Math 210 Take Home Assignment #2

Math 210 Take Home Assignment #2 Due: Wednesday, March 11 NAME:______

Show ALL work! Unless specified otherwise, each question is worth 4 points. 1. Find the domain and range of the following:  y  a) f (x, y)  ln(4  x  y) b) z  arccos   x 

2. Sketch the surface given by the function: g(x, y)  x2  3

3. (5 points) For the function f (x, y)  x 2  2y 2 sketch a contour plot. Include the level curves that correspond to c  0,2,4,6,8 4. (6 points) Evaluate the limit. If the limit does not exist, explain why.

lim y lim x 2  y 2 a) b) (x, y)  (0,0) x 2  y 2 (x, y)  (0,0) x 2  y 2

x  y 5. Find both first partial derivatives: z  ln x  y

6. Find both first partial derivatives: z  e y sin xy 7. (3 points) Find the first partial derivatives with respect to x, y, and z: w  x 2  y 2  z 2

8. Find any relative extrema and/or saddle points using the Second Partials Test: f (x, y)  5x 2  4xy  y 2 16x 10

9. Find an equation of the plane tangent to the surface given by f (x, y)  x2  y2  3 at the point (2, 1, 8). 10. (6 points) Suppose the temperature distribution T (in degrees Celsius) of a heated plate at a point (x, y) 100  in the xy-plane can be modeled by T(x, y)   lnx2  y2  1 x2  y2  9  ln 2  a) Show that T  0°C on the circle x2  y2 1 and T  200°C on the circle x2  y2  4

b) Find the rate of change of T in the direction of the positive x axis at the point (1, 2) and at the point (2,1).

c) Find the rate of change of T in the direction of the positive y axis at the points (2, 0) and (0, 2).

11) The Ideal Gas Law PV  nrT is used to describe the relationship between pressure P, volume V, and temperature T of a confined gas, where n is the number of moles of the gas and r is the universal gas V T P constant. Show that    1 T P V