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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![### Finding the Equation of a Tangent Plane
To find the equation of the tangent plane to the surface at a given point, you'll need to follow these steps:
1. **Surface Equation**:
The surface is given by the function \( f(x, y) \):
\[
f(x, y) = x^2 - 2xy + y^2
\]
2. **Given Point**:
The point provided is \((2, 5, 9)\). Here, \( (x_0, y_0, z_0) = (2, 5, 9) \).
### Steps to Find the Tangent Plane Equation
1. **Compute Partial Derivatives**:
Calculate the first partial derivatives of \( f \) with respect to \( x \) and \( y \):
\[
f_x(x, y) = \frac{\partial}{\partial x}(x^2 - 2xy + y^2) = 2x - 2y
\]
\[
f_y(x, y) = \frac{\partial}{\partial y}(x^2 - 2xy + y^2) = -2x + 2y
\]
2. **Evaluate Partial Derivatives at the Given Point**:
\[
f_x(2, 5) = 2(2) - 2(5) = 4 - 10 = -6
\]
\[
f_y(2, 5) = -2(2) + 2(5) = -4 + 10 = 6
\]
3. **Equation of the Tangent Plane**:
The equation of the tangent plane at the point \((x_0, y_0, z_0)\) is:
\[
z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)
\]
Substituting the values:
\[
z - 9 = (-6)(x - 2) + 6(y - 5)
\]
4. **Simplify the Equation**:
Distribute and combine like terms:
\[
z - 9 = -6x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbbba0909-7b4f-4e80-8e24-06dfda2d061e%2F02cd9d1e-8b20-425c-b245-7193b2f93c10%2Fzis7it_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the Equation of a Tangent Plane
To find the equation of the tangent plane to the surface at a given point, you'll need to follow these steps:
1. **Surface Equation**:
The surface is given by the function \( f(x, y) \):
\[
f(x, y) = x^2 - 2xy + y^2
\]
2. **Given Point**:
The point provided is \((2, 5, 9)\). Here, \( (x_0, y_0, z_0) = (2, 5, 9) \).
### Steps to Find the Tangent Plane Equation
1. **Compute Partial Derivatives**:
Calculate the first partial derivatives of \( f \) with respect to \( x \) and \( y \):
\[
f_x(x, y) = \frac{\partial}{\partial x}(x^2 - 2xy + y^2) = 2x - 2y
\]
\[
f_y(x, y) = \frac{\partial}{\partial y}(x^2 - 2xy + y^2) = -2x + 2y
\]
2. **Evaluate Partial Derivatives at the Given Point**:
\[
f_x(2, 5) = 2(2) - 2(5) = 4 - 10 = -6
\]
\[
f_y(2, 5) = -2(2) + 2(5) = -4 + 10 = 6
\]
3. **Equation of the Tangent Plane**:
The equation of the tangent plane at the point \((x_0, y_0, z_0)\) is:
\[
z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)
\]
Substituting the values:
\[
z - 9 = (-6)(x - 2) + 6(y - 5)
\]
4. **Simplify the Equation**:
Distribute and combine like terms:
\[
z - 9 = -6x +
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