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 .
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!
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,
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
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 re-evaluate 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.