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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# Also at the high school we work with easy examples

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Unformatted text preview: 9876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 2109876543210987654321 nT nT Let y and y ’ b e PERIODIC with period T. Let m and n be whole numbers. We observe how y = sin θ r epeats itself over every inter val of length 2 π . y is said to be PERIODIC with period 2 π . Similar ly, y ’ i s PERIODIC with period 2 π . What is the CHANGE in y over [ 0, 2 π ] ? Z ero. What is the area under y ’ o ver [ 0, 2 π ] ? Z ero. y = sin θ is shown by the solid line. y ’ = cos θ is shown by the lighter line. --1 (0,0 ) 2 Π Π 2 2Π 3Π 3Π 4Π θ y’ y +1 periodic functions E x a m p l e 6: Ex ample 7: A pump delivers at the rate of 60 t l itres/minute. Example a) How many litres will it pump from t = 5 minutes to t = 10 minutes ? b) Starting at t = 0 minutes, how long will it take to pump 3000 litres ? f’(t) = 60 t litres/minute a) t=10 t=10 t=5 t=5 ∫ f ’ (t) . dt = ∫ 10 60 t . dt =[ 60t ] litres 25 2 = [ 6000 − 1500 ] litres = 2250 litres 2 t=T t=T t=0 b) t=0 ∫ f ’(t) . dt = T 60t ∫ 60 t . dt = [ 2 ] 0 2 2 = 60T = 3000 2 T2 = 100 minutes T = 10 minutes This example may be easily adapted to compute the charge on a capacitor and also the time taken to charge or discharge the capacitor given the rate of flow of charge = current i = dq/dt. 189 area of a circle E x a m p l e 8: It is not always necessary to work with r ectangular co-ordinates. Let us find co-ordina dinates the area of a circle using polar co-ordinates (around a point). Here we take polar advantage of the symmetry of the elements of area around the origin. The elements area contiguous elements of area when summed up continuously or integrated form elements area the whole area. y R dr 32109876543210987654321 32109876543210987654321 3210987654321098765432...
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