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Material Type: Exam; Class: Vector Calculus I; Subject: Math; University: Portland Community College; Term: Fall 2007;
Typology: Exams
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MTH 254, Fall Term 2007
Test 1 โ Given October 25, 2007 Name
2
2 2
x = 13 โ y โ z โ 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 r t = 3 t , 4 โ t , 7
at the point
manner. Do not use your calculator on this problem (except as a check). (10 points)
2
r t = t + 2, 5 โ t , 2 t + 2
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)
. 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).
HINT: The left side of the equation is
2 2
x + z.
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
Waldoโs 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.
.
, 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
