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 Final Exam

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

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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

State the Increasing Function Theorem. Using the Mean Value Theorem prove the Increasing Function Theorem.

Problem Number 2

Find the indefinite integral of the function x^ 4 ln(x).

Problem Number 3

Use a graph and symmetry arguments to explain why the integral of cos( 2 x) cos( 4 x), from x = -`pi to x = `pi, is zero.

Problem Number 4

If the equations

1/x dx / dt = 0 x + .5 y + -.8001 and
1/y dx / dt = .5 x + 0 y + -.8001,

model interaction between two species, then

Find the nullclines for each equation and sketch the appropriate slopes on a graph of the nullclines.
Analyze the regions defined by the nullclines to determine the behavior of the solutions.
Sketch several possible solution curves.

Problem Number 5

Assuming that k is constant, solve the differential equation dQ / dt = k Q.

Problem Number 6

Determine whether the antiderivatives of e^( 9 x) / (1 + e^(2 * 9 x)) and sin( 9 x) / (1 + cos(2 * 9 x) ) are in fact different expressions of the same problem. If so specify the problem; if not state why not.

Problem Number 7

Sketch a graph of a continuous function f(x) which is linear from (0, 3) to ( 5, -2.001), then linear to ( 9, 6) and then again concave downward to ( 12, 5). Sketch a graph of its antiderivative F(x) for which F(0)= 3 and label the known points on this graph.

Problem Number 8

A spherical container of radius 14 cm is filled with water to within 4.3 cm of its top. Find the volume of the water.

Problem Number 9

Sketch a graph representing the probability distribution of the velocities of the balls in a billiard ball model, assuming that the mean velocity is 4 . Using your sketch estimate the most likely velocity to two significant figures. Estimate also the probability that a random observation of a given ball will yield a velocity which rounds off to each of the following values: 2, 8 and 14.

Problem Number 10

The rate at which water flows from a uniform cylinder is r(t) cm^3 / sec, where t is clock time in seconds.

Write an expression for the amount of water flowing from the cylinder between clock times t and t + `dt.
Write a Riemann sum with n intervals expressing the amount of water flowing from the cylinder between t = 4 and t = 22.
If r(t) = 82 - 1.4 t, then how much water has flowed out between t = 4 and t = 22?
If there were initially 3600 cm^3 of water in the tank, then what is an expression for Q(t), the amount in the tank at clock time t?

Problem Number 11

The table below represents the cross-sectional area perpendicular to an axis of a solid vs. position on the axis:

position (meters) 1.5 2 2.5 3

c.s. area (m^2) 1 3.2 1.8 2.2

Sketch a reasonable curve through the data points given in the table. Estimate the volume of the solid using LEFT, RIGHT, TRAP and MID, using three intervals for each approximation. Indicate the expected order of accuracy of your approximations, from least accurate to most.

Problem Number 12

Prove whether the integral of f(t) = 1 / t^p, from t = 0 to t = 1, converges or diverges if p < 1.

Problem Number 13

Antidifferentiate ( 3 x + 6) / (x^2 + 6 x - 27) with or without the use of tables.

Problem Number 14

Find the Taylor polynomial of degree 4 for y = 7.94 cos( 3 x), expanding about x=0.

Problem Number 15

Find the area between the x axis and the graph of the polynomial p(x) = ( x - 5) (x + 17)^2 (x - 1).

Your average score on the following assessment problems will replace your score on each the two of the preceding problems on which you score lowest.

Your performance on these problems will also count toward your grade.

mult_029

mult_099

free_036

free_066

free_120

free_161