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Math203 Demo Exam: Vector Calculus and Multivariable Functions, Exams of Advanced Calculus

The instructions and problems for a demo exam in math203, covering topics such as vectors, dot product, contour lines, directional derivatives, and multivariable functions. Students are required to answer all questions, show their work, and perform various calculations and graphical representations.

Typology: Exams

Pre 2010

Uploaded on 08/04/2009

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Math203, Demo Exam #1 (Chapters 12-14)
Directions: Answer all of the questions. The problems will have their weights assigned next
to them. Show all work for maximal credit. No notes, cellular phones, pagers, calculators,
books or friends. Good luck.
1) Definitions.
a. What is a Vector? What does it represent? Provide an example.
b. What does the Dot Product signify? Provide an example.
c. For a single variable function f(x), what does it mean when we state that )(
lim
3
xf
xโ†’
exists?
d. For the multivariable function f(x,y), what does it mean when we state that
),(
lim
2,3
yxf
yx โ†’โ†’
exists?
e. What do contour lines represent?
f. Define the directional derivative and provide an example.
2) For the function,
( ) ( )
22 35
4
1++โˆ’= yxz , perform the following. Be sure to explain
your reasoning.
a. Draw the function on the x-y plane as a set of contours representing static values of
z.
b. Draw the function on the y-z plan as a set of contours representing static values of x.
c. Identify any points which are discontinuous for the above function and describe why.
d. Describe the shape of the function.
e. Graph the function in 3-dimensional space.
3) For the vectors, kjiA
๎˜
๎˜
๎˜
๎˜
23 +โˆ’= and kiB
๎˜
๎˜
๎˜
3โˆ’= , perform the following. Be sure to
explain your reasoning.
a. Graph both vectors in 3 dimensional space and determine them angle between them.
b. Find 45 +โ‹… BA
๎˜
๎˜
. What does this signify?
c. Find
(
)
ABA
๎˜
๎˜
๎˜
โ‹…โˆ’ร— 22 . What does this signify?
d. Find
(
)
BBA
๎˜
๎˜
๎˜
โ‹…. What does this signify?
4) For the function,
(
)
(
)
2
22 435),,( zyxzyxf โˆ’++โˆ’= , perform the following. Be sure to
explain your reasoning.
a. Find x
f,
z
f
โˆ‚
โˆ‚
, f
โˆ‡
. What do each of these represent?
b. Find
)0,1,1(
y
f
โˆ‚
โˆ‚ and the directional derivative at (1,1,0) going in the direction kj
๎˜
๎˜
+.
c. Find yx
f
โˆ‚โˆ‚
โˆ‚2
.
d. Is this function differentiable for the entire region of 3-space?

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Math203, Demo Exam #1 (Chapters 12-14)

Directions: Answer all of the questions. The problems will have their weights assigned next

to them. Show all work for maximal credit. No notes, cellular phones, pagers, calculators,

books or friends. Good luck.

  1. Definitions.

a. What is a Vector? What does it represent? Provide an example.

b. What does the Dot Product signify? Provide an example.

c. For a single variable function f(x), what does it mean when we state that

( )

lim

3

f x

x โ†’

exists?

d. For the multivariable function f(x,y), what does it mean when we state that

(, )

lim

3 , 2

f x y

x โ†’ y โ†’

exists?

e. What do contour lines represent?

f. Define the directional derivative and provide an example.

2) For the function, ( ) ( )

2 2

z = x โˆ’ + y + , perform the following. Be sure to explain

your reasoning.

a. Draw the function on the x-y plane as a set of contours representing static values of

z.

b. Draw the function on the y-z plan as a set of contours representing static values of x.

c. Identify any points which are discontinuous for the above function and describe why.

d. Describe the shape of the function.

e. Graph the function in 3-dimensional space.

  1. For the vectors, A i j k

= 3 โˆ’ + 2 and B i k

= โˆ’ 3 , perform the following. Be sure to

explain your reasoning.

a. Graph both vectors in 3 dimensional space and determine them angle between them.

b. Find 5 A โ‹… B + 4

. What does this signify?

c. Find A ( B A )

2 ร— โˆ’ 2 โ‹…. What does this signify?

d. Find ( A B ) B

โ‹…. What does this signify?

4) For the function, ( ) ( )

2 2 2

f ( x , y , z )= x โˆ’ 5 + y + 3 โˆ’ 4 z , perform the following. Be sure to

explain your reasoning.

a. Find

x

f ,

z

f

, โˆ‡ f. What do each of these represent?

b. Find

( 1 , 1 , 0 )

y

f

and the directional derivative at (1,1,0) going in the direction j k

c. Find

x y

f

2

.

d. Is this function differentiable for the entire region of 3-space?