Precalculus II

Time and Date Stamps (logged): 01:42:03 08-29-2008 ｯｰ浹ｱ浣ｲｯｷ浤ｸ浤ｯｯｷ 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

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

average two values of a function y(t) and multiply 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

Problem Number 2

Use the Chain Rule and the derivative of f(z) = e^z to find the formula for the derivative of y = ln(x).

Problem Number 3

Water is flowing at a constant rate into an urn which is fat at the bottom, fat in the middle and very fat at the top, but narrow about 1/3 of the way up and very narrow about 2/3 of the way up. Sketch the urn, and sketch a possible graph of depth vs. clock time and indicate on your graph when the water level is at the point where the urn is narrowest and at the point where where it is widest.

Problem Number 4

The average value of f(x) = A x^2 between x = 6 and x = 10 is 14. What is the value of A?

Problem Number 5

Find the locations of maxima, minima and inflection points for f(x) = x^3 - k x. Sketch these locations for k = 1, 2, 3 and 4. Sketch the curves corresponding to these values, and describe how k determines the x values at which these points occur, and the values of the function at its extreme points.

Problem Number 6

In terms of two simple examples involving temperature, rate of temperature change and time, explain the difference between the meaning of the slope of a trapezoid and the area of a trapezoid. Be sure to explain your results in terms of units and also in terms of meanings.

In terms of two simple examples involving depth, rate of depth change and time, explain the difference between a situation where you obtain a meaningful result by subtracting two quantities and dividing by a time interval and a situation where you obtain a meaningful result by averaging two quantities and multiplying by a time interval. Be sure to explain your results in terms of units and also in terms of meanings.

Problem Number 7

The area of a rectangle is increasing at a rate of 5 cm^2 / hour. The volume of the cylinder that can be formed by rolling up the rectangle is V(A) = .052 * A^(3/2).

What is dV / dA? What is the meaning of dV / dA?
At what rate, in cm^3/hour, is volume changing when the volume is 2100 cm^3?

Problem Number 8

Is f(x) = `sqrt( 4 + `x ) continuous at x = - 4? Is f(x) differentiable at this point?

Problem Number 9

Sketch a smoothly curving continuous graph with three points A, B and C, each to the right of the preceding, such that the following quantities occur in the given order, from least to greatest:

The slope at C

The average slope between B and C

The slope at A

The average slope between A and B

The slope at B.

Problem Number 10

Determine whether f(x) = 4 / sin( -.5001 x) is continuous; if not find the points at which is it not continuous.

Problem Number 11

One of the following has a constant value. Which is it and what is the constant value? Prove that this value is as you say:

sinh x + cosh x
sinh^2(x) + cosh^2(x)
sinh x - cosh x
sinh^2 x - cosh^2 x

Problem Number 12

On a single set of axes graph f(x) = sin( 2 x) and g(x) = sin( 4 x) from x = 0 to x = 2 `pi. On a second set of axes graph f ' (x) and g ' (x) and compare the graphs, being careful when comparing slopes of f(x) and g(x) at each point.