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Magazine Chapter 9: Problem 19
Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x z=6 ;(2,0,3) $$
Short Answer
Expert verified
The tangent plane equation is \( 3x + 2z = 12 \).
Step by step solution
01
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 \( y \).
02
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: \( abla F = (z, 0, x) \).
03
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.
04
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 \((x_0, y_0, z_0) = (2, 0, 3)\).
05
Substitute Values into the Plane Equation
Substituting the values we have: \( 3(x - 2) + 0(y - 0) + 2(z - 3) = 0 \), which simplifies to \( 3x + 2z = 12 \).
06
Write the Equation
The equation of the tangent plane at the point \((2, 0, 3)\) is \( 3x + 2z = 12 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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, \( abla F = (z, 0, x) \).
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.
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 \( F(x, y, z) = xz - 6 \).
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 \((x_0, y_0, z_0) = (2, 0, 3)\).
