Read the Instructions Carefully

MAT 294III

TEST 2

Fall/2006

Dr. Anmin Zhu

Read the instructions carefully:

1. You are allowed to have one page formula sheet for this test.

2. There are 5 pages and 9 numbered questions.

3. You must show all work in order to receive credit!!

1. True or False

a. ( F ) If (a, b) is a critical point of z = f(x,y), then and .

b. ( T ) If z = f(x,y) is a smooth function then there is a direction at a point P in

which

c. ( F ) The directional derivative of z = f(x,y) at a point in the direction

is .

d. ( F ) If all of the contours of a function z = f(x,y) are parallel lines, then the

graph of z=f(x,y) is a plane.

e. ( F ) The tangent plane to the surface at the point (1, 1, 1) is

parallel to the plane .

2. The level curves of are shown below. Decide the sign (positive, negative, or zero) of the following quantities : __+___,__-___,__0___,__+___, __+___, ___-__ and sketch the surface of .

1 2 3 4 5 6 7 8

A B C

3. a) Let , find and .

b) Let where F, u and v are differentiable,

, , , , ,

, . Find and .

4. The length l, width w, and height h of a box change with time t, at a certain instant the

dimensions are l=100cm and w=h=200cm, and l and w are increasing at rate of 2 cm/s while h

is decreasing at rate of 3 cm/s. At that instant find the rate of change of the surface area of the

box.

5. The Celsius temperature in a region in space is given by . A particle

is at the point .

a)  In which direction should the particle move so that the temperature increases the fastest? and what is that rate.

b)  Find the rate of change of the temperature if the particle moves in the direction toward the point (2,-1,4).

6. a) Find an equation of the tangent plane to the surface at the point (1, -1, -1).

b) Find the tangent line to the level curve when at the point (1, -1).

7. Suppose that the directional derivative of f(x, y) at (1,1) in the direction is and in

the direction is . Find the directional derivative of f in the direction .

8. Design a rectangular milk carton box which holds 432 of milk. The sides of the box

cost 1 cent/ and top and bottom cost 2 cents/. Find the dimensions of the box that

minimize the total cost of materials used.

9. a) Find the local maximum and minimum values and saddle point(s) of the

Function

b) Find the absolute maximum and minimum values of

on the region