Question

Numeric Analysis

1) Letf(x)=- 2x x2-4 a. Approxi the first derivative using central difference and the second derivative mate at x= 1.25 using and h=0.25. (Use 4 decimal digits using backward difference rounding). Approximate f f(x)dx using trapezoidal rule with h-0.5 (Use four decimal digits rounding) c. Approximate S fCx)dxusing 1/3 Simpson's rule with h-0.25 (Use four decimal digits rounding)

Answer 1

F (x) = 2x/x^2 - 4 using centrol difference: (rist derivative) at x_i = 1.25 h = 0.25 x_i = 1.25, x + i + 1 = 1.25 + 0.25 = 1.5 x_i - 1 = 1.25 - 0.25 = 1 f (x_i) = f (1.25) = 2 times 1.25/(1.25)^2 - 4 = -1.0256 f (x_i + 1) = f (1.5) = 2 times 1.5/(1.5)^2 - 4 = -1.7143 f (x_i - 1) = f (1) = 2 times 1/1^2 - 4 = -0.6667 from central difference, f^1 (x_i) = f^1 (1.25) = f (x_i + n) - f (x_i - h)/2 delta x = -1.7143 - (- 0.6667)/2 times 0.25 f^1 (1.25) = -2.0952 Second derivative using backward difference. X_i - 2 = 1.25 - 2 times 0.25 = 0.75 f (x_i - 2) = f (0.75 = 2 times 0.75/(0.75)^2 - 4 = -0.4364 f^11 (x_i) = f (x_i) - 2f (x_i - 1) + f (x_i + 2/