/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Interpret the direction of the g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 3

Interpret the direction of the gradient vector at a point.

Short Answer

Expert verified
Answer: To find the direction of the gradient vector at the point (x0, y0, z0), calculate the gradient vector 鈭噁(x, y, z) by computing its partial derivatives with respect to x, y, and z. Then, evaluate the gradient at the given point, 鈭噁(x0, y0, z0). Analyze the angles the gradient vector forms with the x, y and z axes to determine the direction in which the function increases most rapidly.

Step by step solution

01

Find the gradient of the function

Let's assume we are given a scalar function f(x, y, z). To find the gradient of the function, we need to compute its partial derivatives with respect to x, y, and z. The gradient vector is given by: \nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) Calculate the partial derivatives of the function with respect to x, y, and z.
02

Evaluate the gradient at the point

Let's say we want to evaluate the gradient at the point (x0, y0, z0). To do this, plug the point's coordinates into the gradient vector: \nabla f(x0, y0, z0) = \left(\frac{\partial f}{\partial x}(x0, y0, z0), \frac{\partial f}{\partial y}(x0, y0, z0), \frac{\partial f}{\partial z}(x0, y0, z0)\right)
03

Interpret the direction of the gradient vector

The gradient vector at a point represents the direction in which the function increases most rapidly. The direction of the gradient vector is given by the angle it forms with each axis. If the gradient vector points upward in the x, y, and z-directions, it is interpreted as an increase in the function value in the positive x, positive y, and positive z directions, respectively. Conversely, if the gradient vector points downward in any of the directions, the function value decreases in that direction. With the computed gradient vector \nabla f(x0, y0, z0), analyze the angles it forms with the x, y, and z-axes to determine the direction in which the function increases most rapidly.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Limits at (0,0) may be easier to evaluate by converting to polar coordinates. Remember that the same limit must be obtained as \(r \rightarrow 0\) along all paths to (0,0) Evaluate the following limits or state that they do not exist. $$\lim _{(x, y) \rightarrow(0,0)} \frac{(x-y)^{2}}{\left(x^{2}+y^{2}\right)^{3 / 2}}$$

Maximizing utility functions Find the values of \(\ell\) and \(g\) with \(\ell \geq 0\) and \(g \geq 0\) that maximize the following utility functions subject to the given constraints. Give the value of the utility function at the optimal point. $$U=f(\ell, g)=8 \ell^{4 / 5} g^{1 / 5} \text { subject to } 10 \ell+8 g=40$$

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}$$

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)} c f(x, y)=c \lim _{(x, y) \rightarrow(a, b)} f(x, y)$$

Describe the set of all points at which all three planes \(x+3 z=3, y+4 z=6,\) and \(x+y+6 z=9\) intersect.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.