Calculus II

Time and Date Stamps (logged): 01:32:10 08-29-2008 ｯｰ浚ｱ涓ｯｯｷ浤ｸ浤ｯｯｷ Calculus II

Calculus II Test 2

Completely document your work and your reasoning.

You will be graded on your documentation, your reasoning, and the correctness of your conclusions.

Except where the need for more precision dictates otherwise (e.g., in nuclear physics) all quantities may be rounded to three significant figures. The generating program works in binary and often generates extraneous digits (e.g., 1.5001 for 1.5, 3.6999 for 3.7).

** Write clearly in dark pencil or ink, on one side of the paper only. **

function	general antiderivative
sin(ax)cos(bx)	1/(b^2-a^2) [ a cos(ax)sin(ax) - b sin(ax)cos(bx) ] + c
cos(ax)cos(bx)	1/(b^2-a^2) [ b cos(ax)sin(bx) - a sin(ax)cos(bx) ] + c
sin(ax)cos(bx)	1/(b^2-a^2) [ b sin(ax)sin(bx) + a cos(ax)cos(bx)] + c
p(x) e^(ax)	1/a p(x)e^(ax) - 1/a INTEGRAL(p'(x)e^(ax),x) + c
p(x) sin(ax)	1/a p(x)cos(ax) + 1/a INTEGRAL(p'(x)cos(ax),x) + c
p(x) cos(ax)	1/a p(x)sin(ax) - 1/a INTEGRAL(p'(x)sin(ax),x) + c
1/(sin(x))^m	-1/(m-1) cos(x) / (sin(x))^(m-1) + (m-2)/(m-1) INTEGRAL(1/(sin(x))^(m-2), x) + c
1/sin(x)	1/2 ln \| (cos(x)-1) / (cos(x) + 1) \| + c
1/(cos(x))^m	1/(m-1) sin(x) / (cos(x))^(m-1) + (m-2)/(m-1) INTEGRAL(1/(cos(x))^(m-2), x) + c
1/cos(x)	1/2 ln \| (sin(x)-1) / (sin(x) + 1) \| + c
(bx+c)/(x^2+x^2)	b/s ln \| x^2+x^2 \| + c/a arctan(x/a) + c
(cx + d) / [ (x-a)(x-b) ]	1/(a-b) [ (ac + d) ln \| x-a \| - (bc+d) ln \| x-b \| ] + c
1 / `sqrt( x^2 +- a^2 )	ln \| x + `sqr(x^2 +- a^2 \| + c
`sqrt(a^2 +- x^2 )	1/2 ( x `sqrt(a^2 +- x^2) + a^2 INTEGRAL(1/`sqrt(a^2 +- x^2 ) + c
`sqrt(x^2 - a^2)	1/2 ( x `sqrt(a^2 +- x^2) + a^2 INTEGRAL(1/`sqrt(a^2 +- x^2 ) + c

10-05-2001 16:44:22

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Write on ONE SIDE of paper only
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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:

. . . . . . . . . . . . . . . .

Problem Number 1

A right triangle has its right angle at the origin. Its leg along the x axis has length 50 cm and its leg along the y axis has length 3 cm. At distance x from the y axis the density of the triangle is 1 / ( 1.3 + x) grams / cm^2. Use a Riemann sum to represent the approximate mass of the triangle. What is the precise mass?

Problem Number 2

For the function f(x) = sin(x), evaluated between x = 0 and x = `pi/2, place the following in order:

LEFT(100)
RIGHT(100)
TRAP(100)
MID(100)
the actual value of the integral

Problem Number 3

Find the present value after 8 years of income stream function $ 50,000 * e^( .051 t) per year, where t is in years from the present, provided we expect money to grow at a constant annual rate of 3%, compounded continuously.

Problem Number 4

If p(x) = k x ( 1 - x/ 3), 0 <= x <= 3, represents a probability distribution, then

What is the value of k?
What is the mean value of x?

Problem Number 5

Prove whether the integral of e^-( .9 x) cos^ 10 (x), from x = 1 to infinity, converges or diverges.

Problem Number 6

Antidifferentiate e^t cos( 6 t + 49) with or without the use of tables.

Problem Number 7

Find the integral of the function f(x) = sin(x + `pi), evaluated between x = 0 and x = `pi/2. Then calculate the error obtained by TRAP(3), MID(3) and SIMP(3).

Find the ratios of the errors in MID(3) and SIMP(3) to the error in TRAP(3).
Each estimate will either overestimate or underestimate the integral. For each specify which, and tell why this happens.

Problem Number 8

Antidifferentiate ( 9 x + 4) / (x^2 + 5 x - 24) with or without the use of tables.

Problem Number 9

Find the volume of the solid obtained by rotating the region bounded by the curve y = x^- 1 between x = 0 and x = .36 and the y axis about the line y = -.6001.

Problem Number 10

State the Completeness Axiom, the Nested Interval Theorem and the Intermediate Value Theorem and explain how they are used in proving the central theoretical results of Calculus.