Name                                     MATH 107 PRACTICE MIDTERM III Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.   PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why. A) Let f(x,y) be a function of two variables and P be a point.  If the value of the function tends towards 5 for every line segment that ends at P, then     B)  If  z  =  f(x - y) and f is a continuous function then   PROBLEM 2 Find the dimensions of the rectangular box with the largest volume in the first octant such that one of the vertices is at the origin and the opposite vertex lies on the ellipsoid          x2 + 2y2 + 3z2  =  6 PROBLEM 3 Consider the function                    5         z  =                              xy A.     Use a calculator to sketch the level curves corresponding to z = (-2,-1,0,0.5,1,2,5,10).   Draw them on a whole piece of paper.   B.    Suppose that z represents the altitude function, and you are to travel on this surface above the unit circle in a counter-clockwise direction.  Discuss your travels. PROBLEM 4  Find the following limits if they exist.  A.     B.   PROBLEM 5  Let           f(x,y)  =  x2 + xy,     x(u,v)  = u2v,  and     y(u,v)  =  u - v Find A)       B)    Without calculating any derivatives, write the appropriate chain rule for          C)  Find  PROBLEM 6  If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2, and R3, then             1                1              1               1                     =               +              +                             R              R1             R2             R3                         If the resistances are measured as R1 = 25 ohms, R2 = 40 ohms, and R3 = 50 ohms, with possible errors of 0.5% in each case, use differentials to estimate the maximum error in the calculated value of R.  Solution   PROBLEM 7  Find parametric equations for the tangent line of the curve of intersection of the paraboloid z  =  x2 + y2 and the ellipsoid  4x2 + y2 + z2  =  9 at the point (-1,1,2).