Functions and Change

by Crauder, Bruce; Evans, Benny; Noell, Alan

• ISBN13: 9780395911587
• ISBN10: 0395911583
• Format: Hardcover
• Copyright: 1999-01-01
• Publisher: Cengage Learning

Table of Contents

Preface	xi
Prologue	1	(434)
Typing mathematical expressions	1	(1)
Rounding	2	(1)
Scientific notation	3	(1)
Parentheses and grouping	4	(1)
Minus signs	5	(2)
Special numbers π and e	7	(1)
Chain calculations	8	(9)
Prologue Summary	15	(2)
Functions	17	(54)
Functions given by formulas	17	(9)
Functions of one variable	17	(1)
Functions of several variables	18	(8)
Functions given by tables	26	(11)
Reading tables of values	26	(1)
Filling gaps by averaging	26	(1)
Average rates of change	27	(2)
Spotting trends	29	(8)
Functions given by graphs	37	(17)
Reading graphs	37	(3)
Concavity and rates of change	40	(2)
Inflection points	42	(2)
Drawing a graph	44	(10)
Functions given by words	54	(12)
Comparing formulas and words	54	(3)
Getting formulas from words	57	(3)
Proportion	60	(6)
Chapter Summary	66	(5)
Graphical and Tabular Analysis	71	(78)
Tables and trends	71	(15)
Getting tables from formulas	71	(5)
Optimizing with tables of values	76	(10)
Graphs	86	(20)
Hand-drawn graphs from formulas	86	(1)
Graphing with the calculator	86	(1)
Tracing the graph	87	(1)
Choosing a viewing window in practical settings	88	(5)
Getting limiting values from graphs	93	(13)
Solving linear equations	106	(13)
The basic operations	106	(2)
Reversing the roles of variables	108	(11)
Solving nonlinear equations	119	(13)
The crossing graphs method	119	(1)
The single graph method	120	(12)
Optimization	132	(13)
Optimizing at peaks and valleys	132	(4)
Optimizing at endpoints	136	(9)
Chapter Summary	145	(4)
Straight Lines and Linear Functions	149	(60)
The geometry of lines	149	(9)
Characterizations of straight lines	149	(2)
The slope of a line	151	(1)
Getting slope from points	152	(6)
Linear functions	158	(13)
Constant rates of change	158	(2)
Linear functions and straight lines	160	(3)
Linear equations from data	163	(8)
Modeling data with linear functions	171	(13)
Testing data for linearity	171	(1)
Linear models	172	(3)
Graphing discrete data	175	(9)
Linear regression	184	(12)
The regression line	184	(2)
Uses of the regression line: slope and trends	186	(10)
Systems of equations	196	(9)
Graphical solutions of systems of equations	196	(4)
Algebraic solutions	200	(5)
Chapter Summary	205	(4)
Exponential Functions	209	(56)
Exponential growth and decay	209	(14)
Exponential growth	209	(1)
Exponential decay	210	(3)
Constant proportional change	213	(3)
Unit conversion	216	(2)
Exponential functions and daily experience	218	(5)
Modeling exponential data	223	(12)
Recognizing exponential data	223	(1)
Constructing an exponential model	224	(3)
Growth and decay factor units in exponential modeling	227	(8)
Modeling nearly exponential data	235	(16)
From linear to exponential data	235	(2)
From exponential to linear data: the logarithm	237	(2)
Exponential regression	239	(6)
A note about the growth rate	245	(6)
Another look at exponential regression	251	(10)
How a change in In x affects x	251	(2)
Continuous compounding and the APR	253	(2)
Exponential growth rate and an alternative form	255	(6)
Chapter Summary	261	(4)
A Survey of Other Common Functions	265	(60)
Power functions	265	(13)
Homogeneity property of power functions	267	(4)
Comparing exponential and power functions	271	(7)
Modeling data with power functions	278	(19)
The connection between power data and linear data	278	(1)
Getting a power model from data	279	(4)
Almost power data	283	(4)
Graphing on a logarithmic scale: common versus natural logarithms	287	(10)
Periodic functions and right triangle trigonometry	297	(10)
Period and amplitude	297	(2)
Using the sine function to model periodic phenomena	299	(2)
Surveying and right triangles	301	(6)
Polynomial and rational functions	307	(14)
Quadratics and parabolas	307	(3)
Linear rates of change and second-order differences	310	(5)
The quadratic formula	315	(1)
Higher-degree polynomials and their roots	316	(1)
Rational functions	317	(4)
Chapter Summary	321	(4)
Rates of Change	325	(68)
Velocity	325	(15)
Getting velocity from directed distance	325	(2)
Constant velocity means linear directed distance	327	(7)
When distance is given by a formula	334	(6)
Rates of change for other functions	340	(12)
Examples of rates of change	340	(1)
Common properties of all rates of change	341	(2)
When the rate of change is zero	343	(2)
When the rate of change is constant	345	(7)
Equations of change: liner and exponential functions	352	(10)
Equation of change for linear functions	352	(2)
Instantaneous rates of change	354	(1)
Equation of change for exponential functions	354	(4)
Why equations of change?	358	(4)
Equations of change: graphical solutions	362	(16)
Equilibrium solutions	362	(4)
Sketching graphs	366	(12)
Estimating rates of change	378	(10)
Rates of change for tabulated data	378	(3)
Rates of change for functions given by formulas	381	(7)
Chapter Summary	388	(5)
Mathematics of Population Ecology	393	(42)
Population dynamics: exponential growth	393	(12)
Exponential growth	394	(2)
Doubling times	396	(1)
Empirical data	397	(2)
Discrete generations	399	(6)
Population dynamics: logistic growth	405	(14)
Logistic growth model	405	(1)
Application to harvesting renewable resources	406	(2)
Logistic equations of change	408	(3)
Integral form of the logistic equation	411	(2)
The value of the logistic model	413	(6)
Population structure: survivorship curves	419	(13)
Life tables and survivorship curves	419	(1)
Classifying survivorship curves	420	(5)
Usefulness of survivorship curves	425	(7)
Chapter Summary	432	(3)
Brief Answers to Odd-Numbered Exercises	435	(22)
Index	457