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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![Expand \( f(x) = 5x^2 + 35x + 64 \) as a power series around \( x = -3 \).
\[ f(x) = \quad \boxed{} \quad (x + 3)^2 + \quad \boxed{} \quad (x + 3) + \quad \boxed{} \]
In this exercise, you are given a quadratic function, \(f(x) = 5x^2 + 35x + 64\), and asked to expand it as a power series around \(x = -3\).
The task involves expressing the given function in the form of a power series centered around \(x = -3\), which would look like:
\[ f(x) = a_0 + a_1(x + 3) + a_2(x + 3)^2 + \cdots \]
In the given image:
1. The term \((x + 3)^2\) indicates the coefficient for the squared term after factoring about \(x = -3\).
2. The term \((x + 3)\) represents the linear term after factoring about \(x = -3\).
3. The boxed areas suggest input fields where the coefficients of the respective terms should be entered, having split the polynomial according to the power series format around \(x = -3\).
To solve for the coefficients, follow these steps:
1. Use the translation \( u = x + 3 \).
2. Replace \( x \) with \( u - 3 \) in the original function.
3. Expand and collect terms in terms of \( u \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc5134d69-32d8-4659-bfc6-df6862919637%2F9a32dcd0-807d-47da-a602-d76ab78be872%2Fpqa57db_processed.png&w=3840&q=75)
Transcribed Image Text:Expand \( f(x) = 5x^2 + 35x + 64 \) as a power series around \( x = -3 \).
\[ f(x) = \quad \boxed{} \quad (x + 3)^2 + \quad \boxed{} \quad (x + 3) + \quad \boxed{} \]
In this exercise, you are given a quadratic function, \(f(x) = 5x^2 + 35x + 64\), and asked to expand it as a power series around \(x = -3\).
The task involves expressing the given function in the form of a power series centered around \(x = -3\), which would look like:
\[ f(x) = a_0 + a_1(x + 3) + a_2(x + 3)^2 + \cdots \]
In the given image:
1. The term \((x + 3)^2\) indicates the coefficient for the squared term after factoring about \(x = -3\).
2. The term \((x + 3)\) represents the linear term after factoring about \(x = -3\).
3. The boxed areas suggest input fields where the coefficients of the respective terms should be entered, having split the polynomial according to the power series format around \(x = -3\).
To solve for the coefficients, follow these steps:
1. Use the translation \( u = x + 3 \).
2. Replace \( x \) with \( u - 3 \) in the original function.
3. Expand and collect terms in terms of \( u \).
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