Science:Math Exam Resources/Courses/MATH200/December 2012/Question 05
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Question 05 

The directional derivative of a function at a point P in the direction of the vector is , in the direction of the vector is , and in the direction of the vector is . Find the direction in which the function has the maximum rate of change at the point P. What is the maximum rate of change? 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Remember, that , , .

Hint 2 

Using the expressions from the first hint, make three equations for the gradient of the function , where you write the gradient as

Hint 3 

The maximal rate of change is in the direction of the gradient vector. The maximal rate of change is 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We have the following three statements given The directional derivative is calculated through where is the direction vector. Now we express the gradient of in the points as and obtain
