a. Suppose P3(x) is the polynomial of degree 3 which interpolates f(x) = cos(3x) at xo = -1.5h, x1 = -0.5h, x2 = 0.5h, x3 = 1.5h. Find a reasonable upper bound on the error for co < x < x3. You can be "lazy," that is, bound |x – x;| for each i by |æ3 - xol- %3D %3D |34 car(3E)(34)"| 24 273.375 h9 24 b. [Extra credit] Same question, but now, DON'T be lazy, get the best possible bound on |q(x)| = |(x – x0)(x – 21)(x – x2)(x – x3)|. gla) = (x+ €)(x +€)-4)(x-24) = (x² 4*)( x²4) 96? = 2x g lo) = 4 2(fa) =-A* 76 fd-P,/ * = 4 %3D 3,378h4 W.
a. Suppose P3(x) is the polynomial of degree 3 which interpolates f(x) = cos(3x) at xo = -1.5h, x1 = -0.5h, x2 = 0.5h, x3 = 1.5h. Find a reasonable upper bound on the error for co < x < x3. You can be "lazy," that is, bound |x – x;| for each i by |æ3 - xol- %3D %3D |34 car(3E)(34)"| 24 273.375 h9 24 b. [Extra credit] Same question, but now, DON'T be lazy, get the best possible bound on |q(x)| = |(x – x0)(x – 21)(x – x2)(x – x3)|. gla) = (x+ €)(x +€)-4)(x-24) = (x² 4*)( x²4) 96? = 2x g lo) = 4 2(fa) =-A* 76 fd-P,/ * = 4 %3D 3,378h4 W.
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 is given BUT need full detailed steps and process since I don't understand the concept.
![The content of the image appears to be a mathematical problem related to polynomial interpolation. Here's a complete transcription and explanation:
---
### Problem Statement
1. **Suppose \( P_3(x) \)** is the polynomial of degree 3 which interpolates \( f(x) = \cos(3x) \) at \( x_0 = -1.5h \), \( x_1 = -0.5h \), \( x_2 = 0.5h \), \( x_3 = 1.5h \). Find a reasonable upper bound on the error for \( x_0 < x < x_3 \). You can be "lazy," that is, bound \( |x - x_i| \) for each \( i \) by \( |x_3 - x_0| \).
\[
\left| f(x) - P_3(x) \right| \leq \frac{f^{(4)}(\xi)}{4!} (x-x_0)(x-x_1)(x-x_2)(x-x_3) \leq \frac{3^4 (\cos(3 \xi))}{24} (3h)^4
\]
\[
\leq \frac{3^4}{24} h^4 = 273.375 h^4
\]
2. **[Extra credit]** Same question, but now, DON'T be lazy. Get the best possible bound on \( |q(x)| \equiv |(x-x_0)(x-x_1)(x-x_2)(x-x_3)| \).
- Start with the function:
\[
g(x) = (x + \frac{3}{2}h)(x + \frac{1}{2}h)(x - \frac{1}{2}h)(x - \frac{3}{2}h) = \left( x^2 - \frac{9}{4}h^2 \right) \left( x^2 - \frac{1}{4}h^2 \right)
\]
- Calculate the derivative and evaluate at critical points:
\[
g'(x) = 2x \left[ 2x^2 - \frac{10}{4}h^2 \right] = 0 \quad \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb460c0c-d029-4e90-a450-1d82490780a1%2F9518b7d6-193a-4ca1-b90b-f7e550d7a7c6%2Ffs7l1cz_processed.png&w=3840&q=75)
Transcribed Image Text:The content of the image appears to be a mathematical problem related to polynomial interpolation. Here's a complete transcription and explanation:
---
### Problem Statement
1. **Suppose \( P_3(x) \)** is the polynomial of degree 3 which interpolates \( f(x) = \cos(3x) \) at \( x_0 = -1.5h \), \( x_1 = -0.5h \), \( x_2 = 0.5h \), \( x_3 = 1.5h \). Find a reasonable upper bound on the error for \( x_0 < x < x_3 \). You can be "lazy," that is, bound \( |x - x_i| \) for each \( i \) by \( |x_3 - x_0| \).
\[
\left| f(x) - P_3(x) \right| \leq \frac{f^{(4)}(\xi)}{4!} (x-x_0)(x-x_1)(x-x_2)(x-x_3) \leq \frac{3^4 (\cos(3 \xi))}{24} (3h)^4
\]
\[
\leq \frac{3^4}{24} h^4 = 273.375 h^4
\]
2. **[Extra credit]** Same question, but now, DON'T be lazy. Get the best possible bound on \( |q(x)| \equiv |(x-x_0)(x-x_1)(x-x_2)(x-x_3)| \).
- Start with the function:
\[
g(x) = (x + \frac{3}{2}h)(x + \frac{1}{2}h)(x - \frac{1}{2}h)(x - \frac{3}{2}h) = \left( x^2 - \frac{9}{4}h^2 \right) \left( x^2 - \frac{1}{4}h^2 \right)
\]
- Calculate the derivative and evaluate at critical points:
\[
g'(x) = 2x \left[ 2x^2 - \frac{10}{4}h^2 \right] = 0 \quad \
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