Question Details SCalcET [ ]
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1 72 Gradient II ( ) Due: Fri Oct :00 PM MDT Question Instructions Notes and Learning Goals 1. Question Details SCalcET [ ] Level curves for barometric pressure (in millibars) are shown for 6:00 AM on a day in November. A deep low with pressure 972 mb is moving over northeast Iowa. The distance along the red line from K (Kearney, Nebraska) to S (Sioux City, Iowa) is 300 km. Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. What are the units of the directional derivative? mb/km km/mb mb/km 2 km mb 2. Question Details SCalcET MI. [ ] f(x, y) = 5 sin(xy), (0, 8) direction vector
2 3. Question Details SCalcET XP. [ ] f(p, q) = 2qe p + 4pe q, (0, 0) direction 4. Question Details SCalcET XP. [ ] 2x + 8y f(x, y, z) =, (5, 3, 1) z direction vector 5. Question Details SCalcET XP. [ ] f(x, y, z) = tan(4x + 7y + 4z), (8, 4, 1) direction 6. Question Details SCalcET [ ] The temperature at a point (x, y, z) is given by T(x, y, z) = 400e x2 5y 2 9z 2 where T is measured in C and x, y, z in meters. (a) Find the rate of change of temperature at the point P(4, 1, 2) in the direction towards the point (6, 5, 5). C/m (b) In which direction does the temperature increase fastest at P? (c) Find the maximum rate of increase at P.
3 7. Question Details SCalcET [ ] Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 2x 2 4xy + xyz (a) Find the rate of change of the potential at P(5, 2, 6) in the direction of the vector v = i + j k. (b) In which direction does V change most rapidly at P? (c) What is the at P?
4 8. Question Details SCalcET [ ] Suppose you are climbing a hill whose shape is given by the equation z = x y 2, where x, y, and z are measured in meters, and you are standing at a point with coordinates (100, 120, 1206). The positive xaxis points east and the positive yaxis points north. (a) If you walk due south, will you start to ascend or descend? ascend descend At what rate? vertical meters per horizontal meter (b) If you walk northwest, will you start to ascend or descend? ascend descend At what rate? (Round your answer to two decimal places.) vertical meters per horizontal meter (c) In which direction is the slope largest? What is the rate of ascent in that direction? vertical meters per horizontal meter At what angle above the horizontal does the path in that direction begin? (Round your answer to two decimal places.)
5 9. Question Details SCalcET [ ] If f(x, y) = xy, find the gradient vector f(7, 6) and use it to find the tangent line to the level curve f(x, y) = 42 at the point (7, 6). gradient vector tangent line equation Sketch the level curve, the tangent line, and the gradient vector.
6 10. Question Details SCalcET [ ] If g(x, y) = x 2 + y 2 6x, find the gradient vector g(2, 6) and use it to find the tangent line to the level curve g(x, y) = 28 at the point (2, 6). gradient vector tangent line equation Sketch the level curve, the tangent line, and the gradient vector.
7 11. Question Details SCalcET [ ] Find equations of the following. 2(x 3) 2 + (y 6) 2 + (z 2) 2 = 10, (4, 8, 4) (a) the tangent plane (b) the normal line (x(t), y(t), z(t)) = 12. Question Details SCalcET [ ] Find equations of the tangent plane and the normal line to the given surface at the specified point. x + y + z = 5e xyz, (0, 0, 5) (a) the tangent plane (b) the normal line (x(t), y(t), z(t)) = Assignment Details Name (AID): 72 Gradient II ( ) Submissions Allowed: 10 Category: Homework Code: Locked: Yes Author: Skriletz, Jaimos ( jaimosskriletz@boisestate.edu ) Last Saved: Sep 30, :58 PM MDT Permission: Protected Randomization: Person Which graded: Last Feedback Settings Before due date Question Score Assignment Score Publish Essay Scores Question Part Score Mark Help/Hints Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response
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