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A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).
What is the slope of the line travelling on which would lead him to the lowest elevation.
Correct answer is 'There isn't such a line'. Can you explain this answer?
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A man travelling onf(x, y) = sin(xy). His shadow passing through the o...
Differentiating yields
fxx = -y2.sin(xy)
fyy = -x2.sin(xy)
fxy = -yx.sin(xy)
Observe that (0,0) is an inconclusive point
Hence, he will never reach the lowest elevation(because there isn’t such point.
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A man travelling onf(x, y) = sin(xy). His shadow passing through the o...
Understanding the Function and Shadow
The man is moving along the surface defined by the function f(x, y) = sin(xy). His shadow is cast straight down to the x-y plane, implying the sun is directly overhead.
Exploring the Function's Behavior
- The function f(x, y) = sin(xy) oscillates between -1 and 1.
- As the man travels, the elevation of his position changes according to the sine function, which does not have a global minimum or maximum but varies infinitely.
Shadow's Path Through the Origin
- Since the shadow passes through the origin, the line of travel can be characterized by y = mx, where m is the slope.
- The man can choose different slopes (m) to travel along the surface, affecting his elevation.
Seeking a Minimum Elevation
- To find the lowest elevation, one might consider approaching the problem by minimizing sin(xy).
- However, due to the oscillatory nature of the sine function, it does not settle at a minimum; it perpetually oscillates.
Conclusion: No Optimal Line Exists
- There is no single slope m that can lead him to a lowest elevation because:
- The sine function does not have a global minimum; it oscillates indefinitely.
- As the man moves, the elevation will continuously change without reaching a lowest point.
Thus, the correct assertion is "There isn't such a line" since all potential paths will lead to varying elevations without a definitive minimum.
The man is moving along the surface defined by the function f(x, y) = sin(xy). His shadow is cast straight down to the x-y plane, implying the sun is directly overhead.
Exploring the Function's Behavior
- The function f(x, y) = sin(xy) oscillates between -1 and 1.
- As the man travels, the elevation of his position changes according to the sine function, which does not have a global minimum or maximum but varies infinitely.
Shadow's Path Through the Origin
- Since the shadow passes through the origin, the line of travel can be characterized by y = mx, where m is the slope.
- The man can choose different slopes (m) to travel along the surface, affecting his elevation.
Seeking a Minimum Elevation
- To find the lowest elevation, one might consider approaching the problem by minimizing sin(xy).
- However, due to the oscillatory nature of the sine function, it does not settle at a minimum; it perpetually oscillates.
Conclusion: No Optimal Line Exists
- There is no single slope m that can lead him to a lowest elevation because:
- The sine function does not have a global minimum; it oscillates indefinitely.
- As the man moves, the elevation will continuously change without reaching a lowest point.
Thus, the correct assertion is "There isn't such a line" since all potential paths will lead to varying elevations without a definitive minimum.
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A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer?
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A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer?.
A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A man travelling onf(x, y) = sin(xy). His shadow passing through the origin in a straight line (sun travels with him overhead).What is the slope of the line travelling on which would lead him to the lowest elevation.Correct answer is 'There isn't such a line'. Can you explain this answer?.
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