Math 55 Finals Samplex [UP DILIMAN]

lo*sr Flxel Exeu

Ftnsr Spupsrnn A.Y. 2013-20L4 10 Ocronpn 2013

This exam is for two hours only. Use only black or blue ink. Show all necessary solutions and box ali final answers. Calculators and other electronic devices are not allowed. Any form of cheating in examinations or any act of dishonesty in reiation to studies shall be subject to disciplinary action.

I. Let f(*,y) -

2a,

*

Zyz

*

6r'2

* t2rg.

1' Find the rate of change of / at the point (5,0) in the direction of the vector (g,4). {3 poi,ntsl {5 poi,ntsl

2. Determine and classify the critical points of /.

II.

Use the method of Lagrange multipliers to find the point on the plane is nearest to the

origin.

III.

-

4y

z: -

that {s pointsl

fr:

-

tl,

Rewrite

r'/1 r'/i:fr f \F-"'n'

J, J, J**

zd'zd'Yd'x

in spherical coordinates and then evaluate the resulting triple integral.

=-

* z :3

Set up the iterated triple integral in cylindrical coordinates equal to the volume of the solid in the first octant bounded by the paraboloid L y2, the plane x)2 U, the zy-plane and the pointsl

yz-plane.

IV.

2r

[5 points]

'-!.1 :Yg.> \2ry',W:!9osrv) ---

1. Find all possible potential functions for F. [z points] 2. Compute the work done by F in moving a particle from the point (1,0) to the point (0, [2 pitnt !

_tlz).

VI'

Use Green's Theorern to set up the iterated double integral in polar coordinates that is equai

to

f-

f_

(cos(r)

JC

-

a') dr

*

(eo

+ ,u) dy where C is the circle rz + yz :2r.

fi

VII. Let & bethesurfacewithvectorequation Efu,u):uz t+u2 j+uuhwhere

-1