Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + 2xy – y2 + x = 17, (3, 5) (hyperbola) y =
Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2 + 2xy – y2 + x = 17, (3, 5) (hyperbola) y =
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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![**Implicit Differentiation to Find Tangent Line Equation**
Task: Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
Given equation:
\[ x^2 + 2xy - y^2 + x = 17 \]
Point:
\[ (3, 5) \]
Type of curve:
Hyperbola
Solution:
- Find \(\frac{dy}{dx}\) using implicit differentiation.
- Substitute \(x = 3\) and \(y = 5\) into \(\frac{dy}{dx}\) to find the slope of the tangent.
- Use the point-slope form of a line equation:
\[ y - y_1 = m(x - x_1) \]
where \(m\) is the slope and \((x_1, y_1)\) is the point \((3, 5)\).
\[ y = \]
(Note: Box left empty for the final tangent line equation calculation.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3155c08-bee7-4362-af81-183e39589582%2Fd6fe774a-8cfe-459d-a58d-850b87d159a3%2Fyd64ebu_processed.png&w=3840&q=75)
Transcribed Image Text:**Implicit Differentiation to Find Tangent Line Equation**
Task: Use implicit differentiation to find an equation of the tangent line to the curve at the given point.
Given equation:
\[ x^2 + 2xy - y^2 + x = 17 \]
Point:
\[ (3, 5) \]
Type of curve:
Hyperbola
Solution:
- Find \(\frac{dy}{dx}\) using implicit differentiation.
- Substitute \(x = 3\) and \(y = 5\) into \(\frac{dy}{dx}\) to find the slope of the tangent.
- Use the point-slope form of a line equation:
\[ y - y_1 = m(x - x_1) \]
where \(m\) is the slope and \((x_1, y_1)\) is the point \((3, 5)\).
\[ y = \]
(Note: Box left empty for the final tangent line equation calculation.)
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