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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find the equation of the tangent line at the given point on the curve
![**Problem Statement:**
Find the equation of the tangent line at the given point on the curve.
\[ x^3 + 2xy - y^2 = 11 \quad \text{at} \quad (2,3) \]
---
**Explanation:**
To solve this problem, follow the steps below:
1. **Implicit Differentiation**: Differentiate the given equation implicitly with respect to \(x\).
2. **Solve for \( \frac{dy}{dx} \) **: Find the derivative \( \frac{dy}{dx} \), which represents the slope of the tangent line to the curve at any point \((x, y)\).
3. **Evaluate the Slope**: Substitute the given point \((2, 3)\) into the derivative to find the slope at that particular point.
4. **Point-Slope Form**: Use the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \) to find the equation of the tangent line, where \(m\) is the slope, and \( (x_1, y_1) \) is the given point.
---
Note: No graphs or diagrams are associated with this problem in the provided image.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F198e5e1e-f879-4e60-b997-12ec96a74c50%2Fccc94572-6571-48cf-bc4a-44b1ed71f765%2Frz8j3ef_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the equation of the tangent line at the given point on the curve.
\[ x^3 + 2xy - y^2 = 11 \quad \text{at} \quad (2,3) \]
---
**Explanation:**
To solve this problem, follow the steps below:
1. **Implicit Differentiation**: Differentiate the given equation implicitly with respect to \(x\).
2. **Solve for \( \frac{dy}{dx} \) **: Find the derivative \( \frac{dy}{dx} \), which represents the slope of the tangent line to the curve at any point \((x, y)\).
3. **Evaluate the Slope**: Substitute the given point \((2, 3)\) into the derivative to find the slope at that particular point.
4. **Point-Slope Form**: Use the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \) to find the equation of the tangent line, where \(m\) is the slope, and \( (x_1, y_1) \) is the given point.
---
Note: No graphs or diagrams are associated with this problem in the provided image.
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