# P30 - instructed Then evaluate the approximation KEEPING...

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Project 30 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! Examine the parabolic arch formed by the function f ( x ) = 3 x - 1 2 x 2 on [ 0 , 6 ] . We would like to approximate its area using rectangles. Sketch the parabola below on [ 0 , 6 ] and shade the area beneath it. Do a quick optimization to ﬁnd the maximum height of the parabola (instead of using the vertex formula.) At some point you may have seen a formula for such an area as 2 3 BH where H is the height of the arch and B is the width of the base. What is the area of the arch that you shaded above? The Fundamental Theorem of Calculus (Lecture 32) will tell us that this area is given by F (6) - F (0) where F is ANY antiderivative for f . Find an antiderivative for f and show that this process gives the same answer for the area. Now for the approximations (next page). In each case, sketch the half of the parabolic arch on [ 0 , 3 ] and draw the approximating rectangles as

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Unformatted text preview: instructed. Then evaluate the approximation, KEEPING YOUR WORK WRITTEN AS FRACTIONS. Note that if you have trouble evaluating f ( x ) with fractions, you may ﬁnd it helpful to write the function as 1 2 x (6-x ) . Approximation 1: Choose six rectangles of equal width with x * i = left end-point. Approximation: Approximation 2: Choose three rectangles of equal width with x * i = right endpoint. Approximation: Approximation 3: Choose four rectangles of equal width with x * i = midpoint. Approximation: Based on the pictures, which approximation is clearly an overestimation for the half of the arch on [ 0 , 3 ] ? which is clearly an underestimation? which looks like the best estimation? JUSTIFY your responses with a very brief explanation. How close is the best approximation for the whole arch (of the three you calculated)?...
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## This note was uploaded on 01/13/2010 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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P30 - instructed Then evaluate the approximation KEEPING...

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