assignment 11

course MTH 272

7/5 10 pm

If your solution to a stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm.

Your solution, attempt at solution: If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

011. `query 11

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Question: `q5.5.4 (previous problem was 5.6.2 midpt rule n=4 for `sqrt(x) + 1 on [0,2])

5.5.4 asks for an n = 4 midpoint-rule approximation to the integral of 1 - x^2 on the interval [-1, 1].

Your solution:

Midpoints: -.75, .25, .25, .75

After solving, you get the y values: 4375, .9375, .9375, .4375

You multiply heights by widths.

You do this my multiplying each interval by 1/2. .1/2 * .4375,1/2 * .9375, 1/2 * .9375 and 1/2 * .4375.=1.375

Confidence rating: 2

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Given Solution:

`a Dividing [-1, 1] into four intervals each will have length ( 1 - (-1) ) / 4 = 1/2. The four intervals are therefore

[-1, -.5], [-.5, 0], [0, 5], [.5,1].

The midpoints are -.75, .25, .25, .75. You have to evaluate 1 - x^2 at each midpoint. You get y values .4375, .9375, .9375 and .4375. These values will give you the altitudes of the rectangles used in the midpoint approximation.

The width of each rectangle is the length 1/2 of the interval, so the areas of the rectangles will be 1/2 * .4375,1/2 * .9375, 1/2 * .9375 and 1/2 * .4375, or .21875, .46875, .46875, .21875.

Adding these areas we get total area 1.375.

The curve is concave down so the midpoints will give you values which are a little high. We confirm this by calculating the integral:

The exact integral is integral(1 - x^2, x from 0 to 2). An antiderivative is x - 1/3 x^2; evaluating from -1 to 1 we find that the antiderivative changes from -2/3 to 2/3, a change of 4/3 = 1.333. So the accurate integral is 4/3 = 1.333 and our estimate 1.375 is indeed a little high. ** DER

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Self-critique (if necessary): I did pretty well on this problem, but the end part is confusing to me.

Your self-critique is way too general.

I can clarify if you tell me, phrase by phrase, what you do and do not understand about the given solution.

In any case you did fine with the approximation.

Self-critique Rating:

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Question: `q 5.6. 9 (was 5.6.12) (was 5.6.10 midpt rule n=4 for x^2-x^3 on [-1,0]

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

Intervals: (-1, -3/4), (-3/4, -1/2), (-1/2, -1/4) and (-1/4, 0)

Midpoints= 7/8, -5/8, -3/8, -1/8

Plug in and multiply all the midpoints. You get 735/512, 325/512, 99/512 and 9/512. This is after all the numbers are put with a common denominator.

Confidence rating: 2

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Given Solution:

`a The four intervals are (-1, -3/4), (-3/4, -1/2), (-1/2, -1/4) and (-1/4, 0); in decimal form these are (-1, -.75), (-.75, -.5), (-.5, -.25) and (-.25, 0).

The midpoints of these intervals are-7/8, -5/8, -3/8 and -1/8; in decimal form we get -.875, -.625, -.375, -.125.

The values of the rectangle heights at the midpoints are found by evaluating x^2 - x^3 at the midpoints; we get respectively 735/512, 325/512, 99/512 and 9/512, or in decimal form 1.435546875; 0.634765625; 0.193359375; 0.017578125.

The approximating rectangles each have width 1/4 or .25 so the areas arerespectively 735/2048 325/2048, 99/2048, 9/2048, or in decimal form 0.3588867187; 0.1586914062; 0.04833984375; 0.00439453125. The total area is (735 + 325 + 99 + 9) / 2048 = /2048 = 73/128, or in decimal form approximately .5703.

An antiderivative of the function is x^3 / 3 - x^4 / 4; evaluating from -1 to 0 we obtain 1/3 + 1/4 = 7/12 = .5833... . So the midpoint approximation is low by about .013 units. ** DER

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Self-critique (if necessary): I knew how to do most of the problem.

Self-critique Rating:

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Question: `q Add comments on any surprises or insights you experienced as a result of this assignment.

Thse problems were really long problems. I get how to do the beginning, but after the evaluating of the heights and widths, I get kind of confused.

you've got the basic idea and are executing it correctly

In order to clarify for you, I need more information about what you do and do not understand, per my previous notes.

Self-critique (if necessary)

Self-critique Rating:

&#Good work. See my notes and let me know if you have questions. &#