Functions And Change

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

• ISBN13: 9780618219568
• ISBN10: 0618219560
• Edition: 2nd
• Format: Paperback
• Copyright: 2002-02-28
• Publisher: Brooks Cole

Summary

Intended for students in a college algebra course who are not destined for an engineering-oriented calculus course, Functions and Change, A Modeling Approach to College Algebra, 2/e, has been revised with input from instructors across the curriculum: the sciences, business, economics, and political science. The text builds students' confidence in working with graphs, real data, and linear algebraic manipulations in these various disciplines. Enhancing the text's practical approach are a thorough integration of graphing technology, high-quality, real-world applications, and an emphasis on modeling.

Table of Contents

Preface	vi
Prologue: Calculator Arithmetic	1	(18)
Typing mathematical expressions	1	(1)
Rounding	1	(2)
Parentheses and grouping	3	(1)
Minus signs	3	(1)
Special numbers π and e	4	(1)
Chain calculations	5	(2)
Scientific notation	7	(1)
Another Look: Order of operations	8	(8)
Summary	16	(3)
Functions	19	(72)
Functions given by formulas	19	(16)
Functions of one variable	19	(1)
Functions of several variables	20	(3)
Another Look: Definition of a function	23	(12)
Functions given by tables	35	(17)
Reading tables of values	35	(1)
Filling gaps by averaging	35	(1)
Average rates of change	36	(2)
Spotting trends	38	(3)
Another Look: More on average rates of change	41	(11)
Functions given by graphs	52	(18)
Reading graphs	52	(3)
Concavity and rates of change	55	(1)
Inflection points	56	(1)
Drawing a graph	57	(3)
Another Look: Secant lines	60	(10)
Functions given by words	70	(21)
Comparing formulas and words	70	(3)
Getting formulas from words	73	(2)
Proportion	75	(2)
Another Look: Geometric constructions	77	(11)
Summary	88	(3)
Graphical and Tabular Analysis	91	(84)
Tables and trends	91	(17)
Getting tables from formulas	91	(3)
Optimizing with tables of values	94	(2)
Another Look: Limits	96	(12)
Graphs	108	(19)
Hand-drawn graphs from formulas	108	(1)
Graphing with the calculator	108	(1)
Tracing the graph	109	(1)
Choosing a viewing window in practical settings	109	(3)
Getting limiting values from graphs	112	(3)
Another Look: Shifting and stretching	115	(12)
Solving linear equations	127	(16)
The basic operations	127	(2)
Reversing the roles of variables	129	(3)
Another Look: Equations that are linear in one variable	132	(11)
Solving nonlinear equations	143	(14)
The crossing-graphs method	143	(1)
The single-graph method	144	(3)
Another Look: Solving nonlinear equations by factoring	147	(10)
Optimization	157	(18)
Optimizing at peaks and valleys	157	(3)
Optimizing at endpoints	160	(2)
Another Look: Optimizing with parabolas	162	(10)
Summary	172	(3)
Straight Lines and Linear Functions	175	(80)
The geometry of lines	175	(15)
Characterizations of straight lines	175	(2)
The slope of a line	177	(1)
Getting slope from points	178	(2)
Another Look: Plane geometry, parallel and perpendicular lines	180	(10)
Linear functions	190	(17)
Constant rates of change	190	(2)
Linear functions and straight lines	192	(3)
Linear equations from data	195	(3)
Another Look: Forms of equations	198	(9)
Modeling data with linear functions	207	(16)
Testing data for linearity	207	(1)
Linear models	208	(3)
Graphing discrete data	211	(2)
Another Look: Unevenly spaced data	213	(10)
Linear regression	223	(17)
The regression line	223	(1)
Uses of the regression line: slope and trends	224	(4)
Another Look: Linear regression formula, the error	228	(12)
Systems of equations	240	(15)
Graphical Solutions of systems of equations	240	(3)
Algebraic solutions	243	(1)
Another Look: Solution by elimination, matrices	244	(8)
Summary	252	(3)
Exponential Functions	255	(70)
Exponential growth and decay	255	(17)
Exponential growth	255	(1)
Exponential decay	256	(3)
Constant proportional change	259	(2)
Unit conversion	261	(2)
Exponential functions and daily experience	263	(1)
Another Look: Elementary properties of exponents	264	(8)
Modeling exponential data	272	(16)
Recognizing exponential data	272	(1)
Constructing an exponential model	273	(3)
Growth and decay factor units in exponential modeling	276	(3)
Another Look: Testing unevenly spaced data	279	(9)
Modeling nearly exponential data	288	(19)
From linear to exponential data	288	(1)
From exponential to linear data: the logarithm	289	(1)
Exponential regression	290	(5)
Another Look: Laws of logarithms	295	(12)
Logarithmic functions	307	(18)
The Richter scale	307	(1)
How the common logarithm works	308	(2)
The logarithm as the inverse of the exponential function	310	(2)
Another Look: Solving exponential equations	312	(10)
Summary	322	(3)
A Survey of Other Common Functions	325	(88)
Power functions	325	(16)
Homogeneity property of power functions	327	(4)
Comparing exponential and power functions	331	(2)
Another Look: Homogeneity, more on power and exponential functions	333	(8)
Modeling data with power functions	341	(18)
The connection between power data and linear data	341	(1)
Getting a power model from data	341	(3)
Almost power data	344	(2)
Graphing on a logarithmic scale: Common versus natural logarithms	346	(1)
Another Look: More on power regression	347	(12)
Combining and decomposing functions	359	(15)
Sums, products, and limiting values	359	(2)
Composition of functions	361	(2)
Piecewise-defined functions	363	(2)
Another Look: Chaotic behavior	365	(9)
Quadratic functions and parabolas	374	(17)
Linear rates of change and second-order differences	375	(4)
Quadratic regression	379	(1)
Almost quadratic data	380	(1)
The quadratic formula	381	(1)
Another Look: Completing the square, complex numbers	382	(9)
Higher-degree polynomials and rational functions	391	(22)
Higher-degree polynomials and their roots	391	(1)
Cubic and other polynomial models	392	(1)
Rational functions	393	(3)
Another Look: Factoring polynomials, behavior at infinity	396	(13)
Summary	409	(4)
Trigonometric Functions	413	(38)
Periodic functions	413	(14)
Period and amplitude	413	(2)
Sound waves	415	(1)
The sine and cosine functions	415	(2)
Period and amplitude of sine and cosine functions	417	(1)
Using the sine function to model periodic phenomena	418	(3)
Another Look: Radian angle measure	421	(6)
Modeling periodic data	427	(12)
Vertical and horizontal shifting	427	(3)
Sine models by estimation	430	(3)
Sine regression	433	(6)
Right triangle trigonometry	439	(12)
Right triangle definition of trigonometric functions	439	(3)
Another Look: Trigonometric identities	442	(6)
Summary	448	(3)
Rates of Change	451	(58)
Velocity	451	(13)
Getting velocity from directed distance	451	(2)
Constant velocity means linear directed distance	453	(6)
When distance is given by a formula	459	(5)
Rates of change for other functions	464	(10)
Examples of rates of change	464	(1)
Properties that all rates of change share	465	(2)
When the rate of change is zero	467	(1)
When the rate of change is constant	468	(6)
Estimating rates of change	474	(9)
Rates of change for tabulated data	474	(3)
Rates of change for functions given by formulas	477	(6)
Equations of change: Linear and exponential functions	483	(10)
Equation of change for linear functions	483	(2)
Instantaneous rates of change	485	(1)
Equation of change for exponential functions	485	(3)
Why equations of change?	488	(5)
Equations of change: Graphical solutions	493	(16)
Equilibrium solutions	493	(3)
Sketching graphs	496	(9)
Summary	505	(4)
Mathematics of Population Ecology	509
Population dynamics: Exponential growth	509	(10)
Exponential growth	510	(2)
Doubling times	512	(1)
Empirical data	512	(2)
Discrete generations	514	(5)
Population dynamics: Logistic growth	519	(11)
Logistic growth model	519	(1)
Application to harvesting renewable resources	520	(1)
Logistic equation of change	521	(3)
Integral form of the logistic equation	524	(1)
The value of the logistic model	525	(5)
Population structure: Survivorship curves	530
Life tables and survivorship curves	530	(1)
Classifying survivorship curves	531	(3)
Usefulness of survivorship curves	534	(6)
Summary	540
Brief Answers to Odd-Numbered Exercises	1	(32)
Index	33