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
![**Problem Statement:**
Find the equation of the plane tangent to the surface \( f(x,y) = x \ln(xy) \) at the point \( (1, e) \).
**Solution:**
To find the equation of the tangent plane to the given surface at a specific point, you need to follow these steps:
1. **Determine the function:**
\[
f(x, y) = x \ln(xy)
\]
2. **Find partial derivatives:**
Calculate the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \).
\[
\frac{\partial f}{\partial x} = \ln(xy) + x \cdot \frac{1}{xy} \cdot y = \ln(xy) + 1
\]
\[
\frac{\partial f}{\partial y} = x \cdot \frac{1}{xy} \cdot x = \frac{x}{y}
\]
3. **Evaluate the partial derivatives at the point \( (1, e) \):**
\[
\left. \frac{\partial f}{\partial x} \right|_{(1,e)} = \ln(1 \cdot e) + 1 = \ln(e) + 1 = 1 + 1 = 2
\]
\[
\left. \frac{\partial f}{\partial y} \right|_{(1,e)} = \frac{1}{e}
\]
4. **Find the function value at the point \( (1, e) \):**
\[
f(1, e) = 1 \ln(1 \cdot e) = 1 \ln(e) = 1
\]
Hence, \( f(1, e) = 1 \).
5. **Form the equation of the tangent plane:**
The equation of the tangent plane to the surface at \((x_0, y_0, z_0)\) is given by:
\[
z - z_0 = \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ba0aec5-65e2-43b8-bb6a-70de8eae6b73%2Fe420c0de-53cd-4ced-acb0-919c077354d0%2Flva094a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the equation of the plane tangent to the surface \( f(x,y) = x \ln(xy) \) at the point \( (1, e) \).
**Solution:**
To find the equation of the tangent plane to the given surface at a specific point, you need to follow these steps:
1. **Determine the function:**
\[
f(x, y) = x \ln(xy)
\]
2. **Find partial derivatives:**
Calculate the partial derivatives of \( f(x, y) \) with respect to \( x \) and \( y \).
\[
\frac{\partial f}{\partial x} = \ln(xy) + x \cdot \frac{1}{xy} \cdot y = \ln(xy) + 1
\]
\[
\frac{\partial f}{\partial y} = x \cdot \frac{1}{xy} \cdot x = \frac{x}{y}
\]
3. **Evaluate the partial derivatives at the point \( (1, e) \):**
\[
\left. \frac{\partial f}{\partial x} \right|_{(1,e)} = \ln(1 \cdot e) + 1 = \ln(e) + 1 = 1 + 1 = 2
\]
\[
\left. \frac{\partial f}{\partial y} \right|_{(1,e)} = \frac{1}{e}
\]
4. **Find the function value at the point \( (1, e) \):**
\[
f(1, e) = 1 \ln(1 \cdot e) = 1 \ln(e) = 1
\]
Hence, \( f(1, e) = 1 \).
5. **Form the equation of the tangent plane:**
The equation of the tangent plane to the surface at \((x_0, y_0, z_0)\) is given by:
\[
z - z_0 = \frac{\partial f}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y
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