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Solution Key for Exam 1 - Vector Calculus I | MTH 254, Exams of Calculus

Material Type: Exam; Class: Vector Calculus I; Subject: Math; University: Portland Community College; Term: Fall 2007;

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

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MTH 254, Fall Term 2007
Test 1 โ€“ Given October 25, 2007 Name
1. Figure 1 shows the intersection of the surfaces
2
34
x
yz=โˆ’
and
()
2
2
13 2xyz=โˆ’ โˆ’โˆ’
. Find a
vector-valued function that models
continuous
motion
about this curve. You need to show all relevant work
in a well organized and well documented manner to
earn full credit for this problem. (15 points)
Fi
g
ure 1
pf3
pf4
pf5
pf8
pf9

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MTH 254, Fall Term 2007

Test 1 โ€“ Given October 25, 2007 Name

  1. Figure 1 shows the intersection of the surfaces

2

x = 3 y โˆ’ 4 z and ( )

2 2

x = 13 โˆ’ y โˆ’ z โˆ’ 2. Find a

vector-valued function that models continuous motion

about this curve. You need to show all relevant work

in a well organized and well documented manner to

earn full credit for this problem. (15 points)

Figure 1

2. Find parametric equations for the tangent line to the curve ( )

2 r t = 3 t , 4 โˆ’ t , 7

G

at the point

( 3, 5, 7 ). Make sure that you show all relevant work in a well-organized and well-documented

manner. Do not use your calculator on this problem (except as a check). (10 points)

3. Find the center of the osculating circle for the function ( )

2

r t = t + 2, 5 โˆ’ t , 2 t + 2

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at the

point where t = 1. You may use your calculator for all calculations, but you need to write down

all relevant results in a well-organized and well-documented manner. Make sure that your chain

of calculations is clear and that your final answer is clear. (15 points)

5. Consider r ( ) t = 4sin ( ) t โˆ’ 1, 16 cos ( ) t โˆ’ 8sin ( ) t + 10, 4 cos ( ) t + 2

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. This entire problem is a

non-calculator problem. (5 points each)

a. This curve lies entirely on a right circular cylinder โ€“ what is the equation of this cylinder?

Show your work!

b. The curve lies entirely on a plane โ€“ what is the equation of this plane? Show your work!

HINT: The plane equation contains all three variables and can be found via substitution.

c. The curve lies entirely on a paraboloid โ€“ what is the equation of this paraboloid? Show your

work โ€“ remember, do not use your calculator one this problem (except as a check).

r ( ) t = 4sin ( ) t โˆ’ 1, 16 cos ( ) t โˆ’ 8sin ( ) t + 10, 4 cos ( ) t + 2

G

HINT: The left side of the equation is

2 2

x + z.

  1. Waldo the Human Canon Ball was fired from a cannon whose muzzle was 8 ft above the ground.

Waldo traveled 585 feet horizontally before dropping into a net that was 12.5 ft above the

ground. Waldo left the cannon at a speed of 150 ft/s. How much time transpired between

Waldoโ€™s launch and the instant he landed in the net? (15 points)

Make sure that you show all relevant information and work in a manner consistent with that

discussed and demonstrated in class. You may perform all calculations on your calculator.

9. For some unknown circular motion function, r ( t )

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, you are told โˆ€ t r , โ€ฒโ€ฒ( t )โ‰  0

G

G

.

a. The speed function, y = r โ€ฒ( t )

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, is shown in Figure 3. Explain why this graph illustrates

that even though the motion is taking place around a circle, the velocity and acceleration

vectors for the motion are not always perpendicular. (2 points)

b. Although the velocity and acceleration vectors are notalways perpendicular for this

motion, the speed function in Figure 3 also illustrates that there are times when the

velocity and acceleration vectors for this motion must be perpendicular. Explain why

that is true. (4 points)

Figure 3

y = r โ€ฒ ( ) t

G