Asst 7

course Mth 174

CW鿆yɆassignment #007

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WJИZ{`zd

Physics II

07-06-2007

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22:13:46

query problem 7.6.6 approx using n=10 is 2.346; exact is 4.0. What is n = 30 approximation if original approx used LEFT, TRAP, SIMP?

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The problem is asking: pprox using n=10 is 2.346; exact is 4.0. What is n = 30 approximation if original approx used LEFT, TRAP, SIMP?

I'm not sure how to do this. I understand the other problems we did. But i'm not sure how you find the errors

This is problem 6 in section 7.6 of your text.

LEFT and RIGHT approach the exact value in proportion to the number of steps used.

MID and TRAP approach the exact value in proportion to the square of the number of steps used.

SIMP approachs the exact value in proportion to the fourth power of the number of steps used.

Using these principles we can work out this problem as follows:

** The original 10-step estimate is 2.346, which differs from the actual value 4.000 by -1.654.

If the original estimate was done by LEFT then the error is inversely proportional to the number of steps and the n = 30 error is (10/30) * -1.654 = -.551, approximately. So the estimate for n = 30 would be -.551 + 4.000 = 3.449.

If the original estimate was done by TRAP then the error is inversely proportional to the square of the number of steps and the n = 30 error is (10/30)^2 * -1.654 = -.184, approximately. So the estimate for n = 30 would be -.184 + 4.000 = 3.816.

If the original estimate was done by SIMP then the error is inversely proportional to the fourth power of the number of steps and the n = 30 error is (10/30)^4 * -1.654 = -.020, approximately. So the estimate for n = 30 would be -.02 + 4.000 = 3.98. **

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22:14:01

If the approximation used LEFT then what is your estimate of the n = 30 approximation and how did you get it?

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22:14:04

If the approximation used TRAP then what is your estimate of the n = 30 approximation and how did you get it?

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22:14:16

If the approximation used SIMP then what is your estimate of the n = 30 approximation and how did you get it?

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22:14:23

This problem has been omitted from the present edition and may be skipped: query problem 7.6.10 If TRAP(10) = 12.676 and TRAP(30) = 10.420, estimate the actual value of the integral.

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22:14:25

What is your estimate of the actual value and how did you get it?

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22:14:27

By what factor should the error in the second approximation be less than that in the first, and how does this allow you to estimate the integral based on the difference in the two approximations?

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22:14:29

a < b, m = (a+b)/2. If f quadratic then int(f(x),x,a,b) = h/3 ( f(a) / 2 + 2 f(m) + f(b) / 2).

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22:16:42

query problem 7.7.19 integrate 1 / (u^2-16) from 0 to 4 if convergent

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integrate 1 / (u^2-16) from 0 to 4

-1/4 arctan(-x/4) from 0 to 4

does not converge

If the antiderivative was this arcTan, then it would converge. arctan(0) is 0 and arctan(-1) is -pi/4, and the arctan is continuous from 0 to -1, so the integral would be easily evaluated by its values at the limits.

However this is not a correct antiderivative, as you could verify by taking the derivative of your expression. That derivative would yield 1/16 ( 1 / (x/4)^2 + 1) ).

Your error:

(-4)^2 is not -16. If the integral was 1 / (u^2 + 16) then it would be of the form 1 / (u^2 + a^2), with a = 4 or -4. However that is not the case here.

1 / (u^2-16) = 1 / [(u+4)(u-4)] . Since for 0 < x < 4 we have 1/8 < 1 / (u+4) < 1/4, the integrand is at most 1/4 times 1/(u-4) and at least 1/8 of this quantity, so the original integral is at most 1/4 as great as the integral of 1 / (u-4) and at least 1/8 as great. That is,

1/8 int(1 / (u-4), u, 0, 4) < int(1 / (u^2-4), u, 0, 4) < 1/4 int(1 / (u-4), u, 0, 4).

Thus if the integral of 1 / (u-4) converges or diverges, the original integral does the same. An antiderivative of 1 / (u-4) is ln | u-4 |, which is just ln(4) at the limit u=0 of the integral but which is undefined at the limit u = 4.

We must therefore take the limit of the integral of 1/(u-4) from u=0 to u=x, as x -> 4.

The integral of 1 / (u-4) from 0 to x is equal to ln (4) - ln(x-4) = ln( 4 / (x-4) ).

As x approaches 4 the denominator approaches 0 so the fraction approaches infinity and the natural log approaches infinity. Thus the integral diverges. **

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22:17:33

does your integral converge, and why or why not?

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no the integral does not converge.

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22:17:37

If convergent what is your result?

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22:18:02

Why is there a question as to whether the integral does in fact converge?

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in case you found that it did?

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22:18:05

Give the steps in your solution.

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22:19:24

If you didn't give it, give the expression whose limit showed whether the integral was convergent or divergent.

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lim x->4 -1/4arctan(-4/4) = .1963

lim x->0 -1/4arctan(-0/4) = 0

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22:22:39

describe your graph, including asymptotes, concavity, increasing and decreasing behavior, zeros and intercepts

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starting at (0,0) the graph rapidly increases on the interval (0, 2), the graph peaks at (2, 735.7588), then decreses slower than it increased on the interval (2,5), the entire graph is concave down.

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22:23:40

when our people getting sick fastest and how did you obtain this result?

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people are getting sick fastest on the interval (0,2) - there is a very steep slope here, indicating a quick increase. I obtained this result by looking at the graph and the steepness of the slope.

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22:25:24

How many people get sick and how did you obtain this result?

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2850.81 people get sick. I obtained this result by integrating 1000t e^-0.5t from 0 to 5

1/-0.5 1000t e^-0.5t + 1/0.5 int 1000 e^-0.5t

1/-0.5 1000t e^-0.5t + 1/0.25 1000 e^-0.5t from 0 to 5 which gives the result above.

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22:26:11

What improper integral arose in your solution and, if you have not already explained it, explain in detail how you evaluated the integral.

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e^-0.5t is the improper integral. I already showed how to evaluate.

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See my notes and let me know if you have additional questions on this assignment.

&#You are always welcome to ask self-contained questions about anything. By self-contained questions I mean a question that includes a brief statement of the problem or topic you are asking about (in order to give everyone the best responses I can, I can't take time to look problems up in the text, which I don't carry with me in any case), and a statement of precisely what you do and do not understand about the situation. If it's a problem, you should include a list of things you have tried in attempting to solve (or to understand) the problem. Depending on the problem this might include a description of any diagrams, listings of concepts and topics you think might be helpful, and other relevant information.

This can be relatively brief, but the more you can tell me, the more you will learn in the process, and the more specifically I can address my response. &#