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Section 5.5 o Trapezoidal Rule o Questions on HW

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Warm-Up z Let z Approximate using LRAM w/ n=4 z Is LRAM an overestimate or underestimate? z Approximate using RRAM w/ n=4 z Is RRAM an overestimate or underestimate? z Evaluate z Sz Solutions: LRAM – 23/4, underestimate z RRAM – 31/4, overestimate z Exact value - 20/3

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Trapezoidal Rule x 0 =a y(x) x1x1 x2x2 x3x3 x 4 =b Trapezoidal Rule: ( h = width of subinterval )

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Trapezoidal Rule x 0 =a x1x1 x2x2 x3x3 x 4 =b z Use the trapezoidal rule with n=4 to approximate

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Example 1: z Use the trapezoidal rule with n=4 to approximate

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Example 2: #58 p.319 z The rate at which water flows out of a pipe is given by a differentiable function R of time t. The table below records the rate at 4-hour intervals for a 24-hour period. z a) Use the Trapezoidal Rule with 6a) subdivisions of equal length to approximate. Explain the meaning of your answer in terms of water flow, using correct units. z b) Is there some time t between 0 andb) 24 such that. Justify your answer. z c) Suppose the rate of water flow isc) approximated by Use Q(t) to approximate the average rate of water flow during the 24 hour period. t (hours) R(t) ( gallons/hr) 09.6 410.3 810.9 1211.1 1610.9 2010.5 249.6

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zPart a Example 2: #58 p.319 zThis value approximates the total number of gallons of water that leaked out of the pipe from t=0 to t=24

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zPart b Example 2: #58 p.319 zYes. Since R(t) is differentiable, it is also continuous. By the Mean Value Theorem, there is a number c between 0 and 24 such that

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zPart c Example 2: #58 p.319

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