alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Definite a function fx is an e x p r ession of change

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Unformatted text preview: s us only the CHANGE in height. It does NOT tell us the height . 0 at instant t i . We must know the initial height at time t = 0, which in this case is the constant of integration C 0 . Then : HEIGHT at instant t i = CHANGE in height + INITIAL height = y(t i ) + C 0 2: Example 2 We know the relationship between temperature t measured o o in F and C . o o F (t) = 9 5 t C + 32 / o 9 d F(t) F ’(t) = = /5 dt / ∫ oF ’(t) dt = 9 5 t oC + C 0 o o How do we determine the constant of integration C0 ? We must know F (t) at t = 0. o o We may conduct an experiment. We immerse the F thermometer and the C o o thermometer in a beaker of ice which we know is 0 C . Then we read the F thermometer to get C 0 = 32. So: o F (t) = 9 5 t C + 32 / Example Ex ample 3: A pump delivers with a rate of flow f ’ (t) = 60 litres/minute. o What is the volume of water in the tank after 5 minutes ? f(t) = ∫ f ’ (t) . dt = 60 t + C0 t=5 ∫ f ’ (t) . dt = t=0 t=5 5 t=0 0 ∫ 60 . dt =[60 t ] litres = 300 litres 165 However, this is NOT the volume of water in the tank. This is only the CHANGE in volume. We must know the INITIAL volume, say C0 = 500 litres. Then the volume of water in the tank at t = 5 minutes is : VOLUME (at t = 5 minutes) = CHANGE in volume + INITIAL volume = 300 litres + 500 litres = 800 litres. 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 f ’ (t) = the rate of flow of charge = current i = dq/dt. Let us look at this very interesting in example in a more general way . STOP TIME ∫ ( RATE of FLOW). dt = CHANGE in VOLUME START TIME There are 4 entities involved: f ’(t) = RATE of FLOW, CHANGE in VOLUME, START TIME and STOP TIME. Usually we know f ’ (t) . Now we have 3 possibilities. 1. If we know the START TIME and STOP TIME we may compute the CHANGE in VOLUME for the given RATE of FLOW f ’ (t) . 2. If we know the START TIME and CHANGE in VOLUME we may determine the STOP TIME. Then ( STOP TIME − START TIME) will tell us how much time it took to pump a cer...
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