I have a data set with unequally spaced points: x 0 0.2 0.3 f(x) -2.1 0.3 1.4 I would like to estimate f ′(0.2). Let’s consider h = 0.1 and x0 = 0.2. (a)Derive a second order accurate finite difference method using x0, x0−2h, and x0 + h. The derivation will involve a Taylor polynomial evaluated at x0−2h and x0+h. Clearly show the error term O(h2) result in your derivation. (b) Apply your finite difference formula to estimate f ′(0.2).
I have a data set with unequally spaced points: x 0 0.2 0.3 f(x) -2.1 0.3 1.4 I would like to estimate f ′(0.2). Let’s consider h = 0.1 and x0 = 0.2. (a)Derive a second order accurate finite difference method using x0, x0−2h, and x0 + h. The derivation will involve a Taylor polynomial evaluated at x0−2h and x0+h. Clearly show the error term O(h2) result in your derivation. (b) Apply your finite difference formula to estimate f ′(0.2).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
I have a data set with unequally spaced points:
| x | 0 | 0.2 | 0.3 |
| f(x) | -2.1 | 0.3 | 1.4 |
I would like to estimate f ′(0.2). Let’s consider h = 0.1 and x0 = 0.2.
(a)Derive a second order accurate finite difference method using x0, x0−2h, and x0 + h.
The derivation will involve a Taylor polynomial evaluated at x0−2h and x0+h.
Clearly show the error term O(h2) result in your derivation.
(b) Apply your finite difference formula to estimate f ′(0.2).
![Your formula will take the form:
\[
f'(x_0) = \frac{A f(x_0 - 2h) + B f(x_0) + C f(x_0 + h)}{D h} + \mathcal{O}(h^2)
\]
This equation represents a finite difference approximation for the derivative of a function \(f(x)\) at a point \(x_0\). The terms \(A\), \(B\), \(C\), and \(D\) are coefficients that determine the weights of the function values at different points. The variable \(h\) represents a small step size or interval. The term \(\mathcal{O}(h^2)\) indicates the error term, signifying that the approximation error is on the order of \(h^2\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2de2484e-8c30-4a24-aa43-7ad85b110ab5%2F4559ef17-f140-41ea-9387-c37d1e59d7b1%2F8eemtvx_processed.png&w=3840&q=75)
Transcribed Image Text:Your formula will take the form:
\[
f'(x_0) = \frac{A f(x_0 - 2h) + B f(x_0) + C f(x_0 + h)}{D h} + \mathcal{O}(h^2)
\]
This equation represents a finite difference approximation for the derivative of a function \(f(x)\) at a point \(x_0\). The terms \(A\), \(B\), \(C\), and \(D\) are coefficients that determine the weights of the function values at different points. The variable \(h\) represents a small step size or interval. The term \(\mathcal{O}(h^2)\) indicates the error term, signifying that the approximation error is on the order of \(h^2\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
