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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![### Problem Statement
Find the function \( f \) such that \( f'(x) = 10x - 7 \) and \( f(8) = 0 \).
#### Solution
To find the function \( f(x) \), we need to integrate the given derivative \( f'(x) = 10x - 7 \).
**Step 1: Integrate \( f'(x) \)**
Integrate the expression \( 10x - 7 \):
\[ \int (10x - 7) \, dx \]
The integral is:
\[ f(x) = \int 10x \, dx - \int 7 \, dx \]
\[ f(x) = 10 \left( \frac{x^2}{2} \right) - 7x + C \]
\[ f(x) = 5x^2 - 7x + C \]
Here, \( C \) is the constant of integration.
**Step 2: Determine the constant \( C \)**
We use the initial condition \( f(8) = 0 \):
\[ f(8) = 5(8)^2 - 7(8) + C = 0 \]
\[ 5(64) - 56 + C = 0 \]
\[ 320 - 56 + C = 0 \]
\[ 264 + C = 0 \]
\[ C = -264 \]
**Final Function**
Therefore, the function \( f(x) \) is:
\[ f(x) = 5x^2 - 7x - 264 \]
### Answer
\[ f(x) = 5x^2 - 7x - 264 \]
This is the required function \( f \) that satisfies the given conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febbbc1cc-ba98-4671-a28a-276fb59e8560%2F0599071b-8b01-4e3d-8287-6b09b3b4140c%2Ffaepjn_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Find the function \( f \) such that \( f'(x) = 10x - 7 \) and \( f(8) = 0 \).
#### Solution
To find the function \( f(x) \), we need to integrate the given derivative \( f'(x) = 10x - 7 \).
**Step 1: Integrate \( f'(x) \)**
Integrate the expression \( 10x - 7 \):
\[ \int (10x - 7) \, dx \]
The integral is:
\[ f(x) = \int 10x \, dx - \int 7 \, dx \]
\[ f(x) = 10 \left( \frac{x^2}{2} \right) - 7x + C \]
\[ f(x) = 5x^2 - 7x + C \]
Here, \( C \) is the constant of integration.
**Step 2: Determine the constant \( C \)**
We use the initial condition \( f(8) = 0 \):
\[ f(8) = 5(8)^2 - 7(8) + C = 0 \]
\[ 5(64) - 56 + C = 0 \]
\[ 320 - 56 + C = 0 \]
\[ 264 + C = 0 \]
\[ C = -264 \]
**Final Function**
Therefore, the function \( f(x) \) is:
\[ f(x) = 5x^2 - 7x - 264 \]
### Answer
\[ f(x) = 5x^2 - 7x - 264 \]
This is the required function \( f \) that satisfies the given conditions.
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