=========================preview======================
(math100)[2009](s)final~PPSpider^_10412.pdf
Back to MATH100 Login to download
======================================================
Math100 Introduction to Multivariable Calculus
Spring 2009
C Final Exam C

Name:
Student ID:

Lecture Section:

.
There are FIVE questions in this midterm examination.

.
Answer all questions.

.
You may write on both sides of the paper if necessary.

.
You may use a HKEA approved calculator. Calculators with

symbolic calculus functions
are not allowed.


.
The full mark is 100.


Question Points
Q1
Q2
Q3
Q4
Q5
Total
2
1. The plane de.ned by z =2x + y . 4 and the cylinder de.ned by x2 + y= 1 intersect along an ellipse. Formulate an optimization problem with constraint and use the method of Lagrange multipliers to .nd the coordinates of the highest and lowest points (i.e., points of the highest and lowest z coordinates) on the ellipse. [20 points]
Solution
To maximize/minimize f(x, y)=2x + y . 4 subject to the constraint g(x, y)= x2 + y2 . 1= 0,
we solve
. f = . g .. (2, 1) = (2x, 2y)
11
from which we have x =, y = . Putting them into x2 + y2 =1, we have
2

11 55
+ =1 .. 2 = .. = .
2 42 42



5 5
2
5
1
When =
5 . 4.

, x =

, y =

, z =

2



5 5
2
5
1
When = .
5 . 4.

, x = .

, y = .

, z = .

2




The highest z coordinate is z =5 . 4, and the lowest is z = . 5 . 4.

5
2
,

5
1
The highest point on the ellipse is :
5 . 4.

,
1
5

.

5
2
, .

The lowest point on the ellipse is :
, .

5 . 4

.

2. Consider the vector .eld F(x, y)=(ey + yex) i +(ex + xey . y) j in 2-space.
(a)
Show that F is a conservative vector .eld and .nd an explicit potential function (x, y)for F such that . = F. [12 points]

(b)
Evaluate the line integral F d r, where the curve C is parametrized by


C
r(t)=(t2 . 8) cos t i +(6t +5) j
with t from 0 to /2. [8 points]
Solution
y
(a) Let F(x, y)=(ey + yex) i +(ex + xey . y) j = f i + g j. We .nd .g/.x = .f/.y = ex + e. Therefore, F is conservative.
. .
x
If F = . ,we have = ey + yeand = ex + xey . y. Integrating the .rst equation with
.x .y
respect to x,we have
=(ey + ye x)dx = xey + ye x + h(y).
Putting back to the second equation, we have 1
xey + e x + h(y)= e x + xey . y .. h = . y 2 + C.
2So, a potential function can be chosen as
1 2
= xey + ye x . y.
2 That . = F also shows the given vector .eld to be conservative.
(b) The curve starts from r(0) = (. 8, 5) and ends at r(/2) = (0, 3 +5).
(0,3+5)
1 .
F d r =(xey + ye x . y 2) .
2
C (.8,5)
1 25
.8 +
=(3 +5) . (3 +5)2 +8e 5 . 5e.
22
3. (a) Use a triple integral to evaluate the volume of the solid enclosed by 4
z = ,z =0,y = x, y =3, and x =0.
y2 +1[10 points]
(b) Evaluate the surface integral
(x 2 + y 2)dS

where the surface is the portion of the cone z = 3(x2 + y2)for 0 z 3. [10 points]
Solution
(a) Volume = dV in which the so