Rate of ascent or descent on a hill

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SUMMARY

The discussion focuses on analyzing the rate of ascent or descent on a hill defined by the equation z = 100 - 0.005x² - 0.01y², with a specific point of interest at (60, 40, 966). Participants explore the effects of walking south and northwest on elevation, utilizing calculus concepts such as partial derivatives and the gradient operator. The analysis reveals that walking south results in a descent at a specific rate, while walking northwest also leads to a descent but at a different rate. Additionally, the direction of the steepest slope and the corresponding angle of ascent are determined through gradient calculations.

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  • Understanding of partial derivatives
  • Familiarity with the gradient operator
  • Basic knowledge of multivariable calculus
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  • Study the gradient operator and its applications in optimization
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anik18
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30. Suppose you are climbing a hill whose shape is given by the equation z= 100 -0.005x2 - 0.01 y2, where x,y, and z are measured in meters, and you are standing at a point with coordinates (60,40, 966). The positive x-axis points east and the positive y-axis points north.

(A) If you walk due south, will you start to ascend or descend? At what rate?

(B) If you walk due northwest, will you start to ascend or descend? At what rate?

(C) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?

Book: Calculus: Concepts and Contexts (3rd) by Stewart

thankyou
 
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Please show some attempted workings. For example, do you understand how to take partial derivative? Do you know what the grad operator does?
 

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