P34 - lus Which approximation above was the best...

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Project 34/35 MAC 2233 1. 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. Now for some approximations. In each case, sketch the half of the parabolic arch on [0 , 3] and draw the approximating rectangles as instructed. Then evaluate the approximation, KEEPING YOUR WORK WRITTEN AS FRACTIONS. Note that if you have trouble evalu- ating f ( x ) with fractions, you may find it helpful to write the function as 1 2 x (6 - x ) . Approximation 1: Use three rectangles of equal width and x i = right endpoint of interval i . Approximation: Approximation 2: Use four rectangles of equal width and x i = midpoint of interval i . Approximation:
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Now find the exact area of the entire arch on [0 , 6] using the Fundamental Theorem of Calcu-
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Unformatted text preview: lus. Which approximation above was the best approximation? 2. Sketch the following piecewise-defined function, and evaluate the given definite integral us-ing FTC and properties of the definite integral. Shade the regions whose areas contribute to this definite integral. f ( x ) = e 3 x-1 x < √ x x ≥ Z 1-1 f ( x ) d x = Which region has more AREA: the one to left of the y-axis, or the one to the right ? What is the total shaded AREA? Let f ( b ) = Z b-1 f ( x ) d x . Is the value f ( b ) increasing or decreasing when b =-1 2 ? when b = 1 2 ? 3. Evaluate these definite integrals: Z 1 ( e t + e-t ) 2 d t Z 1 (1 + 2 √ ω ) 2 d ω Z 1 4 t √ 9-t 2 d t Z 1 4 t √ 9-t d t...
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P34 - lus Which approximation above was the best...

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