Quiz_1_Solutions.pdf - Math 141 Quiz One Solutions 1[5...

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Math 141 Quiz One Solutions 1. [5] Evaluate: sec ! 3 ࠵? tan ( 3 ࠵? ) ࠵?࠵? . Let ࠵? = tan ( 3 ࠵? ) and thus sec ! 3 ࠵? tan ( 3 ࠵? ) ࠵?࠵? = !"# ! ( ! ! ) ! + ࠵? 2. [5] Evaluate: ࠵? ࠵? ! ! sin ( ࠵? ! ! ) ࠵?࠵? . Let ࠵? = sin ( ࠵? ! ! ) and thus ࠵? ࠵? ! ! sin ( ࠵? ! ! ) ࠵?࠵? = ! !"# ! ! ! ! + ࠵? 3. [5] Approximate to four decimal places the definite integral: sin ( ࠵? ! ) ! ! ࠵?࠵? You may use whatever method you like. As many of you have noted, this is the “Fresnel Integral” (pronounced “Fra- NEL”) from optics, which doesn’t have a nice antiderivative (it can be evaluated using complex analysis.) The text likes Simpson’s Rule, but your method may vary. The value is approximately 0.8948. 4. [5] Evaluate: ࠵? !""" ! ! ! This was mainly a question to check notation. There are formula you can derive or use to evaluate the sum, but the easiest way is to use “Gauss’ method” (story has it he figured this out in a few minutes when five years old) and note that the first and last term sum to 5001. We can continue to pair terms like this and get that there are 2500 such pairs adding to 5001. Thus the sum is 2500*5001 or 12,502,500.

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