Methods Of Mathematics Applied To Calculus, Probability, And Statistics

by Hamming, Richard W.

• ISBN13: 9780486439457
• ISBN10: 0486439453
• Format: Paperback
• Copyright: 2004-11-30
• Publisher: Dover Publications

Summary

Understanding calculus is vital to the creative applications of mathematics in numerous areas. This text focuses on the most widely used applications of mathematical methods, including those related to other important fields such as probability and statistics. The four-part treatment begins with algebra and analytic geometry and proceeds to an exploration of the calculus of algebraic functions and transcendental functions and applications. In addition to three helpful appendixes, the text features answers to some of the exercises. Appropriate for advanced undergraduates and graduate students, it is also a practical reference for professionals. 1985 ed. 310 figures. 18 tables.

Author Biography

Richard W. Hamming: The Computer Icon
Richard W. Hamming (1915–1998) was first a programmer of one of the earliest digital computers while assigned to the Manhattan Project in 1945, then for many years he worked at Bell Labs, and later at the Naval Postgraduate School in Monterey, California. He was a witty and iconoclastic mathematician and computer scientist whose work and influence still reverberates through the areas he was interested in and passionate about. Three of his long-lived books have been reprinted by Dover: Numerical Methods for Scientists and Engineers, 1987; Digital Filters, 1997; and Methods of Mathematics Applied to Calculus, Probability and Statistics, 2004.

In the Author's Own Words:
"The purpose of computing is insight, not numbers."

"There are wavelengths that people cannot see, there are sounds that people cannot hear, and maybe computers have thoughts that people cannot think."

"Whereas Newton could say, 'If I have seen a little farther than others, it is because I have stood on the shoulders of giants, I am forced to say, 'Today we stand on each other's feet.' Perhaps the central problem we face in all of computer science is how we are to get to the situation where we build on top of the work of others rather than redoing so much of it in a trivially different way."

"If you don't work on important problems, it's not likely that you'll do important work." — Richard W. Hamming

Table of Contents

Preface	xv
I ALGEBRA AND ANALYTIC GEOMETRY
Prologue	3	(16)
The Importance of Mathematics	3	(3)
The Uniqueness of Mathematics	6	(1)
The Unreasonable Effectiveness of Mathematics	7	(2)
Mathematics as a Language	9	(1)
What is Mathematics?	10	(2)
Mathematical Rigor	12	(1)
Advice to You	13	(1)
Remarks on Learning the Course	14	(5)
References	17	(2)
The Integers	19	(31)
The Integers	19	(3)
On Proving Theorems	22	(2)
Mathematical Induction	24	(6)
The Binomial Theorem	30	(7)
Mathematical Induction Using Undetermined Coefficients	37	(6)
The Ellipsis Method	43	(3)
Review and Fallacies in Algebra	46	(3)
Summary	49	(1)
Fractions---Rational Numbers	50	(21)
Rational Numbers	50	(2)
Euclid's Algorithm	52	(1)
The Rational Number System	53	(2)
Irrational Numbers	55	(2)
On Finding Irrational Numbers	57	(3)
Decimal Representation of a Rational Number	60	(5)
Inequalities	65	(3)
Exponents---An Application of Rational Numbers	68	(2)
Summary and Further Remarks	70	(1)
Real Numbers, Functions, and Philosophy	71	(28)
The Real Line	71	(2)
Philosophy	73	(1)
The Idea of a Function	74	(6)
The Absolute Value Function	80	(2)
Assumptions About Continuity	82	(2)
Polynomials and Integers	84	(1)
Linear Independence	85	(6)
Complex Numbers	91	(5)
More Philosophy	96	(1)
Summary	97	(2)
Analytic Geometry	99	(35)
Cartesian Coordinates	99	(2)
The Pythagorean Distance	101	(1)
Curves	102	(3)
Linear Equations---Straight Lines	105	(5)
Slope	110	(4)
Special Forms of the Straight Line	114	(5)
On Proving Geometric Theorems in Analytic Geometry	119	(2)
The Normal Form of the Straight Line	121	(4)
Translation of the Coordinate Axes	125	(1)
The Area of a Triangle	126	(3)
A Problem in Computer Graphics	129	(1)
The Complex Plane	130	(3)
Summary	133	(1)
Curves of Second Degree---Conics	134	(35)
Strategy	134	(1)
Circles	134	(6)
Completing the Square	140	(5)
A More General Form of the Second-Degree Equation	145	(2)
Ellipses	147	(2)
Hyperbolas	149	(3)
Parabolas	152	(3)
Miscellaneous Cases	155	(1)
Rotation of the Coordinate Axes	156	(4)
The General Analysis	160	(2)
Symmetry	162	(2)
Nongeometric Graphing	164	(1)
Summary of Analytic Geometry	165	(4)
II THE CALCULUS OF ALGEBRAIC FUNCTIONS
Derivatives in Geometry	169	(35)
A History of the Calculus	169	(1)
The Idea of a Limit	170	(4)
Rules for Using Limits	174	(1)
Limits of Functions---Missing Values	175	(4)
The Δ Process	179	(6)
Composite Functions	185	(4)
Sums of Powers of x	189	(3)
Products and Quotients	192	(3)
An Abstraction of Differentiation	195	(4)
On the Formal Differentiation of Functions	199	(3)
Summary	202	(2)
Geometric Applications	204	(39)
Tangent and Normal Lines	204	(4)
Higher Derivatives---Notation	208	(2)
Implicit Differentiation	210	(3)
Curvature	213	(4)
Maxima and Minima	217	(10)
Inflection Points	227	(7)
Curve Tracing	234	(7)
Functions, Equations, and Curves	241	(1)
Summary	241	(2)
Nongeometric Applications	243	(39)
Scaling Geometry	243	(3)
Equivalent Ideas	246	(1)
Velocity	247	(3)
Acceleration	250	(2)
Simple Rate Problems	252	(5)
More Rate Problems	257	(7)
Newton's Method for Finding Zeros	264	(5)
Multiple Zeros	269	(1)
The Summation Notation	270	(2)
Generating Identities	272	(3)
Generating Functions---Place Holders	275	(2)
Differentials	277	(2)
Differentials Are Small	279	(2)
Summary	281	(1)
Functions of Several Variables	282	(36)
Functions of Two Variables	282	(8)
Quadratic Equations	290	(5)
Partial Derivatives	295	(2)
The Principle of Least Squares	297	(3)
Least-Squares Straight Lines	300	(4)
n-Dimensional Space	304	(8)
Test for Minima	312	(2)
General Case of Least-Squares Fitting	314	(2)
Summary	316	(2)
Integration	318	(48)
History	318	(1)
Area	318	(2)
The Area of a Circle	320	(2)
Areas of Parabolas	322	(5)
Areas in General	327	(5)
The Fundamental Theorem of the Calculus	332	(8)
The Mean Value Theorem	340	(5)
The Cauchy Mean Value Theorem	345	(1)
Some Applications of the Integral	346	(8)
Integration by Substitution	354	(4)
Numerical Integration	358	(7)
Summary	365	(1)
Discrete Probability	366	(37)
Introduction	366	(1)
Trials	367	(4)
Independent and Compound Events	371	(5)
Permutations	376	(4)
Combinations	380	(2)
Distributions	382	(6)
Maximum Likelihood	388	(3)
The Inclusion-Exclusion Principle	391	(2)
Conditional Probability	393	(3)
The Variance	396	(4)
Random Variables	400	(2)
Summary	402	(1)
Continuous Probability	403	(22)
Probability Density	403	(2)
A Monte Carlo Estimate of Pi	405	(3)
The Mean Value Theorem for Integrals	408	(2)
The Chebyshev Inequality	410	(1)
Sums of Independent Random Variables	411	(2)
The Weak Law of Large Numbers	413	(2)
Experimental Evidence for the Model	415	(1)
Examples of Continuous Probability Distributions	416	(3)
Bertrand's Paradox	419	(2)
Summary	421	(4)
III THE TRANSCENDENTAL FUNCTIONS AND APPLICATIONS
The Logarithm Function	425	(33)
Introduction---A New Function	425	(3)
In x Is Not an Algebraic Function	428	(2)
Properties of the Function In x	430	(3)
An Alternative Derivation---Compound Interest	433	(2)
Formal Differentiation and Integration Involving In x	435	(4)
Applications	439	(4)
Integration by Parts	443	(4)
The Distribution of Numbers	447	(2)
Improper Integrals	449	(5)
Systematic Integration	454	(3)
Summary	457	(1)
The Exponential Function	458	(31)
The Inverse Function	458	(2)
The Exponential Function	460	(4)
Some Applications of the Exponential Function	464	(2)
Stirling's Approximation to n!	466	(5)
Indeterminate Forms	471	(3)
The Exponential Distribution	474	(1)
Random Events in Time	475	(3)
Poisson Distributions	478	(2)
The Normal Distribution	480	(3)
Normal Distribution, Maximum Likelihood, and Least Squares	483	(1)
The Gamma Function	483	(3)
Systematic Integration	486	(2)
Summary	488	(1)
The Trigonometric Functions	489	(37)
Review of the Trigonometric Functions	489	(8)
A Particular Limit	497	(1)
The Derivative of Sin x	497	(1)
An Alternative Derivation	498	(2)
Derivatives of the Other Trigonometric Functions	500	(3)
Integration Formulas	503	(6)
Some Definite Integrals of Importance	509	(5)
The Inverse Trigonometric Functions	514	(3)
Probability Problems	517	(5)
Summary of the Integration Formulas	522	(3)
Summary	525	(1)
Formal Integration	526	(33)
Purpose of This Chapter	526	(1)
Partial Fractions---Linear Factors	527	(7)
Quadratic Factors	534	(6)
Rational Functions in Sine and Cosine	540	(4)
Powers of Sines and Cosines	544	(3)
Integration by Parts---Reduction Formulas	547	(2)
Change of Variable	549	(4)
Quadratic Irrationalities	553	(3)
Summary	556	(3)
Applications Using One Independent Variable	559	(34)
Introduction	559	(1)
Word Problems	559	(2)
Review of Applications	561	(4)
Arc Length	565	(5)
Curvature Again	570	(1)
Surfaces of Rotation	571	(5)
Extensions	576	(3)
Derivative of an Integral	579	(4)
Mechanics	583	(3)
Force and Work	586	(2)
Inverse Square Law of Force	588	(3)
Summary	591	(2)
Applications Using Several Independent Variables	593	(44)
Fundamental Integral	593	(8)
Finding Volumes	601	(3)
Polar Coordinates	604	(5)
The Calculus in Polar Coordinates	609	(5)
The Distribution of Products of Random Numbers	614	(4)
The Jacobian	618	(2)
Three Independent Variables	620	(3)
Other Coordinate Systems	623	(1)
n-Dimensional Space	624	(4)
Parametric Equations	628	(1)
The Cycloid---Moving Coordinate Systems	629	(3)
Arc Length	632	(1)
Summary	632	(5)
IV MISCELLANEOUS TOPICS
Infinite Series	637	(22)
Review	637	(2)
Monotone Sequences	639	(3)
The Integral Test	642	(2)
Summation by Parts	644	(2)
Conditionally Convergent Series	646	(4)
Power Series	650	(3)
Maclaurin and Taylor Series	653	(2)
Some Common Power Series	655	(2)
Summary	657	(2)
Applications of Infinite Series	659	(27)
The Formal Algebra of Power Series	659	(3)
Generating Functions	662	(4)
The Binomial Expansion Again	666	(7)
Exponential Generating Functions	673	(2)
Complex Numbers Again	675	(5)
Hyperbolic Functions	680	(3)
Hyperbolic Functions Continued	683	(1)
Summary	684	(2)
Fourier Series	686	(25)
Introduction	686	(1)
Orthogonality	687	(1)
The Formal Expansion	688	(6)
Complex Fourier Series	694	(4)
Orthogonality and Least Squares	698	(2)
Convergence at a Point of Continuity	700	(2)
Convergence at a Point of Discontinuity	702	(1)
Rate of Convergence	702	(2)
Gibbs Phenomenon	704	(2)
The Finite Fourier Series	706	(4)
Summary	710	(1)
Differential Equations	711	(32)
What Is a Differential Equation?	711	(1)
What Is a Solution?	712	(3)
Why Study Differential Equations?	715	(1)
The Method of Variables Separable	716	(5)
Homogeneous Equations	721	(3)
Integrating Factors	724	(4)
First-Order Linear Differential Equations	728	(7)
Change of Variables	735	(1)
Special Second-Order Linear Differential Equations	736	(2)
Difference Equations	738	(3)
Summary	741	(2)
Linear Differential Equations	743	(28)
Introduction	743	(2)
Second-Order Equations with Constant Coefficients	745	(5)
The Nonhomogeneous Equation	750	(4)
Variation of Parameters Method	754	(5)
nth-Order Linear Equations	759	(4)
Equations with Variable Coefficients	763	(1)
Systems of Equations	764	(1)
Difference Equations	765	(4)
Summary	769	(2)
Numerical Methods	771	(24)
Roundoff and Truncation Errors	771	(4)
Analytic Substitution	775	(1)
Polynomial Approximation	776	(2)
The Direct Method	778	(4)
Least Squares	782	(4)
On Finding Formulas	786	(2)
Integration of Ordinary Differential Equations	788	(4)
Fourier Series and Power Series	792	(2)
Summary	794	(1)
Epilogue	795	(6)
Methods	795	(1)
Methods of Mathematics	796	(2)
Applications	798	(1)
Philosophy	799	(2)
Appendix A: Table of Integrals	801	(16)
Appendix B: Some Geometric Formulas	817	(2)
Appendix C: The Greek Alphabet	819	(1)
Answers to Some of the Exercises	820	(29)
Index	849