Find the equation of tangent line to the curve y = 2 - x 2 (7, ²/7). at the point (7, Express the equation of the tangent line in the form y = mx + b. Simplify completely, entering only reduced fractions.
Find the equation of tangent line to the curve y = 2 - x 2 (7, ²/7). at the point (7, Express the equation of the tangent line in the form y = mx + b. Simplify completely, entering only reduced fractions.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter2: Functions And Graphs
Section2.4: Writing And Graphing Equations Of Lines
Problem 80E
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Question
![### Finding the Equation of a Tangent Line
**Problem Statement:**
Find the equation of the tangent line to the curve \( y = \frac{2}{x} \) at the point \(\left( 7, \frac{2}{7} \right)\).
Express the equation of the tangent line in the form \( y = mx + b \). Simplify completely, entering only reduced fractions.
### Solution Approach:
1. **Determine the derivative of the curve \( y = \frac{2}{x} \):**
The first step involves finding the derivative \( y' \). The derivative of \( y = \frac{2}{x} \) is calculated as follows:
\[
y' = \frac{d}{dx}\left(\frac{2}{x}\right) = -\frac{2}{x^2}
\]
2. **Evaluate the derivative at the given point \( x = 7 \):**
Substitute \( x = 7 \) into the derivative:
\[
y'\bigg|_{x=7} = -\frac{2}{7^2} = -\frac{2}{49}
\]
This value \(-\frac{2}{49}\) is the slope \( m \) of the tangent line at the point \( (7, \frac{2}{7}) \).
3. **Formulate the equation of the tangent line:**
The general form for the equation of a line is \( y = mx + b \). To find \( b \), we use the point \( (7, \frac{2}{7}) \):
\[
\frac{2}{7} = \left(-\frac{2}{49}\right)(7) + b
\]
Simplifying:
\[
\frac{2}{7} = -\frac{14}{49} + b
\]
Since \(\frac{14}{49} = \frac{2}{7}\), we get:
\[
\frac{2}{7} = -\frac{2}{7} + b
\]
Solving for \( b \):
\[
\frac{2}{7} + \frac{2}{7} = b \implies b = \frac](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F78020af4-edac-45d4-a6db-c9f21dc2273f%2Fab207881-f859-437e-8f7f-7c2f34dc3c09%2F2nuw8ty_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Equation of a Tangent Line
**Problem Statement:**
Find the equation of the tangent line to the curve \( y = \frac{2}{x} \) at the point \(\left( 7, \frac{2}{7} \right)\).
Express the equation of the tangent line in the form \( y = mx + b \). Simplify completely, entering only reduced fractions.
### Solution Approach:
1. **Determine the derivative of the curve \( y = \frac{2}{x} \):**
The first step involves finding the derivative \( y' \). The derivative of \( y = \frac{2}{x} \) is calculated as follows:
\[
y' = \frac{d}{dx}\left(\frac{2}{x}\right) = -\frac{2}{x^2}
\]
2. **Evaluate the derivative at the given point \( x = 7 \):**
Substitute \( x = 7 \) into the derivative:
\[
y'\bigg|_{x=7} = -\frac{2}{7^2} = -\frac{2}{49}
\]
This value \(-\frac{2}{49}\) is the slope \( m \) of the tangent line at the point \( (7, \frac{2}{7}) \).
3. **Formulate the equation of the tangent line:**
The general form for the equation of a line is \( y = mx + b \). To find \( b \), we use the point \( (7, \frac{2}{7}) \):
\[
\frac{2}{7} = \left(-\frac{2}{49}\right)(7) + b
\]
Simplifying:
\[
\frac{2}{7} = -\frac{14}{49} + b
\]
Since \(\frac{14}{49} = \frac{2}{7}\), we get:
\[
\frac{2}{7} = -\frac{2}{7} + b
\]
Solving for \( b \):
\[
\frac{2}{7} + \frac{2}{7} = b \implies b = \frac
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