Science:Math Exam Resources/Courses/MATH200/April 2012/Question 04 (c)
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Question 04 (c) 

The temperature at a point (x,y,z) is given by where T is measured in centigrade and x,y,z in metres. Find the maximum rate of decrease at P. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

From part (b) we have the direction which makes the directional derivative smallest. What is the actual value of the directional derivative in this direction? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. When T is decreasing most rapidly from its value at P, the actual rate of change (a scalar) is based on the direction chosen in part (b): This has units of . In plain English, the words can absorb the minus sign: The maximum rate of decrease of temperature at point P is obtained in the direction . 