Question

Task: Using only partial derivatives, directional derivatives, and/or tangent planes, answer the following prompts. 1) Estimate the current height change along the course between the points A = $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ and B = $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, as depicted in the following Figures 1 and 2. For your brainstorming post, identify two different ways to estimate this height change. One with derivatives and one without. (Hint: Setting up these calculations in MATLAB will make your life easier when answering the remaining questions, but these can all be done fairly easily by hand if desired). Figure 1: Input Change from A to B Figure 2: Change Along the Track 2) Harry has found potential locations, Point B' and B", over which to run the course instead of point B, then curving the course around to quickly return to its normal intersection with the y-axis. B' lies directly North of point A, and B" lies directly East of A. Points B' and B" are the same distance away from point A as was point B, as depicted in Figures 3 and 4. (Note: Consider the positive x-axis to be East and the positive y-axis to be North) Figure 3: Input Changes from A Figure 4: Changes Along the Track First, conjecture and then verify whether moving from A to B, B', or B" results in the approximate greatest height change over that isolated region of the course. 3) Because of the topography around the course, Harry can only alter the course after the point A by moving Northeast, Northwest, Southeast, or Southwest, and only by the same horizontal distance as the original change from A to B. Compare the estimated height changes in each direction to determine the new course that would prove the most difficult journey. (Remember to give an initial conjecture based on the provided applet and Figures.)

          Task: Using only partial derivatives, directional derivatives, and/or tangent planes, answer the following prompts.

1) Estimate the current height change along the course between the points A = $\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$ and B = $\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, as depicted in the following
Figures 1 and 2. For your brainstorming post, identify two different ways to estimate this height change. One with derivatives and one
without. (Hint: Setting up these calculations in MATLAB will make your life easier when answering the remaining questions, but these can all
be done fairly easily by hand if desired).

Figure 1: Input Change from A to B
Figure 2: Change Along the Track

2) Harry has found potential locations, Point B' and B", over which to run the course instead of point B, then curving the course around to
quickly return to its normal intersection with the y-axis. B' lies directly North of point A, and B" lies directly East of A. Points B' and B"
are the same distance away from point A as was point B, as depicted in Figures 3 and 4. (Note: Consider the positive x-axis to be East and
the positive y-axis to be North)

Figure 3: Input Changes from A
Figure 4: Changes Along the Track

First, conjecture and then verify whether moving from A to B, B', or B" results in the approximate greatest height change over that
isolated region of the course.

3) Because of the topography around the course, Harry can only alter the course after the point A by moving Northeast, Northwest,
Southeast, or Southwest, and only by the same horizontal distance as the original change from A to B. Compare the estimated height
changes in each direction to determine the new course that would prove the most difficult journey. (Remember to give an initial conjecture
based on the provided applet and Figures.)

Task: Using only partial derivatives, directional derivatives, and/or tangent planes, answer the following prompts.

1) Estimate the current height change along the course between the points A = ((√(3))/(2), (1)/(2)) and B = ((1)/(2), (√(3))/(2)), as depicted in the following
Figures 1 and 2. For your brainstorming post, identify two different ways to estimate this height change. One with derivatives and one
without. (Hint: Setting up these calculations in MATLAB will make your life easier when answering the remaining questions, but these can all
be done fairly easily by hand if desired).

Figure 1: Input Change from A to B
Figure 2: Change Along the Track

2) Harry has found potential locations, Point B' and B", over which to run the course instead of point B, then curving the course around to
quickly return to its normal intersection with the y-axis. B' lies directly North of point A, and B" lies directly East of A. Points B' and B"
are the same distance away from point A as was point B, as depicted in Figures 3 and 4. (Note: Consider the positive x-axis to be East and
the positive y-axis to be North)

Figure 3: Input Changes from A
Figure 4: Changes Along the Track

First, conjecture and then verify whether moving from A to B, B', or B" results in the approximate greatest height change over that
isolated region of the course.

3) Because of the topography around the course, Harry can only alter the course after the point A by moving Northeast, Northwest,
Southeast, or Southwest, and only by the same horizontal distance as the original change from A to B. Compare the estimated height
changes in each direction to determine the new course that would prove the most difficult journey. (Remember to give an initial conjecture
based on the provided applet and Figures.)

Added by Dennis M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

10/18/2022

Step 1

- Use the gradient vector to find the directional derivative in the direction of the line connecting points A and B. - Multiply the directional derivative by the distance between points A and B to estimate the height change. Show more…

Show all steps

Thanks for your feedback!

Task: Using only partial derivatives, directional derivatives, and/or tangent planes, answer the following prompts. Estimate the current height change along the course between the points A=((sqrt(3))/(2),(1)/(2)) and B=((1)/(2),(sqrt(3))/(2)), as depicted in the following Figures 1 and 2. For your brainstorming post, identify two different ways to estimate this height change. One with derivatives and one without. (Hint: Setting up these calculations in MATLAB will make your life easier when answering the remaining questions, but these can all be done fairly easily by hand if desired). Figure 1: Input Change from A to B Figure 2: Change Along the Track Harry has found potential locations, Point B^(') and B^(''), over which to run the course instead of point B, then curving the course around to quickly return to its normal intersection with the y-axis. B^(') lies directly North of point A, and B^('') lies directly East of A. Points B^(') and B^('') are the same distance away from point A as was point B, as depicted in Figures 3 and 4. (Note: Consider the positive x-axis to be East and the positive y-axis to be North) rigure 4: Lnanges Along tne ırack First, conjecture and then verify whether moving from A to B,B^('), or B^('') results in the approximate greatest height change over that isolated region of the course. Because of the topography around the course, Harry can only alter the course after the point A by moving Northeast, Northwest, Southeast, or Southwest, and only by the same horizontal distance as the original change from A to B. Compare the estimated height changes in each direction to determine the new course that would prove the most difficult journey. (Remember to give an initial conjecture based on the provided applet and Figures.) Task: Using only partial derivatives, directional derivatives, and/or tangent planes, answer the following prompts. Figures 1 and 2. For your brainstorming post, identify two different ways to estimate this height change.One with derivatives and one without. (Hint: Setting up these calculations in MATLAB will make your life easier when answering the remaining questions, but these can all be done fairly easily by hand if desired). Figure 1: Input Change from A to B Figure 2: Change Along the Track 2) Harry has found potential locations, Point B' and B" , over which to run the course instead of point B, then curving the course around to quickly return to its normal intersection with the y-axis. B' lies directly North of point A, and B" lies directly East of A. Points B' and B are the same distance away from point A as was point B,as depicted in Figures 3 and 4.(Note:Consider the positive -axis to be East and the positive y-axis to be North) Figure 3: Input Changes from A Figure 4: Changes Along the Track First,conjecture and then verify whether moving from A to B,B',or B" results in the approximate greatest height change over that isolated region of the course. 3) Because of the topography around the course, Harry can only alter the course after the point A by moving Northeast, Northwest, Southeast,or Southwest,and only by the same horizontal distance as the original change from A to B.Compare the estimated height changes in each direction to determine the new course that would prove the most difficult journey.(Remember to give an initial conjecture based on the provided applet and Figures.)