GREEN-SHEET-1995-02



Math 32 (01) TEST 3 (100 Points) Last Name: _________________

(Saleem) Fall 2012 First Name: _________________

You are allowed to use a 3”x5” card of notes & a calculator. Show all work in the spaces provided.

##1(30 points) Consider the surface, z = f(x,y) = [pic]and answer parts (a) through (k):

(a) What is the domain of f? Draw/shade below: (b) Draw a few contours also.

y y

x x

(c) What is the range of f?

(d) Compute the limit of f(x,y) as (x,y) (8,6) along the line y = x−2. Show work below.

(e) Compute the limit of f(x,y) as (x,y) (8,6) along the circle x2+y2=100. First convert everything to polar coordinates x=rcosθ and y = rsinθ. Then take the limit. Show work below.

(g) Is the function f(x,y) = [pic] continuous at (8,6)? ______

(h) Find [pic]at the point (8,6).

(i) Find [pic]at the point (8,6).

(j) The surface z = f(x,y) = [pic]looks like a spherical hill whose base is in the x-y plane. If you were standing on the hill, directly above the point (8,6), which direction would you go initially, in order to get down the hill, fastest? Write your answer in the form [pic]. Find a & b.

##2(20 points). Compute the indicated partial derivatives and simplify completely:

(a) Recall from calculus-1 that the derivative of arctan(x) is 1/(1+x2).

For the function g(x,y,z) = [pic]; Find gy(1,1,1) and gz(1,1,1).

(b) F(x,y,z) = (f(x)+g(y)+h(z))2; Write an expression for Fxx.

(c) f(x,y) = Sin(x+y) + Cos(x−y); Find fx and fxy .

(d) Given an equation: xy ’ ln(xy), use the Implicit Function Theorem, to compute [pic]at the point(2,2). Simplify completely. [Only 50% credit for using implicit differentiation from calculus one]

##3(15 points). (a) Find all the critical points of the function, f(x,y) = [pic]x3 + 4y3 − x4 − y4

(b) Calculate the Hessian, H = [pic]and simplify. Do not plug in numbers or critical points.

(c) Show all work and check whether there is a local min, max or saddle at each critical point .

##4(15 points). Suppose f = ln(x2+y2+z2) where x = t2+v2+w2 , y = t2−v2+w2 and z = t2−v2−w2 Answer the following:

a) Draw a tree diagram in the space provided.

b) Find ft at (t,v,w) = (1,1,1).

c) Find fw at (t,v,w) = (1,1,1).

##5(20 points). The surface z = [pic]looks like a big bowl whose lowest point is directly below the origin. (a) What is the maximum depth of the bowl? ___________

(b) Find [pic]at (3,3)

(c) Find [pic]at (3,3)

(d) What is the directional derivative of z at (3,3), in the direction of the vector [pic]

(d) Write the equation of the tangent plane to the surface at the point that lies directly below (3,3).

---- ---- --- Students should fold here and carefully cut this part --- ---- --- --- --- --- --- ---

“TAKE HOME” (extra credit 10 points) due at class time, Wednesday November 7.

Using Lagrange multipliers, find the points on the sphere, x2+y2+z2 =1, that are closest to and farthest from the point (3,1,-1). For full credit, show all work.

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tree diagram goes here

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