Mathematics A Practical Odyssey

by Johnson, David B.; Mowry, Thomas A.

• ISBN13: 9780534378912
• ISBN10: 0534378919
• Format: Hardcover
• Copyright: 2000-10-26
• Publisher: Brooks Cole

Summary

Discover the many ways mathematics is relevant to your life with MATHEMATICS: A PRACTICAL ODYSSEY and its accompanying online resources. You'll master problem solving skills in such areas as calculating interest and understanding voting systems and come to recognize the relevance of mathematics and to appreciate its human aspect. Included with your purchase is access to the ThomsonNOW, an online tutorial that allows you to work with real math notation in real time, with unlimited practice problems, instant analysis and feedback, and streaming video to illustrate key concepts and Personal Tutor with SMARTHINKING a live, online mathematics tutor.

Table of Contents

1 LOGIC	1	(53)
Deductive vs. Inductive Reasoning	2	(10)
Problem Solving	2	(1)
Deductive Reasoning	3	(1)
Deductive Reasoning and Venn Diagrams	4	(1)
Historical Note: Aristotle	5	(4)
Inductive Reasoning	9	(1)
Exercises 1.1	10	(2)
Symbolic Logic	12	(9)
Statements	12	(1)
Compound Statements and Logical Connectives	13	(1)
The Negation ≈p	14	(1)
Historical Note: Gottfried Wilhelm Leibniz	15	(1)
The Conjunction p q	16	(1)
The Disjunction p q	16	(1)
The Conditional p → q	17	(2)
Exercises 1.2	19	(2)
Truth Tables	21	(13)
The Negation ≈p	21	(1)
The Conjunction p q	21	(1)
The Disjunction p q	22	(3)
The Conditional p &rarrl q	25	(3)
Equivalent Expressions	28	(1)
Historical Note: George Boole	29	(2)
De Morgan's Laws	31	(2)
Exercises 1.3	33	(1)
More on Conditionals	34	(7)
Variations of a Conditional	34	(1)
Equivalent Conditionals	35	(2)
The ``Only If' Connective	37	(1)
The Biconditional p ↔ q	38	(1)
Exercises 1.4	39	(2)
Analyzing Arguments	41	(13)
Valid Arguments	41	(3)
Tautologies	44	(2)
Historical Note: Charles Lutwidge Dodgson	46	(3)
Exercises 1.5	49	(2)
Chapter 1 Review	51	(3)
Sets and Counting	54	(58)
Sets and Set Operations	55	(12)
Notation	55	(2)
Universal Set and Subsets	57	(1)
Intersection of Sets	58	(1)
Mutually Exclusive Sets	58	(1)
Union of Sets	58	(3)
Complement of a Set	61	(1)
Historical Note: John Venn	62	(1)
Shading Venn Diagrams	63	(1)
Set Theory and Logic (prerequisite: Chapter 1)	63	(1)
Exercises 2.1	64	(3)
Applications of Venn Diagrams	67	(11)
Surveys	67	(4)
De Morgan's Laws	71	(2)
Historical Note: Augustus De Morgan	73	(1)
Exercises 2.2	74	(4)
Introduction to Combinatorics	78	(8)
The Fundamental Principle of Counting	78	(3)
Factorials	81	(2)
Exercises 2.3	83	(3)
Permutations and Combinations	86	(13)
With versus Without Replacement	86	(1)
Permutations	86	(3)
Combinations	89	(6)
Historical Note: Chu Shi-chieh	95	(1)
Exercises 2.4	96	(3)
Infinite Sets	99	(13)
One-to-One Correspondence	99	(2)
Historical Note: Georg Cantor	101	(1)
Countable Sets	102	(3)
Uncountable Sets	105	(1)
Points on a Line	106	(2)
Exercises 2.5	108	(2)
Chapter 2 Review	110	(2)
Probability	112	(75)
History of Probability	113	(7)
Roulette	114	(1)
Dice and Craps	115	(1)
Cards	116	(1)
Historical Note: Blaise Pascal	117	(1)
Historical Note: Gerolamo Cardano	118	(1)
Exercises 3.1	119	(1)
Basic Terms of Probability	120	(15)
Finding Probabilities and Odds	122	(2)
Relative Frequency versus Probability	124	(1)
Mendel's Use of Probabilities	125	(1)
Historical Note: Gregor Johann Mendel	126	(2)
Probabilities in Genetics	128	(1)
Genetic Screening	129	(2)
Historical Note: Nancy Wexler	131	(1)
Exercises 3.2	132	(3)
Basic Rules of Probability	135	(9)
Mutually Exclusive Events	136	(1)
Pair-of-Dice Probabilities	136	(3)
More Probability Rules	139	(1)
Probabilities and Venn Diagrams	140	(1)
Exercises 3.3	140	(3)
Fractions on a Graphing Calculator	143	(1)
Combinatorics and Probability	144	(10)
Lotteries	146	(2)
Historical Note: Lotteries	148	(1)
Keno	149	(1)
Cards	150	(2)
Exercises 3.4	152	(2)
Expected Value	154	(7)
Why the House Wins	156	(1)
Decision Theory	157	(1)
Betting Strategies	158	(1)
Exercises 3.5	158	(3)
Conditional Probability	161	(11)
Probabilities and Polls	161	(3)
The Product Rule	164	(1)
Tree Diagrams	165	(4)
Exercises 3.6	169	(3)
Independence; Trees in Genetics	172	(15)
Dependent and Independent Events	172	(3)
Product Rule for Independent Events	175	(1)
Trees in Medicine and Genetics	176	(3)
Hair Color	179	(2)
Exercises 3.7	181	(4)
Chapter 3 Review	185	(2)
Statistics	187	(101)
Population, Sample, and Data	188	(24)
Population versus Sample	188	(1)
Frequency Distributions	188	(2)
Grouped Data	190	(2)
Histograms	192	(2)
Histograms and Relative Frequency Density	194	(3)
Pie Charts	197	(1)
Exercises 4.1	198	(10)
Histograms on a Graphing Calculator	208	(4)
Measures of Central Tendency	212	(10)
The Mean	212	(5)
The Median	217	(1)
The Mode	218	(1)
Exercises 4.2	219	(3)
Measures of Dispersion	222	(15)
Deviations	223	(1)
Variance and Standard Deviation	224	(2)
Alternate Methods for Finding Variance	226	(4)
Exercises 4.3	230	(4)
Measures of Central Tendency and Dispersion on a Graphing Calculator	234	(3)
The Normal Distribution	237	(19)
Discrete versus Continuous Variables	238	(1)
Normal Distributions	238	(2)
Probability, Area, and Normal Distributions	240	(2)
Historical Note: Carl Friedrich Gauss	242	(1)
The Standard Normal Distribution	243	(6)
Converting to the Standard Normal	249	(5)
Exercises 4.4	254	(2)
Polls and Margin of Error	256	(13)
Sampling and Inferential Statistics	256	(1)
Sample Proportion versus Population Proportion	257	(2)
Margin of Error	259	(1)
Historical Note: George H. Gallup	260	(6)
Exercises 4.5	266	(3)
Linear Regression	269	(19)
Linear Trends and Line of Best Fit	270	(3)
Coefficient of Linear Correlation	273	(4)
Exercises 4.6	277	(4)
Linear Regression on a Graphing Calculator	281	(3)
Chapter 4 Review	284	(4)
Finance	288	(80)
Simple Interest	289	(9)
National Debt	292	(1)
Add-On Interest	293	(1)
Credit Card Finance Charge	294	(1)
Historical Note: Credit Card History	295	(1)
Exercises 5.1	296	(2)
Compound Interest	298	(75)
Annual Yield	304	(3)
Exercises 5.2	307	(2)
Doubling Time on a Graphing Calculator	309	(64)
Annuities	373
Calculating Short-Term Annuities	313	(3)
Calculating Long-Term Annuities	316	(1)
Tax-Deferred Annuities	317	(2)
Sinking Funds	319	(1)
Present Value of an Annuity	320	(2)
Exercises 5.3	322	(3)
Annuities on a Graphing Calculator	325	(1)
Amortized Loans	326	(21)
Amortization Schedules	327	(5)
Finding an Unpaid Balance	332	(3)
Exercises 5.4	335	(5)
Amortization Schedules on a Computer	340	(7)
Annual Percentage Rate on a Graphing Calculator	347	(9)
Historical Note: The Truth in Lending Act	349	(1)
Finance Charges	349	(3)
Estimating Prepaid Finance Charges	352	(2)
Exercises 5.5	354	(2)
Payout Annuities	356	(12)
Calculating Short-Term Payout Annuities	356	(1)
Comparing Payout Annuities and Savings Annuities	357	(1)
Calculating Long-Term Payout Annuities	358	(2)
Payout Annuities with Inflation	360	(2)
Exercises 5.6	362	(2)
Chapter 5 Review	364	(4)
Geometry	368	(92)
Perimeter and Area	369	(14)
Polygons	369	(4)
Heron's Formula for the Area of a Triangle	373	(3)
Right Triangles	376	(1)
Circles	377	(2)
Exercises 6.1	379	(4)
Volume and Surface Area	383	(12)
Problem Solving	383	(2)
Surface Area	385	(2)
Spheres	387	(2)
Cones and Pyramids	389	(2)
Exercises 6.2	391	(4)
Egyptian Geometry	395	(11)
Units of Measurement	396	(1)
Empirical Geometry (If It Works, Use It)	397	(2)
The Great Pyramid of Cheops	399	(1)
The Rhind Papyrus	400	(2)
Pi and the Area of a Circle	402	(2)
Exercises 6.3	404	(2)
The Greeks	406	(12)
Thales of Miletus	407	(1)
Pythagoras of Samos	408	(2)
Euclid of Alexandria	410	(1)
Deductive Proof	411	(1)
Historical Note: Archimedes of Syracuse	412	(2)
Archimedes of Syracuse'	414	(1)
Exercises 6.4	415	(3)
Right Triangle Trigonometry	418	(14)
Angle Measurement	419	(1)
Special Triangles	420	(1)
Trigonometric Ratios	421	(4)
Using a Calculator	425	(1)
Finding an Acute Angle	426	(2)
Angles of Elevation and Depression	428	(1)
Exercises 6.5	429	(3)
Conic Sections and Analytic Geometry	432	(13)
Historical Note: Hypatia	433	(1)
Conic Sections	433	(1)
The Circle	434	(1)
The Parabola	435	(3)
The Ellipse	438	(2)
The Hyperbola	440	(4)
Exercises 6.6	444	(1)
Non-Euclidean Geometry	445	(15)
The Parallel Postulate	445	(1)
Girolamo Saccheri	446	(1)
Carl Friedrich Gauss	447	(1)
Janos Bolyai	447	(1)
Nikolai Lobachevsky	448	(1)
Bernhard Riemann	449	(1)
Geometric Models	449	(2)
A Comparison of Triangles	451	(1)
Geometry in Art	452	(3)
Exercises 6.7	455	(1)
Chapter 6 Review	456	(4)
Matrices and Markov Chains	460	(59)
Review of Matrices	467	(6)
Terminology and Notation	461	(1)
Matrix Multiplication	462	(3)
Properties of Matrix Multiplication	465	(1)
Historical Note: Arthur Cayley and James Joseph Sylvester	466	(2)
Identity Matrices	468	(1)
Exercises 7.0	469	(1)
Matrix Multiplication on a Graphing Calculator	469	(4)
Markov Chains	473	(14)
Transition Matrices	474	(1)
Probability Matrices	475	(1)
Using Markov Chains to Predict the Future	476	(6)
Historical Note: Andrei Andreevich Markov	482	(1)
Exercises 7.1	483	(4)
Systems of Linear Equations	487	(21)
Linear Equations	487	(1)
Systems of Equations	487	(3)
Solving Systems: The Elimination Method	490	(1)
Solving Systems: The Gauss-Jordan Method	491	(8)
Exercises 7.2	499	(2)
Technology and the Row Operations	501	(1)
The Row Operations on a Graphing Calculator	502
The Row Operations and Amortrix	000	(507)
Augmented Identity Matrices	507	(1)
Long-Range Predictions with Markov Chains	508	(11)
Exercises 7.3	514	(3)
Chapter 7 Review	517	(2)
Linear Programming	519	(61)
Review of Linear Inequalities	521	(12)
Systems of Linear Inequalities	524	(3)
Finding Corner Points	527	(2)
Exercises 8.0	529	(1)
Graphing Linear Inequalities on a Graphing Calculator	529	(4)
The Geometry of Linear Programming	533	(16)
Creating a Model	534	(3)
Analyzing the Model	537	(2)
Some Graphing Tips	539	(1)
Why the Corner Principle Works	539	(5)
Historical Note: George Dantzig	544	(2)
Exercises 8.1	546	(3)
Introduction to the Simplex Method	549	(11)
Comparison of the Gauss-Jordan and Simplex Methods	555	(1)
Karmarkar's New Method	556	(1)
Historical Note: Hindu Mathematics	557	(1)
Exercises 8.2	558	(2)
The Simplex Method: Complete Problems	560	(20)
Pivoting with the Simplex Method	561	(2)
When to Stop Pivoting	563	(2)
Why the Simplex Method Works	565	(6)
Exercises 8.3	571	(2)
Technology and the Simplex Method	573	(5)
Chapter 8 Review	578	(2)
Exponentials and Logarithmic Functions	580	(74)
Review of Exponentials and Logarithms	581	(10)
Functions	581	(1)
Exponential Functions	581	(1)
Rational and Irrational Numbers	582	(1)
The Natural Exponential Function	583	(2)
Logarithms	585	(1)
Historical Note; Leonhard Euler	586	(1)
Common Logarithms	586	(3)
The Natural Logarithm Function	589	(2)
Exercises 9.0A	591	(1)
Review of Properties of Logarithms	591	(13)
The Inverse Properties	591	(2)
Solving Exponential Equations	593	(2)
The Exponent-Becomes-Multiplier Property	595	(1)
Historical Note: John Napier	596	(2)
The Division-Becomes-Subtraction Property	598	(2)
The Multiplication-Becomes-Addition Property	600	(1)
Solving Logarithmic Equations	601	(1)
Exercises 9.0B	602	(2)
Exponential Growth	604	(16)
Delta Notation	604	(3)
The Exponential Model	607	(6)
Exponential Growth and Compound Interest (for those who have read Chapter 5: Finance)	613	(2)
Exercises 9.1	615	(5)
Exponential Decay	620	(16)
Half-Life	623	(3)
Relative Decay Rate	626	(3)
Historical Note: Marie Curie	629	(1)
Radiocarbon Dating	629	(3)
Historical Note: Willard Frank Libby	632	(2)
Exercises 9.2	634	(2)
Logarithmic Scales	636	(18)
Earthquakes	636	(3)
The Richter Scale	639	(5)
Earthquake Magnitude and Energy	644	(1)
The Decibel Scale	645	(4)
Historical Note: Alexander Graham Bell	649	(1)
Exercises 9.3	650	(2)
Chapter 9 Review	652	(2)
Calculus	654
Review of Ratios, Parabolas, and Functions	655
Ratios, Proportions, and Rates	655
Delta Notation	656
Similar Triangles	657
Parabolas	660
Secant Lines and Their Slopes	661
Locating the Vertex	662
Functions	664
Exercises 10.0	666
The Antecedents of Calculus	668
Ancient Mathematics	668
Greek Mathematics	669
Hindu Mathematics	671
Arabic Mathematics	672
Historical Note: Arabic Mathematics	676
Mathematics during the Middle Ages	677
Mathematics during the Renaissance	677
Historical Note: Rene Descartes	681
Exercises 10.1	682
Four Problems	684
Find the Distance Traveled by a Falling Object	686
Find the Trajectory of a Cannonball	689
Historical Note: Galileo Galilei	690
Find the Line Tangent to a Given Curve	693
Find the Area of Any Shape	695
Exercises 10.2	697
Newton and Tangent Lines	700
Cauchy's Reformulation of Newton's Method	702
Historical Note: Isaac Newton	705
Exercises 10.3	708
Newton on Falling Objects and the Derivative	709
Average Speed and Instantaneous Speed	709
Newton on Gravity	709
The Derivative	712
Interpreting a Derivative	716
Exercises 10.4	718
The Trajectory of a Cannonball	720
The Motion Due to the Explosion	721
The Motion Due to Gravity	722
The Equation for the Cannonball's Motion	722
The Generalized Equation for the Cannonball's Motion (for those who have read Section 6.5)	728
Exercises 10.5	731
Newton and Areas	734
Find the Area of Any Shape	734
The Area Function A(x)	734
Newton and the Area Function	736
Antiderivatives	737
Antiderivatives and Areas	739
Exercises 10.6	743
Conclusion	744
Rates of Change	744
Sketching Curves	744
Finding Maximum and Minimum Values	746
Motion	747
Areas and Volumes	747
Chapter 10 Review	748
Appendices
Appendix A Using a Scientific Calculator	A-1
The Equals Button	A-1
The Clear Button	A-2
The Subtraction Symbol and the Negative Symbol	A-2
Order of Operations and Use of Parentheses	A-3
The Shift Button	A-6
Memory	A-6
Scientific Notation	A-8
Exercises	A-8
Appendix B Using a Graphing Calculator	A-10
The Enter Button	A-10
The 2nd and Alpha Buttons	A-10
Correcting Typing Errors	A-11
The Subtraction Symbol and the Negative Symbol	A-11
The Multiplication Symbol	A-12
Order of Operations and Use of Parentheses	A-12
Memory	A-15
Scientific Notation	A-17
Exercises	A-18
Appendix C Graphing with a Graphing Calculator	A-20
The Graphing Buttons	A-20
Graphing a Line	A-20
Exercises	A-22
Appendix D Finding Points of Intersection with a Graphing Calculator	A-24
Exercises	A-25
Appendix E Dimensional Analysis	A-26
Exercises	A-28
Appendix F Body Table for the Standard Normal Distribution	A-31
Appendix G Answers to Selected Exercises	A-32
Credits	A-73
Index	A-74