.4665 = [2²%-2%+ 44%-45% +...] 2 1 (2) [0-768-(-0.464)+(2-048)-5%(-5-12) 0.04 0.768 +0.464 +1.8773 +4.267) 0.04 [7-3763] = 184.4075 Compute f'(0) and f"(4) from the data. 2-718 7-381 2-0.036 Solution 54598 suice we require f' (0.5) and 6" (3-5) we use Newton's forward formula and Newton's backward formula. Difference bable. 4 y 2-7 1-718 2 4-663 5.097 7.381 8.04-2 8.668 12-705 13-765 20 0824 54.598 2.945 34-512 2.1.807 By Newton's forward formula 21-807+13-765 +7.945 = 43.5177A Find the maximuss and minimum value of y tabulated below. Newton's forward difference formula is y(a) = % +44% + 4/4-1) 21 2024% + 4(4-1) (4-2) 23% +...] =[%+(242% + 2! 3-64+24%+...] Forward defference table 1-0 Choosing to=0, ua dy 3x²-62+2 4x²-18%+221-6 = +-0.25 +2 (25) +-+ (4) + x²-2x-6 (6)] (9)+ 24 =-0-25+(22-1) (2.5)+3x²-6x+219) + 4x²-18x²+222-6(6) =-0.25 +2.5x-1.25 +4.5x²-9x+3+x²-4-6x+5-51-1-5 Now 24 22-100 x00, (x-1) (+1)=0 x=0, 2=1, x=-1 dy dr2 = 3x²-1 at x=0, diy dare = -1 =-ve d2y 42 = 3-1=2=tve x=1 スニー dly dal= 3-12 = tue y is maximum ato minimuns at 2=1 and -1 y(x)=[%+ 24% + 21 4(0) + [0+0] 20 maximum value =0 y(1) = +[% + 4% +++] = [0+(-0.25)] =-0-25 maximum at x=1, 4(1)=-0.25. Consider the following table of data. 0.2 04 0.6 0.8 (2) 0.9798652 0.9177710 0-80803480-6386093/0.3843735 Find &'(0.25) wing Newton's forward difference approximation (0-6) using Stirling's approximation and f'(0.95) wing, Newton's back ward difference approximation. Newton's forward interpolation sormula tor darivatives