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20c-sp95-helton-mid2-a - Math 20C Spring 1995 Exam 2 Helton...

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Math 20C Spring 1995 Exam 2 Helton Give explanations which tell us clearly every step of what you are doing. Let F ( x, y, z ) = ( x 2 + y 2 x 2 ) + x + 100. Suppose the graph of F represents a shiny smooth hill. 1. What are the partial derivatives of F with respect to x and y . 2. What is the gradient? Suppose you are standing over the point x = 1 , y = 2 on the hill. 3. If you drop a marble, in which direction will it start to roll? You only need to specify the direction in the x, y plane which it will take. 4. What is the slope of the hill in the direction of the vector (2 , 4)? 5. Suppose you lay a ruler (1 foot) down on the hill pointing in the northeast direction with one tip at your feet. Approximately how much higher is the other tip? Here we
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