In many algorithms such as Newton-Raphson method for root finding of a nonlinear equation, gradient-based optimization algorithms (e.g., Newton's method, steepest descent method), calculating the derivatives of a function f(x) is often required. Use the basic formulas for the forward finite difference method (FFD), the centered finite difference (CFD), and the backward finite difference (BFD) method to estimate the second derivative of f(x) = e* sin(x²) at x = 0.94 with a step h = 0.05. (points) Use the values given in the following table to numerically calculate the second derivative x f(x) 0.79 1.28759 0.84 1.50214 0.89 1.7334 0.94 1.97894 0.99 2.23521 1.04 2.49738 1.09 2.75912 1.14 3.01249 Determine f" (0.94) numerically using FFD, CFD, and BFC Using FFD, f" (0.94) ~ Using CFD, f" (0.94) ~ Using BFD, f" (0.94) ~
In many algorithms such as Newton-Raphson method for root finding of a nonlinear equation, gradient-based optimization algorithms (e.g., Newton's method, steepest descent method), calculating the derivatives of a function f(x) is often required. Use the basic formulas for the forward finite difference method (FFD), the centered finite difference (CFD), and the backward finite difference (BFD) method to estimate the second derivative of f(x) = e* sin(x²) at x = 0.94 with a step h = 0.05. (points) Use the values given in the following table to numerically calculate the second derivative x f(x) 0.79 1.28759 0.84 1.50214 0.89 1.7334 0.94 1.97894 0.99 2.23521 1.04 2.49738 1.09 2.75912 1.14 3.01249 Determine f" (0.94) numerically using FFD, CFD, and BFC Using FFD, f" (0.94) ~ Using CFD, f" (0.94) ~ Using BFD, f" (0.94) ~
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please provide answers to three sig. digits and do not approximate numbers in intermediate steps
Transcribed Image Text:In many algorithms such as Newton-Raphson method for root finding of a nonlinear equation, gradient-based optimization
algorithms (e.g., Newton's method, steepest descent method), calculating the derivatives of a function f(x) is often required.
Use the basic formulas for the forward finite difference method (FFD), the centered finite difference (CFD), and the backward
finite difference (BFD) method to estimate the second derivative of f(x) = e* sin(x²) at x = 0.94 with a step h = 0.05.
(points)
Use the values given in the following table to numerically calculate the second derivative
X
f(x)
0.79 1.28759
0.84 1.50214
0.89
1.7334
0.94 1.97894
0.99 2.23521
1.04 2.49738
1.09 2.75912
1.14
3.01249
Determine f" (0.94) numerically using FFD, CFD, and BFC
Using FFD, f" (0.94) ~
Using CFD, f" (0.94) ~
Using BFD, f" (0.94) ~
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