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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![**Finding the Equation of the Tangent Line to a Graph**
To find the equation for the line tangent to the graph of the function
\[ f(x) = \frac{2x}{x + 4} \]
at the point \((2, 0.66666666666667)\).
Below the instruction, there is an empty input box where students can input the equation of the tangent line in the form \( y = \).
### Explanation:
1. **Identify the Function**: The function given is \( f(x) = \frac{2x}{x + 4} \).
2. **Given Point on the Graph**: The point at which the tangent line is to be found is \((2, 0.66666666666667)\).
3. **Objective**: Find the equation of the tangent line to the graph of \( f(x) \) at the given point.
### Steps to Find the Tangent Line Equation:
1. **Find the derivative of the function** \( f(x) \) to determine the slope of the tangent line.
2. **Evaluate the derivative at \( x = 2 \) to find the slope at the given point**.
3. **Use the point-slope form of the equation of a line**:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point \((2, 0.66666666666667)\) and \( m \) is the slope found in step 2.
### Input Your Answer:
Students will input their final equation for the tangent line in the designated box provided.
---
This structured explanation helps guide learners through the process of finding the tangent line step-by-step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d39eb1c-3154-4bee-b19a-a1be6277f6e8%2F3651f5a9-6a40-41a4-89e1-4081ad9e7adb%2F2o9hnyo_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding the Equation of the Tangent Line to a Graph**
To find the equation for the line tangent to the graph of the function
\[ f(x) = \frac{2x}{x + 4} \]
at the point \((2, 0.66666666666667)\).
Below the instruction, there is an empty input box where students can input the equation of the tangent line in the form \( y = \).
### Explanation:
1. **Identify the Function**: The function given is \( f(x) = \frac{2x}{x + 4} \).
2. **Given Point on the Graph**: The point at which the tangent line is to be found is \((2, 0.66666666666667)\).
3. **Objective**: Find the equation of the tangent line to the graph of \( f(x) \) at the given point.
### Steps to Find the Tangent Line Equation:
1. **Find the derivative of the function** \( f(x) \) to determine the slope of the tangent line.
2. **Evaluate the derivative at \( x = 2 \) to find the slope at the given point**.
3. **Use the point-slope form of the equation of a line**:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) \) is the point \((2, 0.66666666666667)\) and \( m \) is the slope found in step 2.
### Input Your Answer:
Students will input their final equation for the tangent line in the designated box provided.
---
This structured explanation helps guide learners through the process of finding the tangent line step-by-step.
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