This is a calculus 3 problem. Please explain each step clearly, no cursive writing. **Problem Statement:**Find the equation of the tangent plane to the level surface \( x^2y^2z^2 = 4 \) at the point \((-1, -1, -2)\).**Solution:**To find the equation of the tangent plane to a surface defined by a function \(F(x, y, z)\) at a point \((x_0, y_0, z_0)\), we use the formula for the tangent plane:\[ F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0 \]Where:- \(F_x\), \(F_y\), and \(F_z\) are the partial derivatives of \(F(x, y, z)\) with respect to \(x\), \(y\), and \(z\).Let's define \(F(x, y, z) = x^2y^2z^2 - 4\).1. **Calculate the Partial Derivatives:** - \( F_x = \frac{\partial}{\partial x}(x^2y^2z^2) = 2xy^2z^2 \) - \( F_y = \frac{\partial}{\partial y}(x^2y^2z^2) = 2x^2yz^2 \) - \( F_z = \frac{\partial}{\partial z}(x^2y^2z^2) = 2x^2y^2z \)2. **Evaluate at the Point \((-1, -1, -2)\):** - \( F_x(-1, -1, -2) = 2(-1)(-1)^2(-2)^2 = -8 \) - \( F_y(-1, -1, -2) = 2(-1)^2(-1)(-2)^2 = -8 \) - \( F_z(-1, -1, -2) = 2(-1)^2(-1)^2(-2) = -8 \)3. **Equation of the Tangent Plane:**\[ -8(x + 1) -

Name: This is a calculus 3 problem. Please explain each step clearly, no cur
Uploaded: 2024-03-05T11:52:21.000Z
Channel: Bartleby
Description: This is a calculus 3 problem. Please explain each step clearly, no cursive writing. **Problem Statement:**Find the equation of the tangent plane to the level surface \( x^2y^2z^2 = 4 \) at the point \((-1, -1, -2)\).**Solution:**To find the equation