calclec6

calclec6 - tom.h.wilson tom.wilson@mail.wvu.edu Dept....

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1 tom.h.wilson tom.wilson@mail.wvu.edu Tom Wilson, Department of Geology and Geography Dept. Geology and Geography West Virginia University cos   32 xx x 1 x e 19 5 e Tom Wilson, Department of Geology and Geography 2 x
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2 yx Take the simple function What do we have to differentiate to get x? •What does the result represent? 2 2 x Tom Wilson, Department of Geology and Geography x y=x • What is the area? 2 2 x Don’t forget Waltham’s excel files You’ll find the file integ.xls Tom Wilson, Department of Geology and Geography
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3 derivatives dx 2 ya x 13 x y a x y 2 Tom Wilson, Department of Geology and Geography 5 8 1 x y We did these the other day in the “forward” direction c bx ax y 2 q ax y n 2 Tom Wilson, Department of Geology and Geography
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4 Both have the same derivative c bx ax y 2 2 ya x b x Tom Wilson, Department of Geology and Geography 1 2 1 ) ( x y 2 / 3 2 ) ( bx a y 2 ) 1 2 ( x y Tom Wilson, Department of Geology and Geography
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5 ) 5 )( 3 ( x x y 3 x y Tom Wilson, Department of Geology and Geography 5 x Both these functions have the same derivative 5 3 x x y 2 5 y x Tom Wilson, Department of Geology and Geography
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6 Sometimes we might want to know the area bounded by a function Tom Wilson, Department of Geology and Geography Sometimes we might want to know the area bounded by a function. How could you get excel to compute the area for you? See Waltham’s integ.xls () A fx x  x f x Tom Wilson, Department of Geology and Geography Total Area under this Curve 1 n i f xx
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7 In Waltham’s integ.xls file set n = 2 Notice that in his file you can compare the Tom Wilson, Department of Geology and Geography approximation and exact numbers shown above. The question you are probably asking now is just what is that “exact” number? For y = x 2 what do we have to differentiate Tom Wilson, Department of Geology and Geography what do we have to differentiate to get x 2 ? If we evaluate that then we get the area under the above curve
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8 One way to evaluate the integral is outlined below. The area, as we have said is simply the sum of the product of the function, whatever it happens to be, (in this case x 2 ) and the little x we use to define the length of the base of our rectangles. 1 n i i A yx  n i x ix   Then let  2 i yi x or 2 1 A n i i x x so Tom Wilson, Department of Geology and Geography 23 1 32 1 i n i A ix x i & These kinds of identities are not the sort of thing we geologists relish spending our time on.
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calclec6 - tom.h.wilson tom.wilson@mail.wvu.edu Dept....

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