Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.6: Parametric Equations
Problem 5ECP: Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle...
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![**Finding the Tangent Plane Equation to the Given Surface**
To find an equation for the plane tangent to the surface \(e^{yz} = x \cos(xz) - y\) at the point (2, 1, 0), we need to follow these steps:
1. **Write the surface equation:**
\[
F(x, y, z) = e^{yz} - x \cos(xz) + y
\]
2. **Compute the partial derivatives of \(F(x, y, z)\):**
\[
F_x = \frac{\partial}{\partial x} \left(e^{yz} - x \cos(xz) + y\right)
\]
\[
F_y = \frac{\partial}{\partial y} \left(e^{yz} - x \cos(xz) + y\right)
\]
\[
F_z = \frac{\partial}{\partial z} \left(e^{yz} - x \cos(xz) + y\right)
\]
3. **Evaluate the partial derivatives at the given point (2, 1, 0):**
- Calculate \(F_x(2, 1, 0)\)
- Calculate \(F_y(2, 1, 0)\)
- Calculate \(F_z(2, 1, 0)\)
4. **Use the formula for the tangent plane:**
\[
F_x(x_0, y_0, z_0) (x - x_0) + F_y(x_0, y_0, z_0) (y - y_0) + F_z(x_0, y_0, z_0) (z - z_0) = 0
\]
By performing these calculations, we can derive the specific equation for the tangent plane at the point (2, 1, 0).
\[
\text{Detailed solution steps including the computation of partial derivatives and evaluations are provided below:}
\]
**Step-by-Step Solution:**
1. Calculate \(F_x\):
\[
F_x = -\cos(xz) + xz \sin(xz)
\]
Evaluate \(F_x\) at (2, 1, 0):
\[
F_x(2, 1,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2ba0aec5-65e2-43b8-bb6a-70de8eae6b73%2F4e5a11df-9b3a-451a-b9ac-2e73d699e011%2Fdt9lsy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Tangent Plane Equation to the Given Surface**
To find an equation for the plane tangent to the surface \(e^{yz} = x \cos(xz) - y\) at the point (2, 1, 0), we need to follow these steps:
1. **Write the surface equation:**
\[
F(x, y, z) = e^{yz} - x \cos(xz) + y
\]
2. **Compute the partial derivatives of \(F(x, y, z)\):**
\[
F_x = \frac{\partial}{\partial x} \left(e^{yz} - x \cos(xz) + y\right)
\]
\[
F_y = \frac{\partial}{\partial y} \left(e^{yz} - x \cos(xz) + y\right)
\]
\[
F_z = \frac{\partial}{\partial z} \left(e^{yz} - x \cos(xz) + y\right)
\]
3. **Evaluate the partial derivatives at the given point (2, 1, 0):**
- Calculate \(F_x(2, 1, 0)\)
- Calculate \(F_y(2, 1, 0)\)
- Calculate \(F_z(2, 1, 0)\)
4. **Use the formula for the tangent plane:**
\[
F_x(x_0, y_0, z_0) (x - x_0) + F_y(x_0, y_0, z_0) (y - y_0) + F_z(x_0, y_0, z_0) (z - z_0) = 0
\]
By performing these calculations, we can derive the specific equation for the tangent plane at the point (2, 1, 0).
\[
\text{Detailed solution steps including the computation of partial derivatives and evaluations are provided below:}
\]
**Step-by-Step Solution:**
1. Calculate \(F_x\):
\[
F_x = -\cos(xz) + xz \sin(xz)
\]
Evaluate \(F_x\) at (2, 1, 0):
\[
F_x(2, 1,
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