Chapter 3 & 4

Chapter 3 & 4

1. Chapter 3

   • Section 3.1
     • f(x)=Ca^x
       • x is the number of time periods
         • a is the growth factor
           • C is the initial value when x is 0
           • growth rate is a = 1 + r where a > 1 and r > 0
       • a > 1 for growth
         • 0 < a < 1
       • f(x) = 45(1.03)^x
         • initial value: 45
         • growth factor: 1.03
         • growth rate: .03 or 3%
   • Section 3.2
     • Changing the time period:
       • daily is a..... weekly is a^7
         • weekly is b..... daily is b^1/7
           • monthly is c..... yearly is c^12
             • yearly is d..... monthly is d^1/12
     • Compound Interest: A(t)= P(1 + r/n)^nt
       • Annual = 1
         • Quarterly = 4
           • Semi-Annually = 2
             • Monthly = 12
               • Weekly = 52
                 • Daily = 365
       • A(t)= 700(1+ .04/4)^4x1
         • 728.42
     • Annual Percentage Yield
       • (1 + r/n)1n
       • (1 + .0775/4)^4
         • 1.0797%
   • Section 3.3
     • linear model is the average rate of change
       • f(t)= C+at
         • 45 + 13t
     • exponential model is the percentage rate of change
       • f(t)=Ca^t
         • 45x2^t
   • Section 3.4
     • f(x) =a^x
       • a > 0 and a ≠ 1
       • domain is all real numbers
         • range is all positive real numbers

2. Chapter 4

   • Section 4.1
     • log10(x) = y also written as log(x) = y
     • loga(x) = y is also a^y = x
     • Logarithm Rules
       • loga(1) = 0
         • raise a to the exponent 0 to get 1
         • loga(a)=1
           • raise a to the exponent 1 to get a
           • loga(a^x)=x
             • raise a to the exponent x to get a^x
             • a^loga(x)=x
               • loga(x) is the exponent to which a must be raised to get x
     • loga(x) where a > 0 and a ≠ 1
       • domain: ( 0 , ∞ )
     • x = 0 is the vertical asymptote of f
   • Section 4.2
     • Laws of Logarithms
       • loga AB = logaA + logaB
         • the log of a product is the sum of the logs
           • log9x= log9 + log x
       • logaA/B = logaA-logaB
         • the log of a quotient is the difference of the logs
           • log12/n = log12 - log n
       • logaA^C = ClogaA
         • the log of a power is the exponent times the log of the base
         • log 9^2 = 2log9
     • Change Base Formula
       • logb(x) = loga(x)/loga(b)
       • log(61)321= log 321 / log 61
   • Section 4.3
     • Linear Scale: each mark corresponds to adding a fixed constant
     • Logarithmic Scale: each mark corresponds to multiplying by a fixed factor
     • pH scale ranges from 0-14
       • > 7 is acidic
       • 7 is neutral
       • < 7 is basic
       • pH = - log [ H + ]
         • pH = -log(5.3 x 10^-3
         • 6.5 = -log [H+]
           • [H+] = 10^-6.5
     • Decibel Scale
       • S = k x log (I/Io)
       • Io= 10^-12
       • 10 x log (1.3 x 10^-3 / 10^-12 ) = 91.1
     • Magnitude of an Earthquake
       • M = log (I/S)
   • Section 4.4
     • e = 2.718281828
     • Natural exponential function is f(x) = e^x
     • Natural logarithm function is f(x) = loge(x) =ln x
     • Compounding Continuously
       • A(t) = Pe^rt
         • 25000(1 + .06/2)^2x10 = 45152.78
       • semiannual is 2
         • daily is 365
           • weekly is 52
             • quarterly is 4
     • f(t) = Ce^rt models exponential growth if r > 0 and exponential decay if r < 0
       • t represents time
       • C is the initial value
       • r is the instantaneous growth or decay rate
       • f(t) = 325e^.7t
   • Section 4.5
     • This section was about applying what we have learned and using it to solve problems

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