MATH200 December 2012
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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?
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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.
Remember, that ,
Further, remember that the directional derivative is of a function in direction at the point is
Using the expressions from the first hint, make three equations for the gradient of the function , where you write the gradient as
Use the three equations to find .
The maximal rate of change is in the direction of the gradient vector.
The maximal rate of change is
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We have the following three statements given
The directional derivative is calculated through
is the direction vector.
Now we express the gradient of in the points as and obtain
So, we get that . The function increases most rapidly in the direction of the gradient vector. Hence, the required vector is .
The value of the maximal directional derivative is .
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