course Mth 174 蜐{{Цsyo ȵ쁾assignment #006
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08:48:27 Query problem 7.5.13 (3d edition #10) graph concave DOWN and decreasing (note changes indicated by CAPS)
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RESPONSE --> The exact values, from smallest to largest, are as follows: Right(n) < Trap(n) < Exact < Mid(n) < Left(n).
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08:48:33 list the approximations and their rules in order, from least to greatest
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RESPONSE --> See above.
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08:48:53 between which approximations does the actual integral lie?
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RESPONSE --> The actual integral lies between Trap(n) and Mid(n).
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08:56:12 Explain your reasoning
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RESPONSE --> In this example, the graph is concave down and decreasing. The fact that it is decreasing tells me that the Left(n) function is an overestimate, while the Right(n) is an underestimate. This can be visualized since the Right(n) function uses the divisions on the right, making the estimation stay under the curve. The Left(n) function uses the divisions on the left, making the estimation stay over the curve. The fact that it is concave down indicates that the Trap(n) function will be less since it takes the divisions and draws a straight line between them, creating a series of lines that is underneath the main curve. As for the Mid(n) function, this takes the midpoint and draws a tangent line there. Since the tangent line is over the main curve in this example, the Mid(n) function is an overestimate.
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08:57:42 if you have not done so explain why when a function is concave down the trapezoidal rule UNDERestimates the integral
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RESPONSE --> I believe I did.
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08:57:50 if you have not done so explain why when a function is concave down the midpoint rule OVERrestimates the integral
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RESPONSE --> Done.
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08:58:35 Query NOTE: this problem has been left out of the new edition of the text, which is a real shame; you can skip on to the next problem (was problem 7.5.18) graph positive, decreasing, concave upward over interval 0 < x < h
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RESPONSE --> ?
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09:03:25 why is the area of the trapezoid h (L1 + L2) / 2?
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RESPONSE --> Taking a guess at it, the reason I see is that the Trap(n) function takes the average of two points with a linear function between them.
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09:04:41 Describe how you sketched the area E = h * f(0)
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RESPONSE --> Hmm. I am not quite sure.
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09:04:52 Describe how you sketched the area F = h * f(h)
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RESPONSE --> Lost again.
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09:05:05 Describe how you sketched the area R = h*f(h/2)
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RESPONSE --> Another one bites the dust.
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09:05:27 Describe how you sketched the area C = h * [ f(0) + f(h) ] / 2
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RESPONSE --> This would be the Trap(n) area, right?
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09:05:46 Describe how you sketched the area N = h/2 * [ f(0) + f(h/2) ] / 2 + h/2 * [ f(h/2) } f(h) ] / 2
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RESPONSE --> Hmm.
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09:06:40 why is C = ( E + F ) / 2?
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RESPONSE --> Because E and F are equal to the Left and Right sums, and C is the Trapezoidal sum?
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09:07:01 Why is N = ( R + C ) / 2?
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RESPONSE --> I am not sure. It looks like another variation on the Trapezoidal function.
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09:07:12 Is E or F the better approximation to the area?
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RESPONSE --> I don't know.
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09:07:29 Is R or C the better approximation to the area?
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RESPONSE --> Again, not sure. Two different graphs, or two different sections on one graph?
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10:54:02 query problem 7.5.24 show trap(n) = left(n) + 1/2 ( f(b) - f(a) ) `dx
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RESPONSE --> This is 7.4.25 in 4th edition. According to the book: trap(n) = ( left(n) + right(n) ) / 2 I have used this to work on my transformations. trap(n) = left(n) + (1/2) f(b) 'dx - (1/2) f(a) 'dx (1/2) f(b) 'dx = (1/2) right(n) (1/2) f(a) 'dx = (1/2) left(n) trap(n) = left(n) + (1/2) right(n) - (1/2) left(n) trap(n) = (1/2) left(n) + (1/2) right(n) trap(n) = ( left(n) + right(n) ) / 2 However, I am not sure that these two equations: (1/2) f(b) 'dx = (1/2) right(n) (1/2) f(a) 'dx = (1/2) left(n) are warranted, as I can't think of a way to prove it, since left(n) and right(n) are both based on n. Since n is the number of subintervals in the integral and the number of calculations in left(n) or right(n) depends on n, I can't think of a way to simplify it. Did I miss something?
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10:57:03 Explain why the equation must hold.
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RESPONSE --> I am not sure.
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11:17:14 In terms of a graph describe how trap(n) differs from left(n) and what this difference has to do with f(b) - f(a).
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RESPONSE --> Trap(n) creates the integral based on a trapezoid between each subinterval, whereas left(n) creates the integral based on rectangles based on the left side of the subinterval. As for f(b) - f(a), this equation returns the overall difference in the rate function between b and a. When this is multiplied by (1/2) 'dx, this produces the integral of the difference between b and a. Perhaps this is the relationship: Left(n) is either an overestimate or an underestimate depending on whether f(x) is increasing or decreasing over the interval. If f(x) is increasing, then Left(n) is an underestimate and (1/2) ( f(b) - f(a) ) 'dx adds to it. If f(x) is decreasing, then Left(n) is an overestimate and (1/2) ( f(b) - f(a) ) 'dx removes from Left(n). In either case the small addition or removal by (1/2) ( f(b) - f(a) ) 'dx brings Left(n) closer to the true value.
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