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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![**Question:**
If \( p(x) \) is a polynomial of degree 8, then \( p^{(9)}(x) \) is zero.
- ○ True
- ○ False
**Explanation:**
This question is about differentiating a polynomial. The notation \( p^{(9)}(x) \) represents the ninth derivative of the polynomial \( p(x) \).
A polynomial of degree 8 can be expressed in the form:
\[ p(x) = a_8x^8 + a_7x^7 + \cdots + a_1x + a_0 \]
For a polynomial of degree \( n \), any derivative with order greater than \( n \) results in zero. Therefore, the ninth derivative \( p^{(9)}(x) \) of a polynomial of degree 8 is zero, because all terms will have been differentiated to zero by the time we reach the ninth derivative.
Hence, the correct answer is:
- ◉ True](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7fed27a2-e2b8-433b-9aa6-02621bc09518%2F4525e305-1176-475a-954b-c7f2f4b2d9b4%2Fdm91hw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
If \( p(x) \) is a polynomial of degree 8, then \( p^{(9)}(x) \) is zero.
- ○ True
- ○ False
**Explanation:**
This question is about differentiating a polynomial. The notation \( p^{(9)}(x) \) represents the ninth derivative of the polynomial \( p(x) \).
A polynomial of degree 8 can be expressed in the form:
\[ p(x) = a_8x^8 + a_7x^7 + \cdots + a_1x + a_0 \]
For a polynomial of degree \( n \), any derivative with order greater than \( n \) results in zero. Therefore, the ninth derivative \( p^{(9)}(x) \) of a polynomial of degree 8 is zero, because all terms will have been differentiated to zero by the time we reach the ninth derivative.
Hence, the correct answer is:
- ◉ True
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