Question

Consider each of the following surfaces and the point \(P\) on the surface. a. Find the gradient of \(f\). b. Let \(C^{\prime}\) be the path of steepest descent on the surface beginning. at \(P,\) and let C be the projection of \(C^{\prime}\) on the xy-plane. Find an equation of \(C\) in the xy-plane. c. Find parametric equations for the path \(C\) ' on the surface. $$f(x, y)=4+x(\text { a plane }) ; P(4,4,8)$$

Short answer

Question: Find the gradient of the function \(f(x,y) = 4 + x\), the equation of the projection of the path of steepest descent in the xy-plane (denoted by C), and parametric equations for the path of steepest descent on the surface (denoted by \(C'\)), starting at the point \(P = (4, 4, 8)\). Answer: For the function \(f(x,y) = 4 + x\), the gradient is given by \(\nabla f = \langle 1, 0 \rangle\). The equation of the path of steepest descent (C) in the xy-plane is \(x = 4\). The parametric equations for the path of steepest descent (C') on the surface are given by \(x = 4\), \(y = t\), and \(z = 8\), where \(t\) is any real number.

Step-by-step solution

Find the gradient of f#

To find the gradient of \(f\), we need to compute its partial derivatives with respect to \(x\) and \(y\). The function is given by \(f(x, y) = 4 + x\). Computing the partial derivatives, we get: $$ \frac{\partial f}{\partial x} = 1 \quad \text{and} \quad \frac{\partial f}{\partial y}= 0 $$ Thus, the gradient of \(f\) is given by \(\nabla f=\langle 1, 0 \rangle\).

Find the equation of the path of steepest descent in the xy-plane#

The path of steepest descent on the surface begins at the point \(P = (4, 4, 8)\). Since the gradient of the function is \(\nabla f = \langle 1, 0 \rangle\), the direction of steepest descent is along the x-axis. Thus, the equation of the path of steepest descent in the xy-plane, which is denoted by C, is simply a vertical line with a constant \(x\)-value equal to the \(x\)-coordinate of the point \(P\). In this case, the equation of C is \(x=4\).

Find parametric equations for the path of steepest descent on the surface#

Now we need to find parametric equations for the path of steepest descent on the surface, which is denoted by \(C'\). Since the equation of the path of steepest descent in the xy-plane is \(x = 4\), we can use this to write the parametric equations. Let \(x = 4\), then \(f(x, y) = 4 + x = 4 + 4 = 8\). So, the path of steepest descent on the surface lies on the plane \(z = 8\). Now, we can write parametric equations for \(C'\) as follows: $$ x = 4 \\ y = t \\ z = 8 $$ where \(t\) can be any real number. These parametric equations represent the path of steepest descent on the surface, which is a straight line along the x-axis on the plane \(z = 8\).

Key concepts

Partial Derivatives

Partial derivatives play a crucial role in understanding the behavior of multivariable functions. In essence, they measure how a function changes as each variable is varied independently while all other variables are held constant. For a function of two variables, say, \( f(x, y) \), we can compute the partial derivative with respect to \( x \) by treating \( y \) as a constant, and vice versa.

When working with the function \( f(x, y) = 4 + x \), which describes a plane, the partial derivatives can be calculated easily: \( \frac{\partial f}{\partial x} = 1 \) and \( \frac{\partial f}{\partial y} = 0 \). The former tells us that for each unit increase in \( x \), while keeping \( y \) constant, the function's value increases by one. The latter indicates that changing \( y \) while keeping \( x \) constant does not alter the function's value. These concepts aid in constructing the gradient vector, which defines the direction of steepest ascent.

Gradient Vector

The gradient vector is a core concept in multivariable calculus that provides both a direction and a magnitude for the rate of increase of a function. The vector points in the direction where the function increases most rapidly and its magnitude reflects the rate of this increase.

For the function \( f(x, y) \), the gradient vector is denoted by \( abla f \) and consists of the partial derivatives of \( f \) with respect to all its variables. In our case, with the function \( f(x, y) = 4 + x \), the gradient vector is \( abla f = \langle 1, 0 \rangle \). Notice that the gradient has a non-zero component only in the \( x \)-direction, indicating all the increase for \( f \) happens as \( x \) changes. The steepest descent, which is the direction of greatest decrease, will thus be opposite to the gradient.

Parametric Equations

Parametric equations enable us to describe geometric objects, like lines and curves, in a dynamic way using one or more parameters. They are particularly useful when dealing with paths or trajectories over surfaces.

In the given example, the path of steepest descent on the surface is represented by parametric equations with \( t \) as the parameter: \( x = 4 \), \( y = t \), and \( z = 8 \). These equations define a straight line where \( x \) and \( z \) are held constant regardless of the value of \( t \), which means no matter how \( y \) changes, that line remains at the same height, creating a flat trajectory on the plane \( z = 8 \) that runs parallel to the \( y \)-axis. This simple yet powerful way of describing paths is essential for visualizing movements and changes on surfaces in multi-dimensional space.