University:
Whitman College
Course:
MATH-467 | Numerical Analysis
Academic year:

2022
Views:

367

Pages:

1
Author:

Luna Farrell

A Formula Summary: Derivatives and Integrals Differentiation: • Forward (or backward) Difference Formula: 1 h [f (x + h) − f (x)] − f 00 (c) h 2 2 3 This has type I error: K1 h + K2 h + K3 h + . . . f 0 (x) = • Three point formula: 1 h2 [f (x + h) − f (x − h)] − f 000 (c) 2h 6 2 4 which has a type II error: K1 h + K2 h + . . .. And for an endpoint: f 0 (x) = f 0 (x) = h2 1 [−3f (x) + 4f (x + h) − f (x + 2h)] + f 000 (c) 2h 3 • Second Derivative, three point centered difference: 1 h2 (iv) [f (x − h) − 2f (x) + f (x + h)] + f (c) h2 12 Forward Difference (see the Maple worksheet on our class website) f 00 (x) = 1 [2f (x) − 5f (x + h) + 4f (x + 2h) − f (x + 3h)] + O(h2 ) h2 f 00 (x) = Integration • Trapezoidal Rule: h h3 [y0 + y1 ] − f 00 (c) 2 12 x0 Composite Form, with m + 1 points (m intervals), h = (b − a)/m Z Z b a x1 f (x) dx = m−1 X h (b − a)h2 00 f (x) dx = y0 + ym + 2 yn − f (c) 2 12 n=1 " # • Simpson’s Rule: h h5 [y0 + 4y1 + y2 ] − f (iv) (c) 3 90 x0 Composite Form using 2m + 1 points, a = x0 , b = x2m , evenly spaced: Z Z b a x2 f (x) dx = m m−1 X X h h4 f (x) dx = y0 + y2m + 4 y2i−1 + 2 y2i − (b − a)f (iv) (c) 3 180 i=1 i=1 " #

A Formula Summary Derivatives and Integrals

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