[4] [Centered difference approximation of second derivative] Let fe C4 ([a, b]). (a) Use Taylor's theorem to write ƒ as a third order (cubic) Taylor poly- nomial plus a fourth order (quartic) remainder term. Expand about the point xo. (b) Using the result from (a), evaluate f(x) at the points x = xo+h and x = xo - h; then, add the two results together to derive the centered difference approxi- mation to the second derivative: f(xo+h) — 2ƒ (x0) + ƒ (x − h) h² h² = ƒ” (xo) + ^ (ƒ(¹) (E₁) + ƒ(¹) ({₂)) 4!
[4] [Centered difference approximation of second derivative] Let fe C4 ([a, b]). (a) Use Taylor's theorem to write ƒ as a third order (cubic) Taylor poly- nomial plus a fourth order (quartic) remainder term. Expand about the point xo. (b) Using the result from (a), evaluate f(x) at the points x = xo+h and x = xo - h; then, add the two results together to derive the centered difference approxi- mation to the second derivative: f(xo+h) — 2ƒ (x0) + ƒ (x − h) h² h² = ƒ” (xo) + ^ (ƒ(¹) (E₁) + ƒ(¹) ({₂)) 4!
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
Taylor's theorem
![**Centered Difference Approximation of Second Derivative**
Let \( f \in C^4([a, b]) \).
**(a)** Use Taylor’s theorem to write \( f \) as a third order (cubic) Taylor polynomial plus a fourth order (quartic) remainder term. Expand about the point \( x_0 \).
**(b)** Using the result from (a), evaluate \( f(x) \) at the points:
\[ x = x_0 + h \quad \text{and} \quad x = x_0 - h; \]
then, add the two results together to derive the centered difference approximation to the second derivative:
\[
\frac{f(x_0 + h) - 2f(x_0) + f(x_0 - h)}{h^2} = f''(x_0) + \frac{h^2}{4!} \left( f^{(4)}(\xi_1) + f^{(4)}(\xi_2) \right)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fd47556-f3ce-4f39-818d-be563d9523c8%2F3ce4bf62-ec0d-4185-83c2-e200773260f2%2Fcxgmhy_processed.png&w=3840&q=75)
Transcribed Image Text:**Centered Difference Approximation of Second Derivative**
Let \( f \in C^4([a, b]) \).
**(a)** Use Taylor’s theorem to write \( f \) as a third order (cubic) Taylor polynomial plus a fourth order (quartic) remainder term. Expand about the point \( x_0 \).
**(b)** Using the result from (a), evaluate \( f(x) \) at the points:
\[ x = x_0 + h \quad \text{and} \quad x = x_0 - h; \]
then, add the two results together to derive the centered difference approximation to the second derivative:
\[
\frac{f(x_0 + h) - 2f(x_0) + f(x_0 - h)}{h^2} = f''(x_0) + \frac{h^2}{4!} \left( f^{(4)}(\xi_1) + f^{(4)}(\xi_2) \right)
\]
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