For the function z = f(x, y) = x² − xy² + y³, find the equations of the tangent lines in both the c-direction & the y-direction at the point ( 2, 1, 3). 1.1 2.1 3.1 *Activity 3
For the function z = f(x, y) = x² − xy² + y³, find the equations of the tangent lines in both the c-direction & the y-direction at the point ( 2, 1, 3). 1.1 2.1 3.1 *Activity 3
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
6th Edition
ISBN:9781337111348
Author:Bruce Crauder, Benny Evans, Alan Noell
Publisher:Bruce Crauder, Benny Evans, Alan Noell
Chapter6: Rates Of Change
Section6.1: Velocity
Problem 2SBE: Sign of VelocityWhen directed distance is decreasing, is velocity positive or negative? What is the...
Related questions
Question
![### Finding Tangent Line Equations for Multivariable Functions
**Problem Statement:**
Given the function \( z = f(x, y) = x^2 - xy^2 + y^3 \), find the equations of the tangent lines in both the *x*-direction & the *y*-direction at the point \( (2, 1, 3) \).
**Visualization:**
The 3D plot on the right depicts the surface of the function \( z = f(x, y) = x^2 - xy^2 + y^3 \). The point \( (2, 1, 3) \) is marked on the surface. The plot gives a visual representation of how the surface behaves around the point of interest.
#### Steps to Find Tangent Lines:
1. **Calculate Partial Derivatives:**
- Partial derivative with respect to \( x \):
\[
f_x(x, y) = \frac{\partial}{\partial x}(x^2 - xy^2 + y^3) = 2x - y^2
\]
- Partial derivative with respect to \( y \):
\[
f_y(x, y) = \frac{\partial}{\partial y}(x^2 - xy^2 + y^3) = -2xy + 3y^2
\]
2. **Evaluate Partial Derivatives at the Point (2, 1, 3):**
- Evaluate \( f_x(2, 1) \):
\[
f_x(2, 1) = 2(2) - (1)^2 = 4 - 1 = 3
\]
- Evaluate \( f_y(2, 1) \):
\[
f_y(2, 1) = -2(2)(1) + 3(1)^2 = -4 + 3 = -1
\]
3. **Write Equations for Tangent Lines:**
Using the point (2, 1, 3) and the slopes found from the partial derivatives:
- Tangent line in the *x*-direction:
\[
z = 3 + 3(x - 2)
\]
Simplify:
\[
z = 3 + 3x - 6 \implies z = 3x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ce7c7b2-894f-4c8f-a982-e7518c82686a%2Facb519f7-e236-4682-9ced-22b5ffea9aa2%2F9beveit_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding Tangent Line Equations for Multivariable Functions
**Problem Statement:**
Given the function \( z = f(x, y) = x^2 - xy^2 + y^3 \), find the equations of the tangent lines in both the *x*-direction & the *y*-direction at the point \( (2, 1, 3) \).
**Visualization:**
The 3D plot on the right depicts the surface of the function \( z = f(x, y) = x^2 - xy^2 + y^3 \). The point \( (2, 1, 3) \) is marked on the surface. The plot gives a visual representation of how the surface behaves around the point of interest.
#### Steps to Find Tangent Lines:
1. **Calculate Partial Derivatives:**
- Partial derivative with respect to \( x \):
\[
f_x(x, y) = \frac{\partial}{\partial x}(x^2 - xy^2 + y^3) = 2x - y^2
\]
- Partial derivative with respect to \( y \):
\[
f_y(x, y) = \frac{\partial}{\partial y}(x^2 - xy^2 + y^3) = -2xy + 3y^2
\]
2. **Evaluate Partial Derivatives at the Point (2, 1, 3):**
- Evaluate \( f_x(2, 1) \):
\[
f_x(2, 1) = 2(2) - (1)^2 = 4 - 1 = 3
\]
- Evaluate \( f_y(2, 1) \):
\[
f_y(2, 1) = -2(2)(1) + 3(1)^2 = -4 + 3 = -1
\]
3. **Write Equations for Tangent Lines:**
Using the point (2, 1, 3) and the slopes found from the partial derivatives:
- Tangent line in the *x*-direction:
\[
z = 3 + 3(x - 2)
\]
Simplify:
\[
z = 3 + 3x - 6 \implies z = 3x
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Functions and Change: A Modeling Approach to Coll…
Algebra
ISBN:
9781337111348
Author:
Bruce Crauder, Benny Evans, Alan Noell
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
Functions and Change: A Modeling Approach to Coll…
Algebra
ISBN:
9781337111348
Author:
Bruce Crauder, Benny Evans, Alan Noell
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning