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# P32 - inite integral using FTC and properties of the...

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Project 32 MAC 2311 YOU MUST SHOW YOUR WORK TO RECEIVE FULL CREDIT!! 1. Let’s show that the Fundamental Theorem of Calculus (FTC) does what it claims with an example. Examine the function f ( t ) = t 2 . Fix an unspecified number x 1 (as shown on the axes below), and then draw and shade the region represented by the definite integral x 1 t 2 d t . 0 1 x Show that the integral x 1 t 2 d t is represented by the limit lim n →∞ n i =1 1 + ( x - 1 n ) i 2 ( x - 1 n ) . Evaluate the limit above WITHOUT using FTC ( x is just some unspecified number—treat it like a constant in the sum). Is your answer an antiderivative for f ( x ) as FTC states it should be (you might need to expand and combine like terms to see it)? So what is d dx x 1 t 2 d t ?

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2. Sketch the following piecewise-defined function, and evaluate the given def-
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Unformatted text preview: inite integral using FTC and properties of the deﬁnite integral. Shade the regions whose areas contribute to this deﬁnite integral. f ( x ) = 2 sin( πx ) x < √ x x ≥ Z 1-1 f ( x ) d x = Is your answer positive or negative? What has more area: one “hump” of the function sin( πx ), or the area under the square root function from 0 to 1 ? How much bigger is it? Just looking at your picture and thinking in terms of areas, is the function R x-1 f ( t ) d t increasing or decreasing at x =-1 2 ? at x = 0? at x = 1 2 ? Is it changing more rapidly at x = 1 2 or at x = 1?...
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P32 - inite integral using FTC and properties of the...

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