Time and Date Stamps (logged): 05:22:01 12-05-2008
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Precalculus II
Calculus I Test 1
Completely document your work and your reasoning.
You will be graded on your documentation, your reasoning, and the
correctness of your conclusions.
Test should be printed using Internet Explorer. If
printed from different browser check to be sure test items have not been cut off. If
items are cut off then print in Landscape Mode (choose File, Print, click on Properties
and check the box next to Landscape, etc.).
- Write on ONE SIDE of paper only
- If a distance student be sure to email
instructor after taking the test in order to request results.
Name and Signature of Student
_____________________________
Signed by Attendant, with Current Date and Time:
______________________
If picture ID has been matched with student and name as
given above, Attendant please sign here: _________
Instructions:
- Test is to be taken without reference to text or
outside notes.
- Graphing Calculator is allowed, as is blank paper or
testing center paper.
- No time limit but test is to be taken in one
sitting.
- Please place completed test in Dave Smith's folder,
OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va.,
24212-0828 OR email copy of document to dsmith@vhcc.edu,
OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken. Student may, if proctor deems it feasible, make and retain a copy of the test..
Directions for Student:
- Completely document your work.
- Numerical answers should be correct to 3 significant
figures. You may round off given numerical information to a precision consistent
with this standard.
- Undocumented and unjustified answers may be counted
wrong, and in the case of two-choice or limited-choice answers (e.g., true-false or
yes-no) will be counted wrong. Undocumented and unjustified answers, if wrong, never get
partial credit. So show your work and explain your reasoning.
- Due to a scanner malfunction and other errors some
test items may be hard to read, incomplete or even illegible. If this is judged by
the instructor to be the case you will not be penalized for these items, but if you
complete them and if they help your grade they will be counted. Therefore it is to
your advantage to attempt to complete them, if necessary sensibly filling in any
questionable parts.
- Please write on one side of paper only, and staple
test pages together.
Test Problems:
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Problem Number 1
Write the differential equation expressing the hypothesis that the rate of change of an
investment is proportional to the principal P. Evaluate the proportionality constant
if it is known that the when the principal is 4868 its rate of change is 200. If
this is the t=0 value of the principal, then approximately what will be the principal at t
= 1.6? What then will be the principal at t = 3.2?
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Problem Number 2
Problem: Write a differential equation expressing the statement that the rate at which
illumination changes with respect to distance x from the source is inversely proportional
to the distance.
Problem: If dy / dt = .98 t^2 + .91 (t+y)/(t+1), and if at t = 0 we have y = .45, then
find the approximate value of y when t = .2. Using the new values of y and t, find
approximate value y when t = .4. Continue for two more steps to find the approximate
value of y when t = .8.
(extra credit): Use a predictor-corrector method, with `
Dt = .4 instead of the .5 used above, to find
the approximate value of y when t = .8. Which value do you think is more accurate?
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Problem Number 3
Solve using proportionalities by stating the appropriate proportionality law and
finding the proportionality constant:
- If a sand pile 2.7 meters high has a mass of 8660.521 kg,
then what would we expect to be the mass of a geometrically similar sand pile 6.7 meters
high?
- If there are 1.5309 billion grains of sand exposed on
the surface of the first sand pile, how many grains of sand we expect to be exposed on the
surface of the second?
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Problem Number 4
Determine the functions f(z) and g(x) for which each of the following is a composite of
the form f(g(x)):
5 e ^ ( 9 t^2 - 9 t + 8)
4 tan( 3 x^2)
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Problem Number 5
The number of hours between dawn and dusk is modeled in a certain area by the
function
- h(t) = 2 sin ( (2 `pi / 365) (t - 80) ) + 12,
where h is the number of hours and t the number of days since January 1.
Sketch a graph of this function and show on your graph the points where
- h is maximum (longest days)
- h is increasing fastest
- h is decreasing fastest
- h is minimum.
Determine the approximate values of t for each of these events.
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Problem Number 6
If water depth is changing at the rate = 7 t / ( t + 7) at clock time t, with
rate in cm/sec when t is in sec, then use two 2-interval approximations to estimate the
change in depth between clock times t = 6 sec and t = 15 sec. One of your
approximations should be an overestimate, the other an underestimate.
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Problem Number 7
What are the signs of f(x) and f ' (x) if the function f is negative and
decreasing at a decreasing rate? Sketch possible graphs of f(x) and f ' (x).