Chapters 3 & 4 Concept Map

Chapters 3 & 4 Concept Map

1. Chapter 3

   • Section 3.1
     • Exponential Growth
       • Modeled by: f(x)=Ca^x
         • x= # of time periods
           • a= the growth factor
             • C is the initial value
     • Exponential Decay
       • Modeled by: f(x)=Ca^x
         • Where 0 < a < 1
         • x= # of time periods
           • a= the decay factor
             • Decay rate (r) satisfies a=1+r, so the rate is NEGATIVE
   • Section 3.2
     • Changing the Time Period
       • ex: Jenny had 2 cats. Today (five years later), she has 20 cats.
         • Find the 5 years growth factor
           • later/initial
             • 20/2
               • =10
                 • =1000%
           • Find one-year growth factor, round to three decimal places
             • 10^1/5
               • 1.585
             • Find an exponential function c(t) for the # of cats Jenny has per year starting 5 years ago
               • y=Ca^x
                 • y=2(1.585)^t
     • Half-Life
       • The time it takes for half the atoms to decay
         • a=1/2
       • ex: A bacteria starts with 20,000 bacteria. The half-life is 20 years
         • Find the model, b(t), for a 20 year decay rate
           • How many bacteria are left after 240 years?
             • 240/20=12
               • b(t)=200,000(1/2)^12
                 • 4.88 bacteria
           • a=1/2
             • b(t)=200,000(1/2)^t
               • t=# of 20 year half-lives
     • Compounding Interest
       • A = P(1+(r/n))^nt
         • Ex: $30,000 is invested in a savings account that pays 4% annual interest COMPOUNDED DAILY, what will the account balance be after 6 years?
           • A= ?
           • P=30,000
           • r=.04
           • n=365
           • t= 6
           • A= 30,000(1+(.04/365))^365*6
             • =$38,136.97
     • Compounding Continuously
       • Definition: THE NUMBER e: an irrational number represented by the letter 'e'
       • A=Pe^rt
     • Annual Percentage Yield (APY)
       • The annual growth rate of an investment, given by:
         • (1+(r/n))^nt -1
       • EX: what is the annual percentage yield of an investment that is compounded quarterly at 7.75%?
         • (1+(0.0775/4))^ 4*1
           • =1.0798-1
             • =.0798
               • 7.98%
   • Section 3.3
     • Rate of Change
       • f(x+1) - f(x)
       • If the AVERAGE RATE OF CHANGE for a consecutive outputs is constant, then there is a LINEAR model that fits the data exactly
       • Adding or subtracting
       • Growth or decay factor
     • Percentage Rate of Change
       • For growth or decay models, growth or decay rate (r) is:
         • r = a - 1
       • If the PERCENTAGE RATE OF CHANGE for a consecutive outputs is constant, then there is an EXPONENTIAL MODEL that fits the data exactly
         • Multiplying and dividing
   • Section 3.4
     • Graphs of Exponential Functions
       • f(x) = a^x FOR a > 1
         • Moving AWAY from the x-axis
       • f(x) = a^x FOR 0 < a < 1
         • Moving TOWARDS the x-axis
       • The line y=0 (the x-axis) is a horizontal asymptote of (f)
   • Toolkits A.3 & A.4
     • Exponential Notation
       • 2^4
         • 2*2*2*2
           • =16
     • Rules for Working with Exponents
       • Product Rule
         • To MULTIPLY two powers of the same #, ADD the exponents
           • 2^3*2^6
             • 2*2*2 * 2*2*2*2*2*2
               • =2^9
       • Quotient Rule
         • To DIVIDE two powers of the same #, SUBTRACT the exponents
           • 5^5/5^2
             • 5*5*5*5*5 / 5*5
               • Two 5's cancel out from top & bottom
                 • 5*5*5
                 • OR 5^5-2
                   • =5^3
       • Power Rule
         • To raise a power to ANOTHER power, MULTIPLY the exponents
           • (6^3)^2
             • Two times
             • 6^3 * 6^3
               • = 6^6
     • Powers of Products/Quotient Rules for Exponents
       • To raise a product to a power, raise each factor to the power
         • (7x)^3
           • 7^3 x^3
             • =343x^3
               • Both are in the parenthesis, so they both get the exponent
       • To raise a quotient to a power, raise both numerator and denominator to the power
         • (3/8)^4
           • 3^4/8^4
             • 81/4096
               • Won't reduce because 3 can't go into 8
     • Special Relations
       • If 'a' is any nonzero real number and 'n' is a positive integer, then:
         • a^0=1
         • a^-n= 1/a^n
     • Rational Exponents
       • Radicals:
         • 2^1/3
           • 3√2
       • Exponential Notation:
         • 4√6
           • 6^1/4

2. Chapter 4

   • Section 4.1
     • Determining the Base
       • log5 (625)= c
         • Base: 5
     • Rewrite in Exponential Form
       • log6 (36) = a
         • Exponential form: 6^a = 36
           • Solution: a=2
     • Chase of base formula
       • logbX = logaX/logaB
         • log11 (355)
           • 11^x=355
             • On calculator: log(355)/log(11)
               • =2.449
   • Section 4.2
     • Laws of Logarithms
       • logb(mn)= logb(m)+logb(n)
         • The logarithm of a product is the sum of the logarithms
       • logb(m/n)= logb(m)-logb(n)
         • The logarithm of a quotient is the difference of the logarithms
       • logb(m^n)= n*logb(m)
         • The logarithm of a power is the exponent times the logarithm of the base
   • Section 4.3
     • Linear Scale
       • Each mark on the scale corresponds to adding a fixed #
     • Logarithmic Scale
       • Each mark on the scale corresponds to multiplying by a fixed FACTOR
     • The pH Scale
       • Measures how acidic or basic a substance is
         • Ranges from 0-14
           • A pH of 7 is neutral
       • pH = -log [H+]
     • The Decibel Scale
       • S= k * log (I / I^0)
       • B= 10 * log (I / I^0), where I^0= 10^ -12
   • Section 4.4
     • Natural Exponential function
       • f(x) = e^x
     • Natural Logarithm function
       • f(x) = ln(x) = loge^x
       • ln(x)=y IF AND ONLY IF e^y=x
     • Exponential Growth and Decay
       • The exponential function: f(t)= Ce^rt
         • t represents time
         • r is the INSTANTANEOUS growth rate
         • C is the initial value of f
     • Expressing an Exponential Model in Terms of e
       • The exponential growth or decay model...
         • f(x)= Ca^x
           • (growth or decay factor 'a' time period)
       • ...is equivalent to the exponential model...
         • f(x)= Ce^rx
           • (instantaneous growth or decay rate 'r' per time period)
       • ...where the instaneous growth or decay rate is...
         • r = ln(a)
     • How to find the INSTANTANEOUS growth rate for a culture that starts with 100 bacteria
       • Set the models equal
         • Divide by 100
           • Take the natural logs of each side
             • Power property of logs
               • Divide by x
       • Ca^x
         • 100*2^x
           • 2^x
             • ln(2^x)
               • (inverse of e)
               • x*ln(2)
                 • ln(2)
       • Ce^rx
         • 100e^rx
           • e^rx
             • ln(e^rx)
               • (undoes the exponent)
               • rx
                 • r
                   • Instantaneous
       • =
         • MAIN IDEA:
           • If the growth factor for a population is 'a', then the instantaneous growth rate is ln(a)
   • Section 4.5
     • ex: Convert exponentials to logs
       • 2^x=16
         • log2(16)=x
           • log(16)/log(2)
             • 4=x
     • ex: Get exponents to the same side
       • 3^x= 15*7^x
         • Divide by 7x on both sides
           • (3/7)^x = 15
             • log(3/7)(15)=x
               • log(15)/log(3/7)
                 • x= 3.20
     • ex: Isolate the log first
       • Convert log to an exponent
         • 2logx=3
           • Divide by 2 on both sides
             • log10(x)= 3/2
               • Make 10 the base of the exponent
                 • 10^(3/2)
                   • x=31.62
     • Applying what we learned from other sections to this section

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