Time and Date Stamps (logged): 04:52:01 12-05-2008
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Precalculus II
Precalculus I Final Exam
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.).
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 a difference equation for the principle if the interest rate is 11%, with an
initial principle of $ 18000. Use this difference equation to find the principle after 1, 2, 3
and 4 years.
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Problem Number 2
What are the zeros of the linear polynomials f(x) =
3 x - 9 and g(x) = x + 8?
- What quadratic polynomial q(x) do you get if you
multiply f(x) by g(x)?
- Using the quadratic formula find the zeros of the
quadratic polynomial q(x). Are the zeros identical to those of f(x) and g(x)?
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Problem Number 3
For the function y = f(t)
= t^ 1.5 construct a table of y vs. t for t running from t = 2.5 to t = 2.7 in four equal
increments. Using appropriate transformation(s) on the y column, the t column, or both,
linearize this data set and demonstrate that the data set has in fact been linearized.
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Problem Number 4
The population of a certain organism is governed by the recurrence relation a(0)
= 2, a(1) = 4, a(n) = a(n-1) + 4 a(n-2), where n is the number of the population
transition.
- Find the populations after each of the first 10 transitions.
- Find the ratios a(n) / a(n-1) for n = 3, 4, ..., 10. Fully describe the
behavior of the ratios.
- What would you expect to be the growth factor for an exponential function
f(n) = A b^n which models the population for large values of n?
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Problem Number 5
If f(x) = x^-2.003 and g(x) = .41 ^ x, sketch graphs of f(x) and g(x) between x = -3
and x = 3, showing how you use the appropriate basic points to obtain the shape the graphs
of f(x) and g(x). Show how you combine the graphs to obtain the graph of f(x) +
g(x).
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Problem Number 6
Sketch a graph of y = (x + 4) ^ 4 (x – 3.5) ( 2 x^2 + 4 x + 4).
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Problem Number 7
If a(n) is the amount of drug present just before
the nth dose, then a(n-1) is the amount present just before dose number n-1.
If the dosage is 300 mg and the proportion removed
during a dosage cycle is . 65, then write expressions for the following:
In terms of a(n-1), the drug concentration just
after dose number n-1.
In terms of the preceding expression, the drug
concentration just before dose number n.
The expression you just wrote, along with the
assumed initial condition a(1) = 0 (i.e., just before does #1 the concentration is zero),
constituts a recurrence relation for a(n).
Evaluate your recurrence relation for n = 2, 3, 4
and 5.
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Problem Number 8
What are the basic points of the exponential function y = f(x) = 6 * `2^x? Graph the
function using these points.
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Problem Number 9
A population starts at 9000 and grows at the rate of 4.1 percent per year.
- What will be the population 1, 2, 3 and 10 years later?
- What function P(t) gives population as a function of number t of years?
- What are the growth rate and growth factor of this function?
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Problem Number 10
A graph of the Celsius temperature T(t) of a large dead turkey as it approaches
room temperature is an exponential function of form T(t) = A b^t + c, where t is clock
time in hours. The function has asymptote T = 68, initial temperature T = 0 and
approaches twice as close to room temperature in a period of 4.5 hours. A graph of
the rate, in millions of microorganisms per minute, at which bacteria reproduce is a
quadratic function R(T) of Fahrenheit temperature with vertex at (0, 0) and with the point
( 27, 68).
- Sketch reasonable graphs of these functions and use the graphs to construct the
function R(t) which gives rate of bacteria reproduction as a function of clock time.
- From estimates and interpretations of appropriate slopes on the graphs of R(T)
and T(t) estimate the slope of your R(t) graph at t = 6 and give the interpretation of
this slope.