Calculus II

Time and Date Stamps (logged): 01:37:10 08-29-2008 ｯｰ浚ｶ涓ｯｯｷ浤ｸ浤ｯｯｷ 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

What is the interval of convergence for the series x / .8 + x^2 / 1.6 + x^3 / 3.2 + ... ?

Problem Number 2

Use the Taylor polynomial of a simpler function to find the Taylor polynomial of degree 4 for the function f(x) = e^( 6 x^ 1 ), expanding about x=0.

Problem Number 3

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

Problem Number 4

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

Problem Number 5

Find the approximate solution of the differential equation dy / dx = 1.6 x / y on the interval ( .5, 1.5), if y( .5) = .97. Use increment .25.

Problem Number 6

In an epidemic model the rate at which susceptible people get infected is .004 S I, where S stands for the number of susceptible people and I for the number of infected people. If on any day 33% 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?

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 7

The force of air resistance on a falling object with mass 8 kg has magnitude k v, where k = 2.9 N / (m/s). If the object has initial velocity 16.45 m/s downward, then what function describes its velocity as a function of clock time? Sketch a graph of this function.

Problem Number 8

A simple harmonic oscillator has restoring force constant k, mass m and experiences drag force -c v, where v is its velocity. What is the differential equation for the motion of the oscillator? What are the possible solutions to the equation?