Find the equations of the tangent lines in both the x-direction & the y-direction to the function z = f(x, y) = ¸_x³ −2x²y+2xy² − y³ – 5 12 at the point (2, 3).
Find the equations of the tangent lines in both the x-direction & the y-direction to the function z = f(x, y) = ¸_x³ −2x²y+2xy² − y³ – 5 12 at the point (2, 3).
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Polar Coordinates And Parametric Equations
Section8.CT: Chapter Test
Problem 8CT
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![**Finding the Equations of Tangent Lines**
In this section, we will find the equations of the tangent lines in both the x-direction and the y-direction to the function
\[ z = f(x, y) = \frac{x^3 - 2x^2y + 2xy^2 - y^3 - 5}{12} \]
at the point \((2, 3)\).
### Steps to Follow:
1. **Function Overview**:
The given function is:
\[
z = f(x, y) = \frac{x^3 - 2x^2y + 2xy^2 - y^3 - 5}{12}
\]
2. **Tangent Line in the x-Direction**:
To find the tangent line in the x-direction, calculate the partial derivative of \(f\) with respect to \(x\). This will give the slope of the tangent line in that direction.
3. **Tangent Line in the y-Direction**:
Similarly, for the tangent line in the y-direction, calculate the partial derivative of \(f\) with respect to \(y\). This provides the slope of the tangent line in the y-direction.
4. **Evaluate at \((2, 3)\)**:
Finally, evaluate the partial derivatives at the point \((2, 3)\). Use these slopes and the point to write the equations of the tangent lines.
Note: Standard tangent line equation format is \(z - z_0 = m(x - x_0)\) and \(z - z_0 = m(y - y_0)\) where \(m\) represents the slope.
By the end of this exercise, you will have the equations of the tangent lines for better understanding of the function's behavior at the specified point.
Let's proceed with the calculations and derivations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcef83d19-be6c-42fc-805e-c33d80bf3eeb%2Fcfe49184-ed2c-4f2b-ac27-560985e823bf%2Fkg6btf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Equations of Tangent Lines**
In this section, we will find the equations of the tangent lines in both the x-direction and the y-direction to the function
\[ z = f(x, y) = \frac{x^3 - 2x^2y + 2xy^2 - y^3 - 5}{12} \]
at the point \((2, 3)\).
### Steps to Follow:
1. **Function Overview**:
The given function is:
\[
z = f(x, y) = \frac{x^3 - 2x^2y + 2xy^2 - y^3 - 5}{12}
\]
2. **Tangent Line in the x-Direction**:
To find the tangent line in the x-direction, calculate the partial derivative of \(f\) with respect to \(x\). This will give the slope of the tangent line in that direction.
3. **Tangent Line in the y-Direction**:
Similarly, for the tangent line in the y-direction, calculate the partial derivative of \(f\) with respect to \(y\). This provides the slope of the tangent line in the y-direction.
4. **Evaluate at \((2, 3)\)**:
Finally, evaluate the partial derivatives at the point \((2, 3)\). Use these slopes and the point to write the equations of the tangent lines.
Note: Standard tangent line equation format is \(z - z_0 = m(x - x_0)\) and \(z - z_0 = m(y - y_0)\) where \(m\) represents the slope.
By the end of this exercise, you will have the equations of the tangent lines for better understanding of the function's behavior at the specified point.
Let's proceed with the calculations and derivations.
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