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Let p be an arbitrary polynomial p(x) = a„x" + a,-1x"-1 +..+a,x + ao, a, # 0. (a) Find (d" /dx")[p(x)]. (b) What is (d* /dx*)[p(x)] for k > n?

Let p be an arbitrary polynomial p(x) = a„x" + a,-1x"-1 +..+a,x + ao, a, # 0. (a) Find (d" /dx")[p(x)]. (b) What is (d* /dx*)[p(x)] for k > n?

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Polynomial Differentiation Exercise** Let \( p \) be an arbitrary polynomial defined as: \[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, \, a_n \neq 0. \] **Questions:** (a) Find \(\frac{d^n}{dx^n}[p(x)]\). (b) What is \(\frac{d^k}{dx^k}[p(x)]\) for \(k > n\)? **Explanation:** To solve these differentiation problems, consider the order of derivatives: - For part (a), calculating the \(n\)-th derivative of a polynomial will reduce it to a constant if it is of degree \(n\). - For part (b), any derivative of order greater than \(n\) of a polynomial of degree \(n\) is zero, since all terms will have been fully differentiated.
Transcribed Image Text:**Polynomial Differentiation Exercise** Let \( p \) be an arbitrary polynomial defined as: \[ p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0, \, a_n \neq 0. \] **Questions:** (a) Find \(\frac{d^n}{dx^n}[p(x)]\). (b) What is \(\frac{d^k}{dx^k}[p(x)]\) for \(k > n\)? **Explanation:** To solve these differentiation problems, consider the order of derivatives: - For part (a), calculating the \(n\)-th derivative of a polynomial will reduce it to a constant if it is of degree \(n\). - For part (b), any derivative of order greater than \(n\) of a polynomial of degree \(n\) is zero, since all terms will have been fully differentiated.
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