Find the equation of the tangent plane to the surface z = cos(8x) cos(y) at the point (−2π/2, 3/2, 0).
Find the equation of the tangent plane to the surface z = cos(8x) cos(y) at the point (−2π/2, 3/2, 0).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Find the equation of the tangent plane to the surface \( z = \cos(8x) \cos(y) \) at the point \((-2\pi/2, 3\pi/2, 0)\).
**Solution:**
To solve this problem, we need to find the partial derivatives of the function \( z = \cos(8x) \cos(y) \) with respect to \( x \) and \( y \), and evaluate them at the given point \((-2\pi/2, 3\pi/2)\).
1. **Partial Derivatives:**
- Find the partial derivative of \( z \) with respect to \( x \):
\[
\frac{\partial z}{\partial x} = -8\sin(8x)\cos(y)
\]
- Find the partial derivative of \( z \) with respect to \( y \):
\[
\frac{\partial z}{\partial y} = -\cos(8x)\sin(y)
\]
2. **Evaluate at the Point:**
- Evaluate \(\frac{\partial z}{\partial x}\) at \((-2\pi/2, 3\pi/2)\):
\[
\frac{\partial z}{\partial x}(-2\pi/2, 3\pi/2) = -8\sin(-8\pi/2)\cos(3\pi/2) = 0
\]
- Evaluate \(\frac{\partial z}{\partial y}\) at \((-2\pi/2, 3\pi/2)\):
\[
\frac{\partial z}{\partial y}(-2\pi/2, 3\pi/2) = -\cos(-8\pi/2)\sin(3\pi/2) = 1
\]
3. **Equation of the Tangent Plane:**
The equation of the tangent plane is:
\[
z = f(a, b) + \frac{\partial z}{\partial x}(a, b) (x - a) + \frac{\partial z}{\partial y}(a, b) (y - b)
\]
Substituting the values we get:
\[
z = 0 + 0(x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fe6d10-65c6-4dea-a463-83c487832ab0%2F390efde2-f72a-4608-b6c6-a302651345b6%2Fvsk9l8_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the equation of the tangent plane to the surface \( z = \cos(8x) \cos(y) \) at the point \((-2\pi/2, 3\pi/2, 0)\).
**Solution:**
To solve this problem, we need to find the partial derivatives of the function \( z = \cos(8x) \cos(y) \) with respect to \( x \) and \( y \), and evaluate them at the given point \((-2\pi/2, 3\pi/2)\).
1. **Partial Derivatives:**
- Find the partial derivative of \( z \) with respect to \( x \):
\[
\frac{\partial z}{\partial x} = -8\sin(8x)\cos(y)
\]
- Find the partial derivative of \( z \) with respect to \( y \):
\[
\frac{\partial z}{\partial y} = -\cos(8x)\sin(y)
\]
2. **Evaluate at the Point:**
- Evaluate \(\frac{\partial z}{\partial x}\) at \((-2\pi/2, 3\pi/2)\):
\[
\frac{\partial z}{\partial x}(-2\pi/2, 3\pi/2) = -8\sin(-8\pi/2)\cos(3\pi/2) = 0
\]
- Evaluate \(\frac{\partial z}{\partial y}\) at \((-2\pi/2, 3\pi/2)\):
\[
\frac{\partial z}{\partial y}(-2\pi/2, 3\pi/2) = -\cos(-8\pi/2)\sin(3\pi/2) = 1
\]
3. **Equation of the Tangent Plane:**
The equation of the tangent plane is:
\[
z = f(a, b) + \frac{\partial z}{\partial x}(a, b) (x - a) + \frac{\partial z}{\partial y}(a, b) (y - b)
\]
Substituting the values we get:
\[
z = 0 + 0(x +
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