A circular disk of radius 5 meters centered at the origin has been unevenly heated. The temperature at any point on the disk is given by T(x, y) = 50e-2²-y degrees Celsius, where x and y are measured in meters. (a) Where on the disk is the temperature the greatest? What is the temperature at that point? (b) Find the direction of greatest temperature increase at the point (1, 1). (c) What is the rate of temperature change if you move away from the point (1,1) in the direction of the point (2,0)?

The slope of the tangent line to a plane curve y=f(x) at the point (x,y) is given by - 2 +3 + 16x m=- VX Determine the equation of the curve which passes through the point (16 , 52) The equation of the curve is given by y =

Suppose you are climbing a hill whose shape is given by the equation z = 1600 – 0.005x² – 0.01y², where x, y, and z are measured in meters, and you are standing at a point with coordinates (60, 40, 1566). The positive x-axis points east and the positive y-axis 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.)