### Understanding Gradient Vectors in Contour Plots#### Contour Plot OverviewThe provided image displays a contour plot with labeled points A, B, and C. This plot is utilized to understand the concept of gradient vectors in a scalar field.#### Gradient Vector DirectionsGradient vectors illustrate the direction and rate of the steepest ascent in a scalar field. In a contour plot, these vectors are perpendicular to the contour lines and point towards higher values.#### Multiple-Choice QuestionsBelow the contour plot, there are multiple-choice questions aimed at determining the approximate direction of the gradient vector at each of the points A, B, and C. The directions of the gradient vectors are represented by combinations of unit vectors $\vec{i}$ and $\vec{j}$:1. $\vec{i}$2. $-\vec{i}$3. $\vec{j}$4. $-\vec{j}$5. $\vec{i} + \vec{j}$6. $\vec{i} - \vec{j}$7. $-\vec{i} + \vec{j}$8. $-\vec{i} - \vec{j}$#### Dropdown Menu for SelectionEach labeled point on the contour plot (A, B, and C) has an associated dropdown menu where students can select the appropriate direction of the gradient vector:- **At point C:** [Dropdown menu with options 1 to 8]- **At point A:** [Dropdown menu with options 1 to 8]- **At point B:** [Dropdown menu with options 1 to 8]#### Analyzing the PlotStudents are required to analyze the contour plot and determine the direction of the gradient vector at each point (A, B, C). They should keep in mind that:- **Gradient vectors** point perpendicularly to contour lines.- **Direction** of the gradient vectors is towards increasing values of the scalar field.This task helps reinforce the understanding of gradient vectors, contour plots, and their interrelation in representing scalar fields.

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Answered: At the point C At the point A At the…

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At the point C At the point A At the point B <>

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Coordinate Geometry

Coordinate geometry, also known as analytic geometry or Cartesian geometry in classical mathematics, is a type of geometry that is studied using a coordinate system. Coordinate geometry is a branch of mathematics that describes the position of points on a plane using an ordered pair of numbers called coordinates.

Question

### Understanding Gradient Vectors in Contour Plots

#### Contour Plot Overview
The provided image displays a contour plot with labeled points A, B, and C. This plot is utilized to understand the concept of gradient vectors in a scalar field.

#### Gradient Vector Directions
Gradient vectors illustrate the direction and rate of the steepest ascent in a scalar field. In a contour plot, these vectors are perpendicular to the contour lines and point towards higher values.

#### Multiple-Choice Questions
Below the contour plot, there are multiple-choice questions aimed at determining the approximate direction of the gradient vector at each of the points A, B, and C. The directions of the gradient vectors are represented by combinations of unit vectors $\vec{i}$ and $\vec{j}$:

1. $\vec{i}$
2. $-\vec{i}$
3. $\vec{j}$
4. $-\vec{j}$
5. $\vec{i} + \vec{j}$
6. $\vec{i} - \vec{j}$
7. $-\vec{i} + \vec{j}$
8. $-\vec{i} - \vec{j}$

#### Dropdown Menu for Selection
Each labeled point on the contour plot (A, B, and C) has an associated dropdown menu where students can select the appropriate direction of the gradient vector:

- **At point C:** [Dropdown menu with options 1 to 8]
- **At point A:** [Dropdown menu with options 1 to 8]
- **At point B:** [Dropdown menu with options 1 to 8]

#### Analyzing the Plot
Students are required to analyze the contour plot and determine the direction of the gradient vector at each point (A, B, C). They should keep in mind that:

- **Gradient vectors** point perpendicularly to contour lines.
- **Direction** of the gradient vectors is towards increasing values of the scalar field.

This task helps reinforce the understanding of gradient vectors, contour plots, and their interrelation in representing scalar fields.

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