Using the contour map below, answer the questions that follow. УА 5 4 3 2 L 0 C D 1 80 70 60 2 50- 50 3 B 70 60 80 4 5 X (a) Estimate the average rate of change in the direction of decreasing y between the points A and B. Show all work. (b) Estimate the average rate of change in the direction of increasing a between the points C and D. Show all work. (c) Based on the plot, what point(s) appear to be local maxima? Provide your answers as an ordered pair (x, y). (d) Starting at point A and moving in the direction of vector v = (-1, 1), are you gaining or losing altitude? Provide an explanation on how you know.
Using the contour map below, answer the questions that follow. УА 5 4 3 2 L 0 C D 1 80 70 60 2 50- 50 3 B 70 60 80 4 5 X (a) Estimate the average rate of change in the direction of decreasing y between the points A and B. Show all work. (b) Estimate the average rate of change in the direction of increasing a between the points C and D. Show all work. (c) Based on the plot, what point(s) appear to be local maxima? Provide your answers as an ordered pair (x, y). (d) Starting at point A and moving in the direction of vector v = (-1, 1), are you gaining or losing altitude? Provide an explanation on how you know.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:### Educational Resource on Contour Mapping and Gradient Analysis
#### Using the contour map below, answer the questions that follow.
**Contour Map Description:**
The contour map shown is a graphical representation of scalar values in a two-dimensional plane, where each contour line represents a constant value. The map displays contour lines with values of 50, 60, 70, and 80. The X-axis ranges from 0 to 5, and the Y-axis ranges from 0 to 5. Critical points labeled on the map include:
- Point A: approximately located at (4.2, 3.3)
- Point B: approximately located at (4.3, 4.3)
- Point C: approximately located at (1.1, 3.1)
- Point D: approximately located at (2.1, 3.0)
**Questions:**
(a) **Estimate the average rate of change in the direction of decreasing \( y \) between the points \( A \) and \( B \). Show all work.**
(b) **Estimate the average rate of change in the direction of increasing \( x \) between the points \( C \) and \( D \). Show all work.**
(c) **Based on the plot, what point(s) appear to be local maxima? Provide your answers as an ordered pair \( (x, y) \).**
(d) **Starting at point \( A \) and moving in the direction of vector \( \mathbf{v} = \langle -1, 1 \rangle \), are you gaining or losing altitude? Provide an explanation on how you know.**
### Explanation of Contour Map
The contour map consists of several concentric contour lines representing different altitude levels. Each contour line joins points of equal value, with values provided next to the lines for reference.
- **Contours:** These lines define areas of equal value; for instance, the 50-value contour line encloses a region where the value of the represented quantity is 50.
- **Gradient Direction:** The gradient points in the direction of the greatest rate of increase. Perpendicular to contour lines, moving across closely spaced contour lines indicates a steep slope.
The objective is to analyze this map to determine rate of changes between points, identify local maxima, and understand changes in altitude when movement in specific directions is specified.
**Answering specific questions requires understanding the contour map visualization and measurements between designated points.**
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VIEWStep 4: c) Based on the plot, what point(s) appear to be local maxima;
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