The Calculus Tutoring Book

by Ash, Carol; Ash, Robert B.

• ISBN13: 9780780310445
• ISBN10: 0780310446
• Edition: 1st
• Format: Paperback
• Copyright: 1993-10-19
• Publisher: Wiley-IEEE Press

Summary

This book fills an educational void by adapting unique classroom-tested techniques that students find most congenial...that strip the shroud of mystery from an esoteric subject...that prepare students for applications of calculus in later courses.

Author Biography

Carol Ash is the author of The Calculus Tutoring Book, published by Wiley. Robert B. Ash as written about, taught, or studied virtually every area of mathematics.

Table of Contents

Preface	ix
Functions
Introduction	1	(3)
The Graph of a Function	4	(6)
The Trigonometric Functions	10	(9)
Inverse Functions and the Inverse Trigonometric Functions	19	(4)
Exponential and Logarithm Functions	23	(6)
Solving Inequalities Involving Elementary Functions	29	(3)
Graphs of Translations, Reflections, Expansions and Sums	32	(9)
Review Problems for Chapter 1	38	(3)
Limits
Introduction	41	(4)
Finding Limits of Combinations of Functions	45	(3)
Indeterminate Limits	48	(5)
Review Problems for Chapter 2	51	(2)
The Derivative Part I
Preview	53	(3)
Definition and Some Applications of the Derivative	56	(7)
Derivatives of the Basic Functions	63	(7)
Nondifferentiable Functions	70	(1)
Derivatives of Constant Multiples, Sums, Products and Quotients	71	(7)
The Derivative of a Composition	78	(3)
Implicit Differentiation and Logarithmic Differentiation	81	(3)
Antidifferentiation	84	(11)
Review Problems for Chapter 3	92	(3)
The Derivative Part II
Relative Maxima and Minima	95	(3)
Absolute Maxima and Minima	98	(7)
L'Hopital's Rule and Orders of Magnitude	105	(5)
Indeterminate Products, Differences and Exponential Forms	110	(3)
Drawing Graphs of Functions	113	(3)
Related Rates	116	(4)
Newton's Method	120	(2)
Differentials	122	(6)
Separable Differential Equations	128	(9)
Review Problems for Chapter 4	134	(3)
The Integral Part I
Preview	137	(2)
Definition and Some Applications of the Integral	139	(7)
The Fundamental Theorem of Calculus	146	(5)
Numerical Integration	151	(4)
Nonintegrable Functions	155	(2)
Improper Integrals	157	(6)
Review Problems for Chapter 5	161	(2)
The Integral Part II
Further Applications of the Integral	163	(10)
The Centroid of a Solid Hemisphere	173	(3)
Area and Arc Length	176	(5)
The Surface Area of a Cone and a Sphere	181	(2)
Integrals with a Variable Upper Limit	183	(8)
Review Problems for Chapter 6	188	(3)
Antidifferentiation
Introduction	191	(1)
Substitution	192	(3)
Pre-Table Algebra I	195	(3)
Pre-Table Algebra II: Partial Fraction Decomposition	198	(5)
Integration by Parts	203	(1)
Recursion Formulas	204	(3)
Trigonometric Substitution	207	(2)
Choosing a Method	209	(3)
Combining Techniques of Antidifferentiation with the Fundamental Theorem	212	(5)
Review Problems for Chapter 7	214	(3)
Series
Introduction	217	(3)
Geometric Series	220	(2)
Convergence Tests for Positive Series I	222	(6)
Convergence Tests for Positive Series II	228	(4)
Alternating Series	232	(6)
Power Series Functions	238	(3)
Power Series Representations for Elementary Functions I	241	(7)
Power Series Representations for Elementary Functions II (Maclaurin Series)	248	(4)
The Taylor Remainder Formula and an Estimate for the Number e	252	(2)
Power Series in Powers of x - b (Taylor Series)	254	(5)
Review Problems for Chapter 8	257	(2)
Vectors
Introduction	259	(4)
Vector Addition, Subtraction, Scalar Multiplication and Norms	263	(7)
The Dot Product	270	(6)
The Cross Product	276	(6)
The Scalar Triple Product	282	(3)
The Velocity Vector	285	(5)
The Acceleration Vector	290	(7)
Review Problems for Chapter 9	294	(3)
Topics In Three-Dimensional Analytic Geometry
Spheres	297	(1)
Planes	298	(4)
Lines	302	(5)
Cylindrical and Quadric Surfaces	307	(5)
Cylindrical and Spherical Coordinates	312	(7)
Review Problems for Chapter 10	317	(2)
Partial Derivatives
Graphs and Level Sets	319	(6)
Partial Derivatives	325	(6)
Chain Rules for First-Order Partial Derivatives	331	(3)
Chain Rules for Second-Order Partial Derivatives	334	(3)
Maxima and Minima	337	(9)
The Gradient	346	(9)
Differentials and Exact Differential Equations	355	(8)
Review Problems for Chapter 11	361	(2)
Multiple Integrals
Definition and Some Applications of the Double Integral	363	(7)
Computing Double Integrals	370	(7)
Double Integration in Polar Coordinates	377	(5)
Area and Volume	382	(5)
Further Applications of the Double Integral	387	(4)
Triple Integrals	391	(7)
Triple Integration in Spherical Coordinates	398	(6)
Center of Mass	404	(5)
Review Problems for Chapter 12	408	(1)
APPENDIX
A1 Distance and Slope	409	(1)
A2 Equations of Lines	410	(1)
A3 Circles, Ellipses, Hyperbolas and Parabolas	411	(1)
A4 The Binomial Theorem	412	(1)
A5 Determinants	413	(3)
A6 Polar Coordinates	416	(3)
Solutions to the Problems	419	(108)
Abbreviations Used in the Solutions	527	(1)
List of Symbols	527	(3)
Index	530	(3)
Authors' Biographies	533