(a) Derive a finite difference formula for the first derivative by using three terms plus the remainder of the Taylor series expansion for a set of points that is not equally spaced. The solution should the derivative at point x=xi in terms of X₁, X₁-1, X₁-2, X₁-3, f(xi), f(x-1), f(x-2) and f(xi- 3). (b) Derive a simplified formula in the case of equally spaced points. Submit your hand calculations.
(a) Derive a finite difference formula for the first derivative by using three terms plus the remainder of the Taylor series expansion for a set of points that is not equally spaced. The solution should the derivative at point x=xi in terms of X₁, X₁-1, X₁-2, X₁-3, f(xi), f(x-1), f(x-2) and f(xi- 3). (b) Derive a simplified formula in the case of equally spaced points. Submit your hand calculations.
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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Answer Question number 2 please.
Transcribed Image Text:1. (a) Use the Taylor series expansion to derive a three-point finite difference formula that
evaluates the first derivative dy/dx at point x=x; with given points (x,yi), (X₁+1, Yi+1), (Xi+2, Yi+2),
which are not equally spaced. Use three terms of the series plus the remainder. (b) Derive for
the case when the spacing between the points is equal, a simpler finite difference formula.
(c) Use the data in the table to derive in a MatLab scipt the derivative of the point x=x₁. No
MatLab build-in function is allowed to be used. Submit for (a) and (b) your hand calculations
and for (c) your MatLab files.
Xi
5.49
Xi+1
5.58
Xi+2
5.63
Yi
8.08
Yi+1
8.12
Yi+2
8.15
2. (a) Derive a finite difference formula for the first derivative by using three terms plus the
remainder of the Taylor series expansion for a set of points that is not equally spaced. The
solution should the derivative at point x=xi in terms of xi, Xi-1, Xi-2, Xi-3, f(xi), f(x-1), f(xi-2) and f(xi-
3). (b) Derive a simplified formula in the case of equally spaced points. Submit your hand
calculations.
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