Problem 17 - Advanced Engineering Mathematics

Question

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x^{2}-y^{2}-3 z^{2}=5 ;(6,2,3) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \(12x - 4y - 18z = 10\).

Step 1: Understand the Formula for the Tangent Plane

The equation of the tangent plane to a surface given by an equation \(F(x, y, z) = 0\) at a point \((x_0, y_0, z_0)\) is given by:\[a (F_x(x_0, y_0, z_0))(x-x_0) + (F_y(x_0, y_0, z_0))(y-y_0) + (F_z(x_0, y_0, z_0))(z-z_0) = 0, \]where \(F_x\), \(F_y\), and \(F_z\) are the partial derivatives of \(F\) with respect to \(x\), \(y\), and \(z\), respectively.

Step 2: Find Partial Derivatives

Given \(F(x, y, z) = x^2 - y^2 - 3z^2 - 5\), calculate the partial derivatives:- \(F_x = \frac{\partial}{\partial x}(x^2 - y^2 - 3z^2 - 5) = 2x\)- \(F_y = \frac{\partial}{\partial y}(x^2 - y^2 - 3z^2 - 5) = -2y\)- \(F_z = \frac{\partial}{\partial z}(x^2 - y^2 - 3z^2 - 5) = -6z\).

Step 3: Evaluate Partial Derivatives at the Point

Evaluate the partial derivatives at the point \((6, 2, 3)\):- \(F_x(6, 2, 3) = 2 \times 6 = 12\)- \(F_y(6, 2, 3) = -2 \times 2 = -4\)- \(F_z(6, 2, 3) = -6 \times 3 = -18\).

Step 4: Substitute into the Tangent Plane Equation

Using the point \((6, 2, 3)\) and the evaluated derivatives, substitute into the tangent plane equation:\[12(x - 6) - 4(y - 2) - 18(z - 3) = 0.\]

Step 5: Simplify the Tangent Plane Equation

Expand and simplify the expression:\[12x - 72 - 4y + 8 - 18z + 54 = 0 \]Combine like terms:\[12x - 4y - 18z - 10 = 0.\] Rearrange the terms:\[12x - 4y - 18z = 10.\]

Key Concepts

Partial Derivatives

Partial derivatives are a foundational concept in calculus, particularly when dealing with functions of multiple variables. They represent the rate at which a function changes as you vary one variable, while keeping the other variables constant.
For example, if you have a function of three variables, like the surface equation from our exercise: \( F(x, y, z) = x^2 - y^2 - 3z^2 - 5 \), calculating the partial derivative with respect to \(x\) means differentiating \(F\) while treating \(y\) and \(z\) as constants. This gives \( F_x = 2x \).

• \( F_x \) is the partial derivative of \( F \) with respect to \( x \).
• \( F_y \) is found by differentiating with respect to \( y \), giving \( -2y \).
• \( F_z \) is the derivative with respect to \( z \), giving \( -6z \).

This concept is critical in determining how the function behaves locally and in finding tangent planes to surfaces.

Surface Equation

A surface equation describes a three-dimensional shape and is crucial in fields like differential calculus.
In this exercise, we have the surface equation: \( x^2 - y^2 - 3z^2 = 5 \). This describes a hyperbolic surface in space, defined by the relationship between \(x\), \(y\), and \(z\).
Each point \((x, y, z)\) that satisfies this equation lies on the surface.
For example:

• If \( x = 6 \), \( y = 2 \), and \( z = 3 \) are plugged into the surface equation, they will satisfy the condition of the surface.
• This specific point \( (6, 2, 3) \) helps us find the tangent plane at that location.

Understanding surface equations helps visualize multi-variable functions and analyze their behavior.

Differential Calculus

Differential calculus involves understanding how functions change, which is crucial for finding tangent lines and planes.
The tangent plane concept connects directly to differential calculus principles. It allows us to approximate the surface near any given point.
Here’s how it works:

• The tangent plane provides the best linear approximation to a surface at a given point.
• For the equation \(12(x - 6) - 4(y - 2) - 18(z - 3) = 0\), derived from our earlier calculations, it acts as a flat surface touching our 3D curve at \( (6, 2, 3) \).

The tangent plane reflects how the function behaves locally, giving insight into the slope and curvature at that point.

Engineering Mathematics

Engineering mathematics combines math principles with engineering concepts, applying advanced math to solve complex problems.
Knowledge of tangent planes and their equations is vital in this field. Engineers often analyze surfaces and their behaviors using these concepts.
From the exercise:

• We determined the tangent plane equation \(12x - 4y - 18z = 10\), essential for calculating realistic conditions in engineering scenarios.
• This math technique is used in designing objects, predicting stresses, and ensuring stability.

Understanding how to manipulate surface equations and derive tangents aids engineers in creating safer, more efficient designs.