Problem 74 - Calculus Early Transcendentals

Question

Consider the following surfaces \(f(x, y, z)=0,\) which may be regarded as a level surface of the function \(w=f(x, y, z) . A\) point \(P(a, b, c)\) on the surface is also given. a. Find the (three-dimensional) gradient of \(f\) and evaluate it at \(P\). b. The heads of all vectors orthogonal to the gradient with their tails at \(P\) form a plane. Find an equation of that plane (soon to be called the tangent plane). $$f(x, y, z)=x y+x z-y z-1 ; P(1,1,1)$$

Step-by-Step Solution

Answer

Answer: The equation of the tangent plane is 2x - 2 = 0.

Step 1: Find the gradient of the function f(x, y, z)

Compute the partial derivatives of the given function f(x, y, z) with respect to x, y, and z. $$ f(x, y, z) = x y + x z - y z - 1 $$ The partial derivatives are: $$ \frac{\partial f}{\partial x} = y + z, \\ \frac{\partial f}{\partial y} = x - z, \\ \frac{\partial f}{\partial z} = x - y. \\ $$ The gradient of f is given by the vector: $$ \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) = (y + z, x - z, x - y). $$

Step 2: Evaluate the gradient at point P(a, b, c)

Plug the point P(1, 1, 1) into the gradient expression to get the gradient vector at P. $$ \nabla f(1, 1, 1) = (1 + 1, 1 - 1, 1 - 1) = (2, 0, 0) $$

Step 3: Find the equation of the tangent plane

Now that we have the gradient vector at point P, use the point-normal form of the equation of a plane: $$ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 $$ where (a, b, c) is the normal vector (which is the gradient vector), and (x0, y0, z0) is the given point P(1, 1, 1). $$ 2(x - 1) + 0(y - 1) + 0(z - 1) = 0 $$ Simplify the equation to get the final equation of the tangent plane: $$ 2x - 2 = 0 $$

Key Concepts

Gradient

A gradient in vector calculus is a vector that points in the direction of the greatest rate of increase of a function and its magnitude tells us how steep the function is in that direction. To find the gradient for a given multivariable function like \( f(x, y, z) \), you compute its partial derivatives with respect to each variable, resulting in a vector composed of these derivatives.

For the function \( f(x, y, z) = x y + x z - y z - 1 \), the partial derivatives are:

• \( \frac{\partial f}{\partial x} = y + z \)
• \( \frac{\partial f}{\partial y} = x - z \)
• \( \frac{\partial f}{\partial z} = x - y \)

The gradient is given by the vector\[abla f = \left( y + z, x - z, x - y \right)\]This gradient vector gives us the direction of steepest ascent at any point \( (x, y, z) \) on the surface. Evaluating at point \( P(1, 1, 1) \), the gradient vector is \((2, 0, 0)\), indicating the steepest increase is strictly along the x-axis from point \( P \).

Gradients are crucial when analyzing surfaces because they help predict how changes in input variables will affect the function's output, enabling us to understand the surface's behavior.

Tangent Plane

In geometry, the tangent plane to a surface at a given point is the plane that best approximates the surface near that point. If you imagine the surface as a sheet, the tangent plane would be like the flat part of paper lying directly on that sheet at the chosen point. The gradient vector provides the "normal vector" for the plane, which is essential to define the tangent plane.

The equation for a tangent plane involves the point-normal form\[a(x - x_0) + b(y - y_0) + c(z - z_0) = 0\]where \((a, b, c)\) is your normal vector and \((x_0, y_0, z_0)\) is the point through which the plane passes. Here, \((a, b, c)\) is found from the gradient vector. For our exercise, the normal vector at \( P(1, 1, 1) \) is \((2, 0, 0)\).

By substituting the point into the plane equation, the resulting formula is\[2(x - 1) + 0(y - 1) + 0(z - 1) = 0\]which simplifies easily to \(2x - 2 = 0\) or \(x = 1\). This plane, \(x = 1\), is flat along the y and z axes, perfectly tangential to the surface at point \( P \).

Tangent planes are used in calculus for linear approximations and are fundamental in understanding the local behavior of surfaces.

Partial Derivatives

Partial derivatives are a core concept in multivariable calculus used to understand how a function changes as one of its input variables changes, while the other variables are held constant. In the context of a function \( f(x, y, z) \), partial derivatives give the rate of change of \( f \) in the direction of each independent variable.

They are computed by differentiating the function with respect to one variable at a time, treating all other variables as constants. For the function \( f(x, y, z) = x y + x z - y z - 1 \), the partial derivatives are:

• The partial derivative with respect to \(x\) is \( y + z \).
• The partial derivative with respect to \(y\) is \( x - z \).
• The partial derivative with respect to \(z\) is \( x - y \).

These derivatives are interpreted as follows:

• \( \frac{\partial f}{\partial x} = y + z \) tells us how \( f \) changes as \( x \) changes, with \( y \) and \( z \) constant.
• \( \frac{\partial f}{\partial y} = x - z \) relates how \( f \) varies with \( y \) changing, \( x \) and \( z \) fixed.
• \( \frac{\partial f}{\partial z} = x - y \) indicates the variation in \( f \) as \( z \) changes, holding \( x \) and \( y \) constant.

Understanding partial derivatives is essential for grasping how functions behave and is utilized in finding an equation of the tangent plane and computing the gradient vector.