\f10. For f(x, y, z) = sin(aty+=) xtytz calculate the following (a) Compute Of /Oz. (b) Does lim f(x, y, z ) exists? Explain why /why not. (x , y, =) - (0,0,0)\f3. In a room. the temperature is given by T : x. :9) degrees Celsius where :5 gives the distance from the heater ( in meters) and t is the elapsed time (in minutes) since the heat has been turned on. A person standing 5 meters from the heater 3 minutes after it has been turned on observes the following: (1) At this distance. the temperature is increasing by 2" C per minute. and (2) Walking awayr from the heater, the temperature decreases by 1206' per meter as time is held constant. Estimate how much colder or warmer it would be 6 meters from the heater after 2.5 minutes. 4. Consider the surface with equation 4x2 - 3y2 + 22 = 25 (a) Find a vector normal to the surface at (3, 2, 1). (b) Find the equation of the tangent plane at the point (3, 2, 1).5. Please do all parts a-c. (a) Find a unit vector u, having the same direction as: 3i + 4j. (b) Find the measure of the angle between a = and b = (c) Find a x b given a = and b =6. Suppose that the depth. h, of a lake below the point (3;,y) on the surface is given by Hazy) : ley 3:3 '20. The point (3:, y) is the location of the duck on the surface of the lake where a: and y are measured in feet on the surface of the lake and h is the depth. measured in feet below the point (I. y). Suppose a duck is at the point (3. 2). (a) What will the instantaneous rate of change in the depth of the lake if the duck heads in the direction towards the point (6, 7)? \"ill the depth be increasing or decresing? No decimals in your answer please (in other words, provide an EXACT answer). (13) In what direction should the duck swim from the point (3, 2) in order for the depth to decrease most rapidly? Your answer should be in vector format. (c) How fast will the depth increase in the direction given in Part I)? No decimals in your answer please (in other words. provide an EXACT answer). (d) In what direction should the duck swim so that the depth remains the same? 7. Find an equation of the plane determined by the points P(4, 3, 1): Q(6. 4. 7); and R(1. 2. 2) 8. The depth: in feet, of a lake at a point :1: miles east and 3; miles north of a buoy is given by h($_._ y) : 175 i 30:):2 7 20312. (a) A rowboat is 1 mile east and 2 miles south of the buoy. At what rate is the depth changing with respect to distance in the direction of the buoy? (b) The boat starts moving toward the buoy at a rate of 4 mph. At what rate is the depth of the lake beneath the boat changing with respect to time?