MAT1505-03/04 15F Final Exam Print Name (Last, First) _______________________________|____Show all work, including mental steps, in a clearly organized way that speaks for itself. Use proper mathematical notation, identifying expressions by their proper symbols (introducing them if necessary), and use EQUAL SIGNS and arrows when appropriate. Always SIMPLIFY expressions. BOX final short answers. LABEL parts of problem. Keep answers EXACT (but give decimal approximations for interpretation when appropriate). Indicate where technology is used and what type (Maple, GC).1. For the parametrized curve: 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 :
a) Write down an integral formula for the arclength of this curve, simplifying the sum of squares inside the square root integrand with Maple, then evaluate the integral exactly with Maple. [The simple exact result follows from a perfect square simplification using a half-angle identity, after using the cosine difference formula to simplify it.] How does its numerical value compare to the circumference of the comparison circle of radius 2.5 centered at 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, namely 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? Explain.b) Using a technology plot with equal tickmarks on both axes, make a hand sketch of this curve and mark off the points where the tangent line is horizontal (H), or vertical (V) by putting a bullet mark at those points and labeling them as H or V. Also mark off the point P on the curve where 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. Draw in a piece of the tangent line at P. [Label your tickmarks and axes!]c) Evaluate the slope LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRI2R5RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRi82JVEjZHhGJ0YyRjVGMkY1LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZCLyUpYmV2ZWxsZWRHUSZmYWxzZUYnL0Y2USdub3JtYWxGJw== at the point P. Does this value seem to agree with your sketch? Write an equation for this tangent line.d) Now find the exact coordinates LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPkY+ of the points labeled H and V, using Maple to solve the necessary conditions, using symmetry to add any points Maple misses, and annotate your sketch with the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ5RidGNEY3Rj5GPkY+ values. [Hint: the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= parameter values are all multiples of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbWZyYWNHRiQ2KC1JI21pR0YkNiVRJyYjOTYwO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1JI21uR0YkNiRRIjNGJ0Y1L0YzUSV0cnVlRicvRjZRJ2l0YWxpY0YnLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZHLyUpYmV2ZWxsZWRHRjRGNQ==.] What happens to the limiting slope at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=?e) Evaluate the coordinates of these points numerically and verify that they correspond to the points you marked off in part b).2. The parametrized curve 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 traces out the polar coordinate graph 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, which is an ellipse centered at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSI0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiLEYnRjQvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHUSV0cnVlRicvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JFEiMEYnRjRGNEY0RjQ=.a) Make a rough sketch of your technology plot of this curve. Evaluate the area of the inscribed (LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RIilGJ0Y5L0Y8RjFGPi9GQUYxRkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GWUY5, circumscribed LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNUYnRj5GPkY+Rj4= circles, and their average area, as well as the area of the circle with the average radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSJyRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiPUYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkY+Rj4= , and give the corresponding numerical values.b) Use the polar coordinate area formula to evaluate the area of the ellipse exactly. Is its value closer to the average circle area or the area of the circle of average radius?c) Write down a generic formula the slope dy/dx of the tangent line in polar coordinates, and use Maple to evaluate and simplify that formula for this particular polar curve. Write down the result. Use it to find the exact values of the polar coordinates of the upper vertex of the ellipse (where the tangent line is horizontal), and give the one decimal place degree equivalent of the polar angle. Evaluate the corresponding Cartesian coordinates exactly (this can be done easily by hand). Do they agree with their obvious values in the Cartesian parametrized curve representation?
(see next page)3. We showed in a homework exercise that for an ellipse 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 with semiaxes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJiRidGL0YyRjUtSSNtbkdGJDYkUSIwRidGOUY5, the area enclosed by the ellipse is 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. The eccentricity 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measures the deviation of the ellipse from a circle, corresponding to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=.The minor semiaxis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= expressed in terms of the major semiaxis LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the eccentricity LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is 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. If we look at the family of ellipses with unit area 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 so that solving for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= we find 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, the limit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= corresponds to unit area circle radius 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with circumference 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.a) These ellipses can be parametrized by
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 where 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSJiRidGL0YyRjUtSSNtbkdGJDYkUSIwRidGOUY5.Show that their circumferences can be written
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b) Use the binomial series Taylor expansion of the integrand about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk= to write out the first 3 nonzero terms of the integrand, then integrate term by term (use Maple). This shows how the circumference begins to change as we increase the eccentricity.c) Optional.
Now expand the relation 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 in a similar binomial Taylor expansion up to the first 3 nonzero terms, and multiply out the product expression for the circumference LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, neglecting terms higher than LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUkjbW5HRiQ2JFEiNEYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRj4=.Obtain the result: 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5Rjk=. What is the value of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=?
This shows that within this family of curves enclosing unit area, increasingly deforming the circle (with circumference LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUkjbW5HRiQ2JFEiMEYnL0YzUSdub3JtYWxGJ0Y+Rj4tRiw2I1EhRidGPg==) initially leads to increasing the circumference from its minimal value. The circle in fact is the minimal perimeter curve enclosing a fixed area in the plane, and this calculation confirms this fact within this family of curves.solution1.Eliminated for pressure relief:
f) Since 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 changes sign as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEidEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= increases, use the formula LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JKG1zdWJzdXBHRiQ2Jy1GNjYtUSsmSW50ZWdyYWw7RidGOS9GPFEmdW5zZXRGJy9GP0ZWL0ZBRjEvRkNGVi9GRUYxL0ZHRlYvRklGVi9GS1EmMC4wZW1GJy9GTkZobi1GIzYnLUYsNiVRImFGJ0YvRjJGLy8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRjFGMi1GIzYnLUYsNiVRImJGJ0YvRjJGL0Zfb0Zib0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy8lL3N1YnNjcmlwdHNoaWZ0R0ZbcC1JKG1mZW5jZWRHRiQ2Ji1GIzYpLUYsNiVRInlGJ0YvRjItRl9wNiQtRiM2JC1GLDYlUSJ0RidGL0YyRjlGOS1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIRmduRmluLUYsNiVRInhGJ0YvRjItRjY2LVEiJ0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4xMTExMTExZW1GJ0ZpbkZmcEY5RjkvJSVvcGVuR1EifGdyRicvJSZjbG9zZUdGanFGXXEtRjY2LVEwJkRpZmZlcmVudGlhbEQ7RidGOUZVRlcvRkFGVkZZL0ZFRlZGZW5GZm5GZ25GaW5GanBGOQ== to evaluate the area inside this curve, and compare it with the circle of radius 2.5 centered at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNictSSNtb0dGJDYtUSomdW1pbnVzMDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yMjIyMjIyZW1GJy8lJ3JzcGFjZUdGSC1JI21uR0YkNiRRIy41RidGNC1GMTYtUSIsRidGNEY3L0Y7USV0cnVlRidGPEY+RkBGQkZEL0ZHUSYwLjBlbUYnL0ZKUSwwLjMzMzMzMzNlbUYnLUZMNiRRIjBGJ0Y0RjRGNEY0.ForLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRJnBsb3RzRidGL0YyL0YzUSdub3JtYWxGJy1JI21vR0YkNi1RIjpGJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQ=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eliminating the parameter just to prove it can be done (bob's idea of fun)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We have to find the zeros of these two functions. 3 zeros of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGL0YyRjlGOUY5, 2 zeros of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiWUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIidGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4xMTExMTExZW1GJy8lJ3JzcGFjZUdRJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ0RidGL0YyRjlGOUY5 other than the common zeros at the endpoints which require further examination of their limiting ratio.n 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=2.test 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We take a 3-4-5 ellipse with integer values of the three parameters LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJiRidGNEY3RjotRjE2JVEiY0YnRjRGN0Y+Rj4tRjs2LVEiPUYnRj5GQC9GREZCRkVGR0ZJRktGTS9GUFEsMC4yNzc3Nzc4ZW1GJy9GU0Zqbi1GLDYkLUYjNigtSSNtbkdGJDYkUSI1RidGPkY6LUZhbzYkUSIzRidGPkY6LUZhbzYkUSI0RidGPkY+Rj5GPg==.This leads to the polar representation 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiaEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the displacement between the center of the ellipse and one focus.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The polar representation is 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using the parametrized curve approach as a check: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how do these compare in 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O0Y+RkBGQkZERkZGSEZhb0Zjby1GaG42JFEjNDBGJ0Y5RmZxLUYsNiVRJGNvc0YnRmlwRjktRlQ2JC1GIzYlRlhGZHBGOUY5RmRwRjlGOS1GIzYlLUZfcTYoLUZobjYkUSIxRidGOS1GIzYlRmduRi9GMi8lLmxpbmV0aGlja25lc3NHRmVzLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlx0LyUpYmV2ZWxsZWRHRj1GL0YyRltvRmRwRjktRiM2JS1GUTYlLUZUNiQtRiM2Ky1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkhGYW9GY28tRmhuNiRRIjVGJ0Y5Rl5vLUZobjYkUSI0RidGOS1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIRmhvRk1GaHItRlQ2JC1GIzYoRlgvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRidGZHAvJSlyZWFkb25seUdGMS8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYnRjlGOUYvRjJGOUZmc0Zbb0YvRjJGaHNGanNGXXRGX3RGZHBGOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEnYXNzdW1lRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNictRiw2JVEidEYnRi9GMi1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVElcmVhbEYnRi9GMi8lK2V4ZWN1dGFibGVHRkVGQUZBLUY+Ni1RIjtGJ0ZBRkNGRkZIRkpGTEZORlBGUi9GVlEsMC4yNzc3Nzc4ZW1GJy1JKG1zdWJzdXBHRiQ2Jy1GPjYtUSsmSW50ZWdyYWw7RidGQUZDL0ZHRkUvRklGMUZKL0ZNRjFGTkZQRlIvRlZGVC1GIzYkLUkjbW5HRiQ2JFEiMEYnRkFGQS1GIzYmLUZpbzYkUSIyRidGQS1GPjYtUSJ+RidGQUZDRmJvRkhGSkZMRk5GUEZSRmVvLUYsNiVRJyYjOTYwO0YnL0YwRkVGQUZBLyUxc3VwZXJzY3JpcHRzaGlmdEdGW3AvJS9zdWJzY3JpcHRzaGlmdEdGW3AtRiw2JVElZHNkdEYnRi9GMi1GLDYjUSFGJ0ZfcS1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR1EmMC41ZW1GJy8lJmRlcHRoR0ZncS8lKmxpbmVicmVha0dRJWF1dG9GJy1GPjYtUTAmRGlmZmVyZW50aWFsRDtGJ0ZBL0ZEUSZ1bnNldEYnL0ZHRmRyL0ZJRmRyL0ZLRmRyL0ZNRmRyL0ZPRmRyL0ZRRmRyRlJGZW9GOkZlbkZBLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEmZHlkeENGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJm1mcmFjR0YkNigtRiM2Jy1GLDYlUSJZRidGL0YyLUY2Ni1RIidGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5RJjAuMGVtRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ0RidGL0YyLyUrZXhlY3V0YWJsZUdGPUY5RjlGYG9GOS1GIzYnLUYsNiVRIlhGJ0YvRjJGVy1GaW42JC1GIzYlRl1vRi9GMkY5Ri9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYHAvJSliZXZlbGxlZEdGPUZgb0Y5LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkmbWZyYWNHRiQ2KC1GIzYnLUYsNigtSSNtb0dGJDYtUTAmRGlmZmVyZW50aWFsRDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJnVuc2V0RicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GIzYmRjItRjM2LVEifkYnRjYvRjpRJmZhbHNlRicvRj1GUy9GP0ZTL0ZBRlMvRkNGUy9GRUZTL0ZHRlNGSEZLLUkjbWlHRiQ2JVEnJiM5NTI7RicvJSdpdGFsaWNHRlNGNkY2LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zfby8lKWJldmVsbGVkR0ZTRk8tSShtZmVuY2VkR0YkNiQtRiM2KS1GZW42JVEiUkYnL0ZpblEldHJ1ZUYnL0Y3USdpdGFsaWNGJy1GZW82JC1GIzYlRlovJStleGVjdXRhYmxlR0ZTRjZGNkZPLUZlbjYlUSRzaW5GJ0ZobkY2RmBwRmRwRjZGNkZkcEY2LUYjNictRmVuNiNRIUYnLUYjNiYtRiw2KEYyLUYjNiZGMkZPLUZlbjYlUSgmdGhldGE7RidGaG5GNkY2RmpuRl1vRmBvRmJvRk8tRmVvNiQtRiM2KEZpby1GZW82JC1GIzYkRmRxRjZGNkZPLUZlbjYlUSRjb3NGJ0ZobkY2RltyRjZGNkY2RltxRlxwRl5wRmpuRl1vRmBvRmJvLUYzNi1RIjtGJ0Y2RlIvRj1GXXBGVUZWRldGWEZZRkgvRkxRLDAuMjc3Nzc3OGVtRictRmVuNiVRJmR5ZHhQRidGXHBGXnAtRjM2LVEqJmNvbG9uZXE7RidGNkZSRlRGVUZWRldGWEZZL0ZJRmdyRmZyLUZlbjYlUSlzaW1wbGlmeUYnRlxwRl5wLUZlbzYkLUYjNiUtRmVuNiVRIiVGJ0ZccEZecEZkcEY2RjZGZHBGNg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JVEnJiM5NDU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnRjIvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZJLUYsNiVRJnNvbHZlRicvRjBRJXRydWVGJy9GM1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2KC1GLDYlUSZudW1lckYnRk9GUS1GVDYkLUYjNiUtRiw2JVEmZHlkeFBGJ0ZPRlEvJStleGVjdXRhYmxlR0YxRjJGMi1GNjYtUSI9RidGMkY5RjtGPUY/RkFGQ0ZFRkdGSi1JI21uR0YkNiRRIjBGJ0YyRlxvRjJGMi1GNjYtUSI7RidGMkY5L0Y8RlBGPUY/RkFGQ0ZFL0ZIUSYwLjBlbUYnRkotRjY2LVEifkYnRjJGOUY7Rj1GP0ZBRkNGRUZpby9GS0Zqby1GLDYlUSZldmFsZkYnRk9GUS1GVDYkLUYjNiUtSSZtZnJhY0dGJDYoLUYjNiktRiw2JVEiJUYnRk9GUS1GNjYtUScmc2RvdDtGJ0YyRjlGO0Y9Rj9GQUZDRkVGaW9GXnAtRmJvNiRRJDE4MEYnRjJGXnEtRiw2JVEoZGVncmVlc0YnRk9GUUZcb0YyLUYjNiUtRiw2JVEnJiM5NjA7RidGL0YyRk9GUS8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYXIvJSliZXZlbGxlZEdGMUZcb0YyRjJGXG9GMg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkobWZlbmNlZEdGJDYmLUYjNi8tSSNtaUdGJDYlUSJSRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiQtRiM2JC1GMTYlUScmIzk0NTtGJy9GNVEmZmFsc2VGJy9GOFEnbm9ybWFsRidGQ0ZDLUkjbW9HRiQ2LVEifkYnRkMvJSZmZW5jZUdGQi8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZZLUYxNiVRJGNvc0YnRkFGQ0Y6LUZGNi1RIixGJ0ZDRkkvRkxGNkZNRk9GUUZTRlVGVy9GZW5RLDAuMzMzMzMzM2VtRidGMC1GLDYkLUYjNiQtRjE2JVEoJmFscGhhO0YnRkFGQ0ZDRkNGRS1GMTYlUSRzaW5GJ0ZBRkNGX28vJStleGVjdXRhYmxlR0ZCRkNGQy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zpb0ZD3.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUY2Ni1RIidGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRicvRk5RJjAuMGVtRidGK0ZPLUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktGVUZNLUYsNiVRImRGJ0YvRjJGNUZPRmVuRk8tRjY2LVEiOkYnRjlGO0Y+RkBGQkZERkZGSEZKRk0tRiw2JVEjUlJGJ0YvRjJGNS1GLDYlUScmIzk1MjtGJy9GMEY9RjktRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSEZaRlQtSSZtZnJhY0dGJDYoLUYjNiZGKy1GNjYuUSJ+RidGL0YyRjtGPkZARkJGREZGRkhGWkZURmVuRjktRiM2KS1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZ3BGKy1GNjYtRlxwRjlGO0Y+RkBGQkZERkZGSEZaRlQtRiw2JVEkY29zRidGYW9GOS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRKCZ0aGV0YTtGJ0Zhb0Y5RjlGOUY5LyUubGluZXRoaWNrbmVzc0dGYnAvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGanEvJSliZXZlbGxlZEdGPS8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUkjbWlHRiQ2JVEjUlJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSInRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMTExMTExMWVtRicvJSdyc3BhY2VHUSYwLjBlbUYnLUklbXN1cEdGJDYlLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEidEYnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5LUYjNiUtSSNtbkdGJDYkUSIyRidGOUYvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZG9GK0ZQLUY2Ni1RIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkgvRktGTy9GTlEsMC4yNzc3Nzc4ZW1GJy1GLDYlRjpGL0YyLUZUNiQtRiM2JS1GLDYlUSIlRidGL0YyRmVuRjlGOUZmby1GLDYlUSlkc2R0aGV0YUYnRi9GMi1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktGXHBGW3AtSSZtc3FydEdGJDYjLUYjNiUtRiw2I1EhRictSSZtZnJhY0dGJDYoLUYjNiYtRiw2JVEpc2ltcGxpZnlGJ0YvRjItRlQ2JC1GIzYmLUYsNiVRJm51bWVyRidGL0YyRl9wRmVuRjlGOUZlbkY5LUYjNiYtRiw2JVEmZGVub21GJ0YvRjItRlQ2JC1GIzYlRmNwRi9GMkY5Ri9GMi8lLmxpbmV0aGlja25lc3NHUSIxRicvJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGYnMvJSliZXZlbGxlZEdGPUY5RmVuRjk=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEnc2VyaWVzRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiwtRiw2JVEpZHNkdGhldGFGJ0YvRjItSSNtb0dGJDYtUSIsRicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVRImVGJ0YvRjItRj42LVEiPUYnRkFGQy9GR0ZFRkhGSkZMRk5GUC9GU1EsMC4yNzc3Nzc4ZW1GJy9GVkZqbi1JI21uR0YkNiRRIjBGJ0ZBRj0tRj42LVEifkYnRkFGQ0ZobkZIRkpGTEZORlBGUi9GVkZULUZdbzYkUSIzRidGQS8lK2V4ZWN1dGFibGVHRkVGQUZBRmdvRkE=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkmbWZyYWNHRiQ2KC1GIzYmLUkjbWlHRiQ2JVEiZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0Y0RjcvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTy1GMTYlUSJkRidGNEY3L0Y4USdub3JtYWxGJy1GIzYpLUkjbW5HRiQ2JFEiMUYnRlUtRjs2LVEoJm1pbnVzO0YnRlVGPkZBRkNGRUZHRklGSy9GTlEsMC4yMjIyMjIyZW1GJy9GUUZbb0YwLUY7Ni1GPUZVRj5GQUZDRkVGR0ZJRktGTUZQLUYxNiVRJGNvc0YnL0Y1RkBGVS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYxNiVRKCZ0aGV0YTtGJ0Zib0ZVRlVGVUZVLyUubGluZXRoaWNrbmVzc0dGZm4vJStkZW5vbWFsaWduR1EnY2VudGVyRicvJSludW1hbGlnbkdGX3AvJSliZXZlbGxlZEdGQC1GMTYjUSFGJy8lK2V4ZWN1dGFibGVHRkBGVQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY5LUkjbWlHRiQ2JVEiY0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUkmbXNxcnRHRiQ2Iy1GIzYmLUklbXN1cEdGJDYlLUYsNiVRImFGJ0YvRjItRiM2JC1JI21uR0YkNiRRIjJGJ0Y5RjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GYW8tRlU2JS1GLDYlUSJiRidGL0YyRlpGam5GOS1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnRk0tRiw2JVEiZUYnRi9GMkY1LUkmbWZyYWNHRiQ2KEYrLUYjNiRGV0Y5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZbcS8lKWJldmVsbGVkR0Y9RmhvLUYsNiVRImhGJ0YvRjJGNS1GNjYtUSomdW1pbnVzMDtGJ0Y5RjtGPkZARkJGREZGRkhGYG9GYm9GK0Zoby1GLDYlUSJkRidGL0YyRjUtRmJwNigtSShtZmVuY2VkR0YkNiQtRiM2Ji1GZ242JEZocEY5Rl1vLUZVNiVGXnBGWkZqbkY5RjktRiM2JEZickY5RmZwRmlwRlxxRl5xLUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkhGXHAvRk5GXXBGKy1JJ21zcGFjZUdGJDYmLyUnaGVpZ2h0R1EmMC4wZXhGJy8lJndpZHRoR0ZdcC8lJmRlcHRoR0Zfcy8lKmxpbmVicmVha0dRKG5ld2xpbmVGJy1GW3M2JkZdc0Zgc0Zicy9GZXNRJWF1dG9GJy1GLDYjUSFGJy8lK2V4ZWN1dGFibGVHRj1GOQ==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJw==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 you have completed the exam, please read and sign the dr bob integrity pledge and hand this test sheet in on top of your answer sheets as a cover page, with the first test page facing up:"During this examination, all work has been my own. I give my word that I have not resorted to any ethically questionable means of improving my grade or anyone else's on this examination and that I have not discussed this exam with anyone other than my instructor, nor will I until after the exam period is terminated for all participants."
Signature: Date: