Calculus II

Time and Date Stamps (logged): 17:12:20 06-10-2020 ｰｶ涓ｱ浤ｯｯｵ涓ｯ浤ｯｱｯ 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 16:44:21

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:

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

Problem Number 1

A right triangle has its right angle at the origin. Its leg along the x axis has length 46 cm and its leg along the y axis has length 5 cm. At distance x from the y axis the density of the triangle is 1 / ( 1.8 + 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 a certain definite integral MID(10) = 93.64 and MID(30)= 91.94. What is your best estimate of the exact value of the integral?

Problem Number 3

Antidifferentiate ( 9 x + 10) / (x^2 + 4 x - 32) with or without the use of tables.

Problem Number 4

Find the volume of the solid obtained by rotating the curve y = e^(- 1.2 x), between x = 0 and x = .77, about the line x = -1.301.

Problem Number 5

Show that LEFT(n) = RIGHT(n) + (f(a) - f(b) ) `dx, and also that TRAP(n) = LEFT(n) + RIGHT(n).

Problem Number 6

Sketch a graph representing the probability distribution of the velocities of the balls in a billiard ball model, assuming that the mean velocity is 9. Using your sketch construct a sketch for the cumulative distribution function. Explain how to use your cumulative distribution graph to estimate the most likely velocity.

Problem Number 7

Use Riemann Sums to obtain the integral required to solve the following, and evaluate the integral: A tank is in the shape of an upright right circular cylinder with base diameter 9 meters and altitude 29 meters. The tank originally contains a fluid with weight density 8000 Newtons / m^3, to a depth of 14.6 meters. How much work will be required to pump all the fluid to the height of the top of the container?

Problem Number 8

Prove whether the integral of e^-( .3 x) cos^ 6 (x), from x = 1 to infinity, converges or diverges.

Problem Number 9

State the Racetrack Principle, and use the Increasing Function Theorem to prove it.

Problem Number 10

Antidifferentiate sin^ 10( 4 x) cos^ 8( 4 x) with or without the use of tables.