Determine the points at which the graph of the function has a horizontal tangent line. f(x) = x - 3 (x, y) = (x, y) = (smaller x-value) (larger x-value)
Determine the points at which the graph of the function has a horizontal tangent line. f(x) = x - 3 (x, y) = (x, y) = (smaller x-value) (larger x-value)
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
![**Title: Understanding Horizontal Tangents for Rational Functions**
**Objective:**
Determine the points at which the graph of the function has a horizontal tangent line.
**Function Analysis:**
Given the function:
\[ f(x) = \frac{x^2}{x - 3} \]
**Task:**
Find the coordinates \((x, y)\) where the horizontal tangent line occurs for the given function.
**Instructions:**
Please fill in the values of \((x, y)\) for both the smaller and larger x-values.
**Form:**
- \((x, y) = {}\) (smaller x-value)
- \((x, y) = {}\) (larger x-value)
Ensure you analyze the function and solve for \(x\) where the derivative \( f'(x) = 0 \) to identify the points where the horizontal tangents occur.
**Additional Notes:**
A horizontal tangent line indicates that at those particular points, the slope of the function is zero. This is determined by setting the derivative of the function to zero and solving for \(x\). Then, place the values of \(x\) found back into the original function to find the corresponding \(y\) values.
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---
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Transcribed Image Text:**Title: Understanding Horizontal Tangents for Rational Functions**
**Objective:**
Determine the points at which the graph of the function has a horizontal tangent line.
**Function Analysis:**
Given the function:
\[ f(x) = \frac{x^2}{x - 3} \]
**Task:**
Find the coordinates \((x, y)\) where the horizontal tangent line occurs for the given function.
**Instructions:**
Please fill in the values of \((x, y)\) for both the smaller and larger x-values.
**Form:**
- \((x, y) = {}\) (smaller x-value)
- \((x, y) = {}\) (larger x-value)
Ensure you analyze the function and solve for \(x\) where the derivative \( f'(x) = 0 \) to identify the points where the horizontal tangents occur.
**Additional Notes:**
A horizontal tangent line indicates that at those particular points, the slope of the function is zero. This is determined by setting the derivative of the function to zero and solving for \(x\). Then, place the values of \(x\) found back into the original function to find the corresponding \(y\) values.
[Textboxes are provided for user input]
**Subscription:**
Stay updated for more mathematical problems and solutions by subscribing to our educational platform. Discover in-depth explanations and techniques to enhance your learning experience.
For more information or personalized assistance, contact us through our educational portal.
---
*Note: There are no graphs or diagrams in the provided image to describe.*
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