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 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
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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:

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Problem Number 1

Describe the overall strategy for proving that a function continuous on an interval is integrable on that interval. Include a precise statement of the theorem.

Problem Number 2

A conical tower has base radius 8 meters and altitude 7.3 meters. It is filled with water to a depth of 3.6 meters. Find the volume of the water.

Problem Number 3

For a normal distribution, with p(x) = 1 / [ `stdDev `sqrt(2 `pi) ] * e^[ -(x - `mean)^2 / ( 2 * `stdDev^2) ], `mean and `stdDev can be regarded simply as numbers. `mean is often denoted by the Greek letter `mu (small mu) and `stdDev by the Greek letter `sigma (small sigma), and you may use this notation if it is familiar to you.

Use the substitution u = (x - `mean) / `stdDev to show that the number of occurrences between `mean - `stdDev and `mean + `stdDev is always the same, regardless of the values for `mean and `stdDev.

Problem Number 4

Find the indefinite integral of the function x^ 2 ln(x).

Problem Number 5

Sketch a graph of a continuous function f(x) which is linear from (0, 2) to ( 2, -3.001), then linear to ( 10, 7) and then again linear to ( 14, 2). Sketch a graph of its antiderivative F(x) for which F(10)= 18 and label the known points on this graph.

Problem Number 6

Use Riemann Sums to obtain the integral required to solve the following, and evaluate the integral: How much work is required to lift a weight of 300 pounds a vertical distance of 23 feet using a chain that weighs 3 lb/ft?

Problem Number 7

If an object's temperature moves 35% closer to the 24 degree temperature of a room in any 6-minute time period, then if the object's initial temperature is 42.8 degrees, what function models the temperature as a function of time? What will be the temperature after 34 minutes?

Problem Number 8

What is the interval of convergence for the series x * .6^ 3.5 + x^2 * 1.8^ 3.5 + x^3 * 3.6^ 3.5 + ... ?

Problem Number 9

In an epidemic model the rate at which susceptible people get infected is .0025 S I, where S stands for the number of susceptible people and I for the number of infected people. If on any day 26% of the infected people either die or recover, and if we assume that dead and recovered people can no longer infect or be infected, what differential equation models the rate at which people become infected? What equation models the rate at which people are removed from the susceptible to the infected category?

For a population of 1000 people sketch the slope field on a graph of I vs. S.
Determine the value of S at which the number I peaks.
Sketch the trajectories corresponding initial S populations of 1000, 900, 800, 700, ..., 100.

Problem Number 10

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

Problem Number 11

Prove whether the integral of 1/( x^ .7 + x^(1/ .7) ), from x = 0 to x = 1, converges or diverges.

Problem Number 12

A ball dropped from a height of h feet, or a ball bouncing to a maximum height of h feet, requires 1/4 `sqrt(h) seconds to cover the distance. If a ball is dropped to the floor from a height of 42 feet, how long does it take it to come to rest?

Problem Number 13

Antidifferentiate sin^ 6( 4 x) with or without the use of tables.

Problem Number 14

Find formulas for the velocity and position of an object dropped on a small planet where the acceleration is 7 m/s^2.

Problem Number 15

If we increase the number of intervals in a trapezoidal approximation by 45, by what factor do we expect our error to change? By what factor do we expect error to change if we are using Simpson's Rule?

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.

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mult_051

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