This is a draft cheat sheet. It is a work in progress and is not finished yet.
Exponential Growth
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population of bacteria is known to increase in size by 50% every 2 hours. There are 2000 bacteria in the population at 12 noon |
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i) the constant of proportionality correct to four decimal places k=0.5 t=2 a= 2000 : a= 2000ekt : 3000=2000e2(k): k= 0.2027
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ii) the time when population will reach 8000 a= 2000e0.2027t : 8000=2000e0.2027t : t= 6.84 : after 12 pm
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Binomial
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Binomial experiment has 7 trials. prob. of successes is 0.4. what is the probability that: |
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X=3 0.2903 (BinPDF)
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X is at least 3 0.58009 (BinCDF)
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X is more than 5 (go from 5.5)= 0.018842
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Rate of Change
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ROC is modelled by d/dx= -x2+ e0.4x, where A is the area, x is the time- days from June 1st. on june 1st there was 6000m infested |
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i) the ROC in area on June 5th. ie when x=4 -(4)2 + e0.4(4)= -11.047 : 11m/sq
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ii) the date when ROC is a minimum f(x)0= 2x+0.4e0.4x solve: x=9.7 =10th June
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iii) what is the total change in area infested between June 1st and June 12th inclusive integral from 0 to 12 of -x2 + e0.4x = -274.7 thus decreases by 275 m/sq
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iv) what is the total area infested by end of june 15th 6000 + integral from 0 to 15 of -x2 + e0.4x = 5881.1 m/sq
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