Problem 19 - Advanced Engineering Mathematics

Question

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x z=6 ;(2,0,3) $$

Step-by-Step Solution

Answer

The tangent plane equation is \( 3x + 2z = 12 \).

Step 1: Understand the Problem

We are tasked with finding the equation of the tangent plane to the surface defined by the equation \( xz = 6 \) at the point \((2, 0, 3)\). The surface implicitly defines \( z \) as a function of \( x \) and \( y \).

Step 2: Find the Gradient

The gradient of a function \( F(x, y, z) = xz - 6 \) is \( abla F = (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}) \). This gives: \( abla F = (z, 0, x) \).

Step 3: Evaluate the Gradient at the Given Point

Substitute the point \((2, 0, 3)\) into the gradient: \( abla F(2, 0, 3) = (3, 0, 2) \). This provides the normal vector to the tangent plane at the point.

Step 4: Use the Point-Normal Form for Plane Equation

The equation of a plane with normal vector \((A, B, C)\) passing through point \((x_0, y_0, z_0)\) is given as \( A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \). Here, \( A = 3, B = 0, \) and \( C = 2 \) with \((x_0, y_0, z_0) = (2, 0, 3)\).

Step 5: Substitute Values into the Plane Equation

Substituting the values we have: \( 3(x - 2) + 0(y - 0) + 2(z - 3) = 0 \), which simplifies to \( 3x + 2z = 12 \).

Step 6: Write the Equation

The equation of the tangent plane at the point \((2, 0, 3)\) is \( 3x + 2z = 12 \).

Key Concepts

Gradient

The gradient is a fundamental concept in calculus that helps us find the rate at which a function is changing at any given point. For a function of several variables, the gradient is a vector that contains the partial derivatives of the function with respect to each of its variables.

• For a function \( F(x, y, z) = xz - 6 \), the gradient \( abla F \) is calculated as \( (\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}) \).
• For this example, \( abla F = (z, 0, x) \).

The gradient gives the direction of the steepest ascent in the function's graph. In our problem, calculating the gradient helps us to determine the normal vector of the tangent plane at the specific point.

Normal Vector

A normal vector is a vector that is perpendicular to a surface or plane. In the context of a tangent plane, it is crucial because it allows us to correctly describe the plane's orientation in space.

• In our problem, after calculating the gradient \( abla F = (z, 0, x) \), we evaluate it at the given point \((2, 0, 3)\) which yields \((3, 0, 2)\).
• This vector, \((3, 0, 2)\), acts as the normal vector for the tangent plane.

Using this normal vector, we can easily construct the equation of the tangent plane by applying it in the point-normal form. The normal vector is key since any plane in three-dimensional geometry can be defined using one.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of a function when it is given in an implicit form rather than the usual explicit form. This means not solving for one variable in terms of the others.

• For the equation \( xz = 6 \), \( z \) is implicitly defined as a function of \( x \) and \( y \).
• Rather than solving \( z \), we differentiate directly using partial derivatives from the function \( F(x, y, z) = xz - 6 \).

This method simplifies the process of finding rates of change with respect to all variables involved without isolating just one variable. It is especially useful when functions are intertwined.

Point-Normal Form

The point-normal form is a straightforward method for finding the equation of a plane. It requires a known point on the plane and a normal vector to describe the plane's orientation.

• The formula for the point-normal form is \( A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \).
• For our example, the normal vector is \((A, B, C) = (3, 0, 2)\) and the point is \((x_0, y_0, z_0) = (2, 0, 3)\).

When the values are substituted into the formula, it results in \( 3(x - 2) + 0(y - 0) + 2(z - 3) = 0 \), simplifying to \( 3x + 2z = 12 \). This is the equation of the tangent plane, providing a practical use of the point-normal form.