Calculus II

Time and Date Stamps (logged): 12:34:21 05-21-2012 ｰｱ浚ｳ浤ｰｯｴ浤ｰ浤ｯｰｱ Calculus II

Calculus II Test 3

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

10-03-2001 19:58:37

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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

Every minute 3.2 grams of salt, along with 2.3 gallons of water, flow into a reservoir containing 38 gallons of water. The salt is immediately and thoroughly mixed with the water, and 2.3 gallons of the solution flow out of the reservoir. If the reservoir initially contains 258.4 grams of salt, then what function gives the concentration of salt as a function of time?

Problem Number 2

A simple harmonic oscillator has restoring force constant .44 N/m, mass 7 kg and experiences drag force - 13.55 N s / meter * v, where v is its velocity. What is the differential equation for the motion of the oscillator? What is the solution to the equation?

Problem Number 3

In a predator-prey model we assume that in the absence of predators the prey will grow exponentially with an annual growth rate of .34, while in the absence of this prey the predators will decline with an annual growth rate of -.5301. If the interaction rate between predator and prey is the product of the number of predators and the number of prey, then we assume that the rate at which prey are consumed is .0018 times the interaction rate. We assume also that the annual growth rate of the predators increases by .000009 times the number of prey. Write the differential equations for the rates of change of the populations of predators and prey.

If there are initially 318.2 predators and 87740 prey,

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 4

Solve the equation dy/dt = .033 y / (t + 5), y( 2) = 57.

Problem Number 5

Use the Taylor polynomial of a simpler function to find the Taylor polynomial of degree 4 for the function f(x) = sin( 4 x^-1.002 ), expanding about x=0.

Problem Number 6

Use a comparison with an improper integral to determine whether the series 1/2 + 1 / (3*2) + 1 / (4*3) + 1 / (5*4) + ... converges.

Problem Number 7

What is the interval of convergence for the series x / 4 + x^2 / 12^2 + x^3 / 36^3 + ... ?

Problem Number 8

Sketch the direction field for the differential equation dy / dx = .38 x / y. Sketch some solution curves corresponding to this field.