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Precalculus II
Calculus I Test 2
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
Explain the difference between a situation in which you would do each of the following;
if there is no such situation for a given item tell why:
- average two values of a function y(t) and divide by the corresponding difference in t
values
- subtract two values of a function y(t) and divide by the corresponding difference in t
values
- subtract two values of a function y(t) and multiply by the corresponding difference in t
values
- average two values of a function y(t) and multiply by the corresponding difference in t
values.
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Problem Number 2
The depth of water in a certain uniform cylinder is given by the depth vs. clock time
function y = .019 t2 + -2.2 t + 78.
What is the average rate at which depth changes between clock times t = 15.9 and t =
31.8?
- What is the clock time halfway between t = 15.9 and t = 31.8, 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 = 15.9 and t =
31.8?
If the rate of depth change is given by dy/dt = .028 t + -1.5 then how much depth change
will there be between clock times t = 15.9 and t = 31.8?
- Give the function that represents the depth.
- Give the specific function corresponding to depth 50 at clock time t = 0.
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Problem Number 3
Give the equations of the tangent lines to y = sin(-2.501 x) and y = 3 x at x = 0.
- Sketch the tangent lines and show how they can be used to determine the limiting
value of sin(-2.501 x ) / ( 3 x) at x = 0.
- Show that l'Hopital's Rule applies to the limiting value of sin(-2.501 x) / ( 3 x) at
x = 0. Then use the Rule to find this limit.
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Problem Number 4
Find dy/dx if arctan(x^ 1.5 y^ 5) = x^ 1.5 y^ 5.
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Problem Number 5
Yearly income grows at the rate of R(t) ) dollars / month, where t is in months and
where R(0) = 63.8, R( 4) = 51.97, R( 8) = 37.03 and R( 16) = 102.19. Write an
integral to express the increase in annual income from t = 0 and t = 16. Estimate the
value of this integral.
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Problem Number 6
The volume of a muscle is increasing at a rate of 5 cm^3 / month. The
lifting strength of the muscle is L(V) = 1.05 * V^(2/3) pounds, where V is the volume of the
muscle in cm^3.
- What is dL / dV?
- What is the meaning of dL / dV when V = 2000 cm^3?
- At what rate, in pounds/month, is lifting strength therefore increasing when the
muscle volume is 2000 cm^3?
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Problem Number 7
Answer by calculating an appropriate limit using l'Hopital's Rule:
- Which of the functions y = 811 x^ 5 or y = x^ 12 dominates for large x?
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Problem Number 8
Use both 2-interval and a 5-interval approximations to find the left and right
Riemann sums of the function y = 3 * 2^( .17 t) from t = 3 to t = 9.
- Based on these approximations make your best estimate of the integral.
- What would be the difference between left- and right-hand sums if we used 100
intervals for the approximation?