Math 141 chapter 3 and 4

Math 141 chapter 3 and 4

1. Chapter 3

   • 3.2 Exponential Models: Comparing Rates
     • Mike has 2 cats, five years later he has 20 cats.
       • 5 year growth factor
         • 20=2(a)^1=10
         • One year growth Factor
           • 20=2(a)^5
       • Start= 15 bacteria, after 15 minutes there was 45 bacteria
         • 15 minute growth factor
           • 45=15a,=3
           • One hour growth factor
             • 45=15(a), 3^4, =81
             • Find exponential function for # of bacteria per hour
               • B=15(81)^t, t=1,215
     • Half-Life
       • A=G(1/2)^t/h
         • G=initial Amount, t=time, h= half-life
       • Start with 20,000 Bacteria, the half-life is 20 years
         • b(t)=20,000(1/2)^t
           • =4.88 Bacteria
     • Later/Initial
     • Compound Interest
       • A=P(1+r/n)^nt
         • 30,000 in savings account, pays 4% annual, after 6 years?
           • Monthly
             • A=30,000(1+.004/12)^12?6
       • Compound Continuously
         • A=Pe^rt
           • 4000 is invested, with a bank that pays 2.4 annual compounded continuously what will it be after 9 years?
             • A=4000e^(.025 x 9) =5009.29
   • 3.3 Comparing Linear and Exponential Growth
     • Rate of change
       • f(x+1)-f(x)
       • Percentage Rate of Change
         • r=a-1
     • Average Rate of change/Percentage rate of change
       • f(x)
         • 6x5^2=30
           • 30-6=24
             • Average rate of change
             • 30-6/6=24/6=400%
               • percentage rate of change
     • x=0,1,2,3,4
       • f(x)=2000,400,80,16,3.2
         • 400=2000xa^1
     • • decreasing, exponential decay, f(6)=10, estimate the y-intercept=80
     • Linear vs. Exponential
       • t=0,1,2,3,4,5
       • j(t)=16,24,32,40,48,56
       • j(t)=8t+16
         • Linear
       • t=0,1,2,3,4,5
         • j(t)=16,24,36,54,81,121.5
           • 16=Ca^t
             • 24=16a^1
               • a^1=1.5
           • j(t)=16(1.5)^t
   • 3.4 Graphs of Exponential Funtions
     • • C=(0,1)
         • a>1
         • Domain:(-Infinity,Infinity)
           • Range:(0,Infinity)
     • Make sure to carefully read the graph and to think about the domain and the range
       • Decay= a<1
       • Growth= a>1

2. chapter 4

   • 4.5 Exponential Equations: Getting Information from a model
     • 2^x=16, log216=x
     • 11^x=54=log(11)54=x
     • 2 x 6^x=25, = log6(25/2)=x
     • 2logx=3, = x=10^3/2
     • 7.5 billion population and increasing by 1.13% every year
       • p(t)=7.5(1+.0113)^t
         • What will it be in 2050?
           • 7.5(1.0113)^34
   • 4.4 Natural Exponential & Logarithm Functions
     • Natural Exponential Function= f(x)=e^x
       • Natural Logarithm function= f(x)= ln(x)=log(e)(x)
     • ln(1), ln(1)=x, loge(1)=x, 1=e^x
       • ln(e^10), ln(e^10)=x, loge(e^10)=10 e^10 = e^x
     • ln1=0, because loge1=0
       • ln e =1 because loge e=1
         • ln e^x = x
           • e^ln x =x
     • f(t)=Ce^rt
       • 500 CFU/ml with a growth rate of 40% per hour
         • f(t)=500e^.40t
           • After 10 hours?
             • f(t)=500e^.40(10)=27,299.1
     • f(x)=Ca^x
       • f(x)=Ce^rx
         • r=ln(a)
           • Instantaneous growth or decay rate
         • Instantaneous growth or decay rate r per time period
       • Exponential growth or decay model
   • 4.3 Logarithmic Scales
     • pH=-log[H+]
     • When pH increases by one, [H+] decrease by factor of 10
       • When a pH increases by one, [H+] decrease by factor of 10
     • [H+] =3.16 x 10^-8M
       • pH=-log10(3.16 x 10^-8) = 7.5, Basic
     • B=10 x log(I/I-12)
       • If the intensity was 100 W/m^2
         • B= 10 x log (100/10^-12) = 140 db
     • M=log(I/S)
       • San Fran=8.3, Colombia-Ecuador was 4x as intense
         • 8.3=log(I/S)
         • m=(4I/S)
         • m=log(4) + log(I/S) =8.3
           • log (4)+8.3=8.9
   • 4.2 Laws of logarithms
     • logb(mn)=logb(m)+logb(n)
       • logb(m/n)=logb(m)-logb(n)
         • logb(m^n)= n x logb(m)
           • log5(3^2)=2 x log5(3)
         • log(12/n)= log10(12) - log10(h)
       • log3(9x)= log3(9) + log3(x)
     • log(h)+log(e)= log(he)
       • log15(25)+log15(9)=log15(25x9)=log15(225)

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