You may have observed that the Lagrange interpolating p(x) always goes to ±oo when x → ±00. This may be explained by the following facts: • The Lagrange interpolating polynomial p(x) has degree N - 1 if it interpolates N points: M-I . Given any polynomial q(x) = aмx + am−1xM−1 + . + a₁x' + a then q(x) lim = aм· x-100 xM 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 lim→∞ 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. ) b. Show how the two bullet points above prove that the Lagrange interpolating polynomial p(x) always goes to ±∞o when x → ±00. c. We may ask the question of whether p(x) goes to +∞o or -∞o when x → ∞o. Suppose p(x) interpolates the function f(x) at the points x₁, ... XN, so that p(xn) = f(xn), n = 1,... N. Show how the sign of p(x) as x → ∞ can be determined from the values of x1, ...x and the values of f(x1),... f(xN). Similarly determine the sign of p(x) as x → ∞o.

You may have observed that the Lagrange interpolating p(x) always goes to ±oo when x → ±00. This may be explained by the following facts: • The Lagrange interpolating polynomial p(x) has degree N - 1 if it interpolates N points: M-I . Given any polynomial q(x) = aмx + am−1xM−1 + . + a₁x' + a then q(x) lim = aм· x-100 xM 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 lim→∞ 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. ) b. Show how the two bullet points above prove that the Lagrange interpolating polynomial p(x) always goes to ±∞o when x → ±00. c. We may ask the question of whether p(x) goes to +∞o or -∞o when x → ∞o. Suppose p(x) interpolates the function f(x) at the points x₁, ... XN, so that p(xn) = f(xn), n = 1,... N. Show how the sign of p(x) as x → ∞ can be determined from the values of x1, ...x and the values of f(x1),... f(xN). Similarly determine the sign of p(x) as x → ∞o.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 54E

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