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8. Suppose that the temperature in degrees Celsius at each point of the coordinate plane is given by T(x,y) = 3x2 + 4y2 +5. (Assume that x and y are in meters.) a) If we leave the point (3, 4) heading to the point (4, 3) how fast is the temperature changing as we move in the direction of (4,3)? b) After reaching the point (4,3), in what direction should we go if we want to cool of as fast as possible? c) Suppose that we want to move away from (4,3) along a path of constant temperature. What is an equation of path our path and what is the temperature along this path? d) A person at the origin may go any direction and will experience the same rate of change of temperature. Explain why. What is the rate? 9. Determine the maximum of f(x, y, z) = 2x + 3y + 5z subject to the constraint g(x,y,z) = x2 + y2 + z2 = 19 using Lagrange multipliers.
8. Suppose that the temperature in degrees Celsius at each point of the coordinate plane is given by T(x,y) = 3x2 + 4y2 +5. a) If we leave the point (3, 4) heading to the point (4, 3) how fast is the temperature changing as we move in the direction of (4,3)? b) After reaching the point (4,3), in what direction should we go if we want to cool of as fast as possible? (Assume that x and y are in meters.) c) Suppose that we want to move away from (4, 3) along a path of constant temperature. What is an equation of path our path and what is the temperature along this path? d) A person at the origin may go any direction and will experience the same rate of change of temperature. Explain why. What is the rate? 9. Determine the maximum of f(x,y,z) = 2x + 3y + 5z subject to the constraint g(x,y,z) = x2 + y2 + z2 = 19 using Lagrange multipliers.