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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![### Problem Statement
**Find an equation for the tangent plane to the surface**
\[ z^2 - xy^2 = 0 \]
**at the point**
\[ \left( \frac{9}{4}, 2, 3 \right) .\]
---
The task involves determining the equation of the tangent plane to the given surface \( z^2 - xy^2 = 0 \) at the specified point.
To find this, we need to use the concept of partial derivatives and evaluate them at the given point.
### Steps to Solve:
1. **Implicit Function Differentiation**:
- Differentiate the equation \(z^2 - xy^2 = 0\) with respect to \(x\), \(y\), and \(z\).
2. **Partial Derivatives**:
- Compute the partial derivatives \(\frac{\partial F}{\partial x}\), \(\frac{\partial F}{\partial y}\), and \(\frac{\partial F}{\partial z}\) where \(F(x, y, z) = z^2 - xy^2\).
3. **Evaluate at Given Point**:
- Evaluate these partial derivatives at the given point \(\left( \frac{9}{4}, 2, 3 \right)\).
4. **Tangent Plane Equation**:
- Use the values from step 3 to write the equation of the tangent plane in the form:
\[ \frac{\partial F}{\partial x} (x - x_0) + \frac{\partial F}{\partial y} (y - y_0) + \frac{\partial F}{\partial z} (z - z_0) = 0 \]
where \((x_0, y_0, z_0)\) is the given point.
---
### Visual Aid:
There are no graphs or diagrams provided with the problem statement. If needed, a 3D graph representing the surface \(z^2 - xy^2 = 0\) and the tangent plane at the given point could be very helpful in understanding the geometric relationship.
---
By following these steps, one should be able to find the desired equation for the tangent plane at the given point. This method reinforces concepts in multivariable calculus, particularly in the application of partial derivatives and implicit differentiation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F086767bd-31d2-4529-b304-0e1d54a1ff2d%2F4d96c6c0-c156-416c-b659-45d9b99b25f8%2Ffaavlsb_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Find an equation for the tangent plane to the surface**
\[ z^2 - xy^2 = 0 \]
**at the point**
\[ \left( \frac{9}{4}, 2, 3 \right) .\]
---
The task involves determining the equation of the tangent plane to the given surface \( z^2 - xy^2 = 0 \) at the specified point.
To find this, we need to use the concept of partial derivatives and evaluate them at the given point.
### Steps to Solve:
1. **Implicit Function Differentiation**:
- Differentiate the equation \(z^2 - xy^2 = 0\) with respect to \(x\), \(y\), and \(z\).
2. **Partial Derivatives**:
- Compute the partial derivatives \(\frac{\partial F}{\partial x}\), \(\frac{\partial F}{\partial y}\), and \(\frac{\partial F}{\partial z}\) where \(F(x, y, z) = z^2 - xy^2\).
3. **Evaluate at Given Point**:
- Evaluate these partial derivatives at the given point \(\left( \frac{9}{4}, 2, 3 \right)\).
4. **Tangent Plane Equation**:
- Use the values from step 3 to write the equation of the tangent plane in the form:
\[ \frac{\partial F}{\partial x} (x - x_0) + \frac{\partial F}{\partial y} (y - y_0) + \frac{\partial F}{\partial z} (z - z_0) = 0 \]
where \((x_0, y_0, z_0)\) is the given point.
---
### Visual Aid:
There are no graphs or diagrams provided with the problem statement. If needed, a 3D graph representing the surface \(z^2 - xy^2 = 0\) and the tangent plane at the given point could be very helpful in understanding the geometric relationship.
---
By following these steps, one should be able to find the desired equation for the tangent plane at the given point. This method reinforces concepts in multivariable calculus, particularly in the application of partial derivatives and implicit differentiation.
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