Concept map Chapter 3-4

Concept map Chapter 3-4

1. Chapter 3

   • The Growth factor: Exponential growth is modeled by a function of the form- f(x)=Ca^x; a>1
   • The growth factor and rate: For an exponential growth model, the factor a is greater than 1. The growth r is positive- a=1+r; r>0
   • Exponential Decay: The decay factor is a, where a is a positive number less than 1, and the graph has a general slope down- f(x)=Ca^x ; 0<a>1
   • Percent rate of change: For the growth or decay model f(x)=Ca^x the growth or decay rate is r; r=f(x+1)-f(x)/f(x), r is expressed as a percentage
   • Logistic Growth: A function of the form f(x)=C/1+bxa^-1
   • Exponential functions: An exponential function with base a is a function of the form f(x)= a^x, where a>0 and a doesnt equal 1, the domain of f is a set of all real numbers
   • Half lives: The time it takes for half the atoms to decay; N(t)= N(base 0)(1/2)^t, Ex: 20,000(1/2)^t

2. Chapter 4

   • Log Base 10: Log base 10 of x is defined by , log10x=y if and only if 10^y=x
   • Log base a: If a is a positive number then the log base a of x is defined by, logax=y if and only if a^y=x
   • Basic Properties of log's: Log(a)1=0, log(a)a=1, log(a)a^x=x,
   • Graphs of log functions: The graph of the log function f(x)= log(a)x for a>1
   • Laws of Logs: log(a)AB= log(a) A+ log(a)B, log(a)A/B= log(a)A- log(a)B, log(a)A^c= Clog(a)A, Ex: log (base 3) (9x)= 2+ log(base 3)(x)
   • Change of Base Formula: log(b)X= log(a)x/log(a)b; Ex: log(base 11)(355)=x, log(355)/log(11)=2.449
   • Natural exponential function: The natural exponential function is the exponential function f(x)= e^x
   • Basic properties of natural log: ln 1=0; ln e=1; ln e^x=x; e^ln x=x
   • Continuously Compounded interest: If an amount P is invested at an annual interest rate r compounded continuosly, then the amount A(t) of the investment after t years is given by the formula; A(t)=Pe^rt
   • Exponential Growth and Decay: The exponential formula, f(t) =Ce^rt; models exponential growth (if r>0) or exponential decay (if r<0)
   • Exponential model in terms of e: The exponential growth or decay model f(x)=Ca^x is equivalent to the exponential model f(x) = Ce^rx, where the instantaneous growth or decay rate is, r=;n(a); Ex: 45(.85)^x
   • pH scale: Measures how acidic or basic a substance is, pH= -log[H+], Ex: pH=-log(10^-7)= -7
   • The decibel scale: Scientist test human responses to stimuli (light, sound) using logs. S=K x log (I/I0)
   • The Richter scale: Used to compare the size of earthquakes ; M= log(I/S)