Science:Math Exam Resources/Courses/MATH200/December 2011/Question 03 (b)
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Question 03 (b) 

A bee is flying along the curve of intersection of the surfaces and in the direction for which z is increasing. At time t = 2, the bee passes through the point (1,1,0) at speed 6. (b) The temperature T at position (x,y,z) at time t is given by . Find the rate of change of temperature experienced by the bee at time . 
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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 

The rate of change of the temperature is the derivative of the temperature, which is the partial derivative of the temperature plus the directional derivative of in the direction of the velocity vector v of the bee,

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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To evaluate the change in temperature, , experienced by the bee with respect to time at time 2, we need to evaluate the full derivative of with respect to and substitute in the value . Writing the expression for the full time derivative gives: where is the velocity vector of the bee. Evaluating the terms in the above equation we find: Note that we don't need to reevaluate since we can just use our results from part (a) here. At time , the bee is at , so the above terms satisfy, at . Using this and the fact that , we evaluate the change in temperature: Therefore, the bee is experiencing a change in temperature of 10 temperature units per second at t = 2. 