Problem 20 - 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 equation of the tangent plane is \( 3x + 2z = 12 \).

Step 1: Understand the Problem

We need to find the equation of the tangent plane to the surface given by the equation \( xz = 6 \) at the point \((2, 0, 3)\). The equation of a tangent plane to a surface \( F(x, y, z) = 0 \) at a point \((x_0, y_0, z_0)\) is given by \( F_x(x_0, y_0, z_0) (x - x_0) + F_y(x_0, y_0, z_0) (y - y_0) + F_z(x_0, y_0, z_0) (z - z_0) = 0 \).

Step 2: Identify the Function and Point

The surface can be written as \( F(x, y, z) = xz - 6 = 0 \). We are looking for the tangent plane at the specific point \((x_0, y_0, z_0) = (2, 0, 3)\).

Step 3: Calculate Partial Derivatives

First, calculate the partial derivatives of \( F(x, y, z) = xz - 6 \) with respect to \( x \), \( y \), and \( z \). Thus, we have:- \( F_x = z \)- \( F_y = 0 \)- \( F_z = x \)

Step 4: Evaluate Partial Derivatives at the Given Point

Substitute the point \((2, 0, 3)\) into the partial derivatives:- \( F_x(2, 0, 3) = 3 \)- \( F_y(2, 0, 3) = 0 \)- \( F_z(2, 0, 3) = 2 \)

Step 5: Formulate the Tangent Plane Equation

Substitute the evaluated partial derivatives and the point \((x_0, y_0, z_0)\) into the equation of the tangent plane:\[ 3(x - 2) + 0(y - 0) + 2(z - 3) = 0 \]Simplifying this, we get:\[ 3x + 2z = 12 \]

Step 6: Simplify the Equation

Rearrange the equation to its simplest form:\[ 3x + 2z = 12 \]This is the equation of the tangent plane.

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in multivariable calculus. They help us understand how a function changes as we tweak one variable while keeping the others constant. This is similar to taking the derivative in single-variable calculus but in the context of functions with several variables. So, imagine a mountain range.

• The partial derivative in the x-direction tells us how steep the incline is when moving directly horizontally across the range.
• The partial derivative in the y-direction would tell the story of the steepness when you are moving forward or backward through the range.

In our exercise, the function representing the surface is given by \( F(x, y, z) = xz - 6 \). We need to find how changes in each of the variables affect the surface at a specific point. That's where the partial derivatives come in. Thus, we calculate:

• \( F_x \) tells us how the surface changes as \( x \) changes, which is simply \( z \).
• \( F_y \) shows no change here with respect to \( y \), resulting in zero: \( F_y = 0 \).
• \( F_z \) indicates the change due to \( z \), which is \( x \).

Calculating these derivatives gives us the necessary tools to describe the behavior of the surface precisely at any point.

Multivariable Calculus

Multivariable calculus extends traditional calculus to functions of two, three, or even more variables. It's a powerful tool in fields like physics, engineering, statistics, and economics, where simple single-variable calculus isn't sufficient. In multivariable calculus, we are interested in quantities such as:

• Partial derivatives: Derivatives with respect to each individual variable, showing how a function changes.
• Gradient: A vector composed of all the partial derivatives, indicating the direction of the steepest ascent of a function.
• Tangent planes: Planes that just touch the surface of a graph at a point, not cutting through it.

This exercise involves finding the tangent plane to a surface defined by \( xz = 6 \). We approach the problem by first calculating the partial derivatives of the function representing the surface. These derivatives are then evaluated at the given point. The result gives us coefficients that help form the tangent plane equation, using the formula:\[ F_x(x_0, y_0, z_0) (x - x_0) + F_y(x_0, y_0, z_0) (y - y_0) + F_z(x_0, y_0, z_0) (z - z_0) = 0 \]This equation provides a way to determine the best plane "touching" the surface exactly at the point, effectively summarizing the local shape of the surface.

Surface Equation

In this context, a surface equation like \( xz = 6 \) describes a relationship between variables \( x, y, \) and \( z \) that form a surface in a 3D space. Such equations are pivotal in understanding how different variables interact to form complex shapes. When dealing with a surface, we often want to find characteristics like slopes, tangents, and changes in altitude.To gain insights into these surfaces:

• A surface equation represents a condition that the points \( (x, y, z) \) must satisfy to lie on the surface.
• Tangent planes are flat surfaces that lightly touch the curved surface at a point, providing a linear approximation at that location.

For \( xz = 6 \), this surface does not depend on \( y \), showing symmetry in the y-axis direction. It essentially means any change in \( y \) does not affect the overall shape at the considered points. Finding a tangent plane helps approximate how the surface behaves locally, especially useful in further calculations and analyses.Thus, after computing the necessary partial derivatives and using the specific point, the tangent plane equation \( 3x + 2z = 12 \) gives a simple but thorough insight into the structure of the surface where they touch momentarily at \( (2, 0, 3) \).