You may have observed that the Lagrange interpolating p(x) always goes to ±∞ when x → ±∞. This may be explained by the following facts: ⚫ The Lagrange interpolating polynomial p(x) has degree N - 1 if it interpolates N points: ⚫ Given any polynomial q(x) = x + aм-1xΜ−¹ + ... + a₁ x² + do then lim x-100x q(x) = aм- Question 2 a. Prove the above limit equation. You may assume that the limit of a sum is the sum of the limits (when they exist), and that limx→∞ x = 0 if m<0. (This is a Calc 1 exercise. If you don't remember how to prove it, find a Calc 1 book. )
You may have observed that the Lagrange interpolating p(x) always goes to ±∞ when x → ±∞. This may be explained by the following facts: ⚫ The Lagrange interpolating polynomial p(x) has degree N - 1 if it interpolates N points: ⚫ Given any polynomial q(x) = x + aм-1xΜ−¹ + ... + a₁ x² + do then lim x-100x q(x) = aм- Question 2 a. Prove the above limit equation. You may assume that the limit of a sum is the sum of the limits (when they exist), and that limx→∞ x = 0 if m<0. (This is a Calc 1 exercise. If you don't remember how to prove it, find a Calc 1 book. )
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 make sure you show work and show steps. Please do part A and B
Transcribed Image Text:You may have observed that the Lagrange interpolating p(x) always goes to ±∞o when x →±oo. This may be explained by the following facts:
•
The Lagrange interpolating polynomial p(x) has degree N - 1 if it interpolates N points:
⚫ Given any polynomial q(x) = aмx + aм-1xM- +...+axa then
Question 2
q(x)
lim
= aм.
x-100 xM
= 0 if m<0. (This
a. Prove the above limit equation. You may assume that the limit of a sum is the sum of the limits (when they exist), and that lim→∞ x" =
is a Calc 1 exercise. If you don't remember how to prove it, find a Calc 1 book. )
b. Show how the two bullet points above prove that the Lagrange interpolating polynomial p(x) always goes to ±o when x → ±0.
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