Problem 175 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point. \(x y+y z+z x=11, P(1,2,3)\)

Step-by-Step Solution

Answer

The tangent plane is \(5x + 4y + 3z = 22\).

Step 1: Understand the Problem

We need to find the equation of the tangent plane to the surface given by the equation \(x y + y z + z x = 11\) at the point \(P(1,2,3)\). The general form of the tangent plane to a surface \(F(x, y, z) = 0\) is given by \(F_x(x_0, y_0, z_0)(x-x_0) + F_y(x_0, y_0, z_0)(y-y_0) + F_z(x_0, y_0, z_0)(z-z_0) = 0\), where \((x_0, y_0, z_0)\) is a point on the surface.

Step 2: Calculate Partial Derivatives

First, calculate the partial derivatives of \(F(x, y, z) = x y + y z + z x - 11\). - \(F_x = \frac{\partial F}{\partial x} = y + z\)- \(F_y = \frac{\partial F}{\partial y} = x + z\)- \(F_z = \frac{\partial F}{\partial z} = x + y\).

Step 3: Evaluate Partial Derivatives at the Point

Substitute the point \(P(1,2,3)\) into the partial derivatives:- \(F_x(1,2,3) = 2 + 3 = 5\)- \(F_y(1,2,3) = 1 + 3 = 4\)- \(F_z(1,2,3) = 1 + 2 = 3\).

Step 4: Write the Equation of the Tangent Plane

Use the formula for the tangent plane with the evaluated derivatives at \(P(1,2,3)\):\[5(x - 1) + 4(y - 2) + 3(z - 3) = 0\].Simplify the equation:\[5x + 4y + 3z = 22\].

Step 5: Conclusion

The equation of the tangent plane to the surface \(x y + y z + z x = 11\) at the point \(P(1,2,3)\) is \(5x + 4y + 3z = 22\). This is the final answer.

Key Concepts

Partial Derivatives

Partial derivatives play a crucial role when dealing with functions of several variables. A partial derivative measures how a function changes as one of its variables changes, while keeping the other variables constant. When considering a surface defined by a function of three variables, such as \(F(x, y, z) = x y + y z + z x - 11\), we need to take partial derivatives with respect to each variable: \(x, y,\) and \(z\).

• Partial Derivative with respect to \(x\) is \(F_x = y + z\): This tells us how the function \(F\) changes with small changes in \(x\).
• Partial Derivative with respect to \(y\) is \(F_y = x + z\): This indicates the change in \(F\) when \(y\) is altered, keeping \(x\) and \(z\) constant.
• Partial Derivative with respect to \(z\) is \(F_z = x + y\): This shows how \(F\) changes when \(z\) varies, while \(x\) and \(y\) remain fixed.

By understanding partial derivatives, we gain insight into the directional changes of a surface, which are particularly useful for finding tangent planes at specific points.

Surface Equation

The surface equation in the exercise is given by \(x y + y z + z x = 11\). This equation represents a surface in three-dimensional space. To better understand the concept of a tangent plane, consider how the surface looks around a particular point, here \((1,2,3)\).

• This equation implicitly describes a relationship between the variables \(x, y,\) and \(z\) which constrains them to lie on the surface.
• Our task involves finding the tangent plane at a specific point, essentially providing a linear approximation of the surface near that point.

To do this, we make use of the partial derivatives of the surface equation. They guide us in calculating the slope of the plane that best approximates the surface locally. Hence, the surface equation is central to navigating and solving problems involving tangent planes.

Point Evaluation

Point evaluation is the process of substituting a point into our derived expressions to solve or simplify them. In the problem, once we have the partial derivatives, we substitute the coordinates of point \(P(1,2,3)\) into each partial derivative to evaluate them:

• \(F_x(1,2,3) = 2 + 3 = 5\)
• \(F_y(1,2,3) = 1 + 3 = 4\)
• \(F_z(1,2,3) = 1 + 2 = 3\)

This step is key to finding the normal vector of the tangent plane. The evaluated partial derivatives give us the coefficients needed for the tangent plane equation. After substituting the point into the derivatives, the tangent plane equation can be formed: \[5(x - 1) + 4(y - 2) + 3(z - 3) = 0\], which simplifies to \[5x + 4y + 3z = 22\]. Point evaluation makes the abstract concrete by providing specific values critical for interpreting geometrical and analytical outcomes.