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Precalculus II
Calculus 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.).
- 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
Find the derivative of y = -5 cos (-4 `pi e^( 8 x) ) ).
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Problem Number 2
Use the Chain Rule and the derivative of f(z) = sin(z) to find the formula for
the derivative of y = sin^-1(x).
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Problem Number 3
The profit function for a certain product is equal to -$ 9800 when x = number of
units produced is zero. The profit function remains negative for x = 0 to x = 1350
units, when its value is zero. The profit function then increases to a maximum of
$ 136900 at x = 2241 units and returns to zero at x = 3010 units, from which point the profit
function becomes concave downward. Sketch possible cost and revenue functions
consistent with this description and indicate the folowing:
- all x values at which the graphs cost and revenue functions meet.
- the marginal cost and marginal revenue when profit is maximized.
- the value of x for which profit is increasing most rapidly with respect to number
of units produced.
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Problem Number 4
Problem: Derive the expression for the instantaneous rate of change of the function
y(t) = a t^2 + b t + c at clock time t.
Problem: If the rate of depth change is rate(t) = .056 t + -2.3, then what is the depth
function if the depth at clock time t = 0 is 78? At what instant does the flow cease, and
what is the depth at that instant?
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Problem Number 5
Is f(x) = sin(1 / (x- 4) ) continuous at x = 4? Is f(x) differentiable at
this point?
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Problem Number 6
Optimize the volume of a cylindrical container closed at top and bottom, given
that its area is 410 cm^2.
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Problem Number 7
Give the equations of the tangent lines to y = sin( 2.5 x) and y = -4 x^2 at x =
0.
- Sketch the tangent lines and show how they can be used to determine the limiting
value of sin( 2.5 x ) / (-4 x^2) at x = 0.
- Show that l'Hopital's Rule applies to the limiting value of sin( 2.5 x) / (-4 x^2)
at x = 0. Then use the Rule to find this limit.
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Problem Number 8
The depth of water in a certain uniform cylinder is given by the depth vs. clock time
function y = .024 t2 + -2.7 t + 92.
What is the average rate at which depth changes between clock times t = 15.2 and t =
30.4?
- What is the clock time halfway between t = 15.2 and t = 30.4, and what is the rate of
depth change at this instant?
- What function represents the rate r of depth change at clock time t?
- What is the value of this function at the clock time halfway between t = 15.2 and t =
30.4?
If the rate of depth change is given by dy/dt = .177 t + -2.7 then how much depth change
will there be between clock times t = 15.2 and t = 30.4?
- Give the function that represents the depth.
- Give the specific function corresponding to depth 220 at clock time t = 0.
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Problem Number 9
Identify maxima, minima and points of inflection for f(x) = 5 x - 8 ln(x).
Code `t
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Problem Number 10
The average value of f(t) on the interval from t = 5 to t = 11 is 9 and the average
value of f(t) from t = 11 to t = 15 is 6.5. What is the average value of f(t) on the
interval from t = 5 to t = 15?
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Problem Number 11
If f(x) is measured in Newtons and x in meters then what are the units of the integral
int( f(x), x, a, b )?
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Problem Number 12
The rate at which water flows out of a small hole in a container is proportional
to the square root of the depth of the water relative to the hole. If no water is
added to the container, water water level changes as water exits through the holes.
Suppose the water level is initially 1.57 meters above one hole and .661 meters above a
second, and that both holes are of the same size. The following events occur:
- At the instant the water level reaches 1.296 meters above the lower hole, the first
hole is suddenly plugged.
- At the instant the water level reaches 1.099 meters above the lower hole, the first
hole is suddenly unplugged and the second plugged.
- At the instant the water level reaches 1.002 meters above the lower hole, both holes
are again unplugged.
Sketch a graph showing the behavior of water depth vs. clock time.
- Is this function continuous?
- Is this function differentiable?
Sketch a graph showing the behavior of the rate of depth change vs. clock time.
- Is this function continuous?
- Is this funciton differentiable?
Sketch a graph showing the behavior of the rate at which the rate of depth
change changes vs. clock time.
- Is this function continuous?
- Is this funciton differentiable?