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Numerade Educator

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Problem 4 Medium Difficulty

Draw the graph of $ f(x) = \sin \left (\frac{1}{2} x^2 \right) $ in the viewing rectangle $ [0, 1] $ by $ [0, 0.5] $ and let $ I = \displaystyle \int_0^1 f(x)\ dx $.
(a) Use the graph to decide whether $ L_2 $, $ R_2 $, $ M_2 $, and $ T_2 $ underestimate or overestimate $ I $.
(b) For any value of $ n $, list the numbers $ L_n $, $ R_n $, $ M_n $, $ T_n $, and $ I $ in increasing order.
(c) Compute $ L_5 $, $ R_5 $, $ M_5 $, and $ T_5 $. From the graph, which do you think gives the best estimate of $ I $?

Answer

a. underestimate
b.$L_{n}<M_{n}<I<T_{n}<R_{n}$

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Video Transcript

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