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: Finding the Equation of the Tangent Plane**
**Problem Statement:**
**Objective:**
Identify the equation for the tangent plane to the surface defined by the equation:
\[ x^2y^2z = 12 \]
at the specific point:
\[ (-2, 1, 3) \]
**Approach:**
To solve this problem, you will need to utilize the concept of gradients to find the normal vector to the surface at the given point. Then, apply the point-normal form of the equation of a plane.
**Solution Steps:**
1. **Compute the Gradient:**
- For the function \( F(x, y, z) = x^2y^2z - 12 \), find the partial derivatives with respect to \( x \), \( y \), and \( z \).
2. **Evaluate at the Point:**
- Substitute \( (-2, 1, 3) \) into the gradient to find the normal vector.
3. **Construct the Equation of the Plane:**
- Use the point-normal form: \[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]
- Here \((a, b, c)\) is the normal vector and \((x_0, y_0, z_0) = (-2, 1, 3)\).
**Conclusion:**
This analytical process allows for the determination of a tangent plane equation at any point on a given surface, providing insights into the surface's local geometric properties.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9096d41b-294f-4faf-9bb4-866edd76e996%2F096fe43c-dba5-4712-8791-8e8d1e625411%2Fy2rxk89_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Finding the Equation of the Tangent Plane**
**Problem Statement:**
**Objective:**
Identify the equation for the tangent plane to the surface defined by the equation:
\[ x^2y^2z = 12 \]
at the specific point:
\[ (-2, 1, 3) \]
**Approach:**
To solve this problem, you will need to utilize the concept of gradients to find the normal vector to the surface at the given point. Then, apply the point-normal form of the equation of a plane.
**Solution Steps:**
1. **Compute the Gradient:**
- For the function \( F(x, y, z) = x^2y^2z - 12 \), find the partial derivatives with respect to \( x \), \( y \), and \( z \).
2. **Evaluate at the Point:**
- Substitute \( (-2, 1, 3) \) into the gradient to find the normal vector.
3. **Construct the Equation of the Plane:**
- Use the point-normal form: \[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]
- Here \((a, b, c)\) is the normal vector and \((x_0, y_0, z_0) = (-2, 1, 3)\).
**Conclusion:**
This analytical process allows for the determination of a tangent plane equation at any point on a given surface, providing insights into the surface's local geometric properties.
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