A Companion To Calculus

by Ebersole, Dennis C.; Schattschneider, Doris; Sevilla, Alicia; Somers, Kay

• ISBN13: 9780495011248
• ISBN10: 049501124X
• Edition: 2nd
• Format: Paperback
• Copyright: 2005-07-22
• Publisher: Cengage Learning

Summary

Improve your algebra and problem-solving skills with A COMPANION TO CALCULUS! Every chapter in this companion provides the conceptual background and any specific algebra techniques you need to understand and solve calculus problems related to that topic. Verbal descriptions, diagrams, graphs, pictures, symbolic formulas, and numerical data are all used to reinforce communicating and understanding in different modes.

Table of Contents

Introduction	1	(7)
Symbols and Notation	2	(2)
Modes of Communication	4	(2)
Chapter 0 Exercises	6	(2)
Cartesian Coordinates	8	(31)
The Cartesian Coordinate Plane	9	(5)
Graphs	14	(4)
Lines and Their Equations	18	(8)
Parallel and Intersecting Lines	26	(4)
Distance between Two Points	30	(2)
The Circle	32	(7)
Exercises	35	(4)
Functions	39	(49)
Function Notation	41	(3)
Domain and Range of a Function	44	(5)
Set Notation for Real Numbers	45	(1)
Interval Notation for Real Numbers	45	(4)
Different Ways to Represent Functions	49	(4)
The Graph of a Function	53	(8)
Special Classes of Functions	61	(13)
Linear Functions	61	(2)
Piecewise Linear Functions	63	(4)
Quadratic Functions	67	(2)
Other Functions	69	(5)
Transformations of Graphs	74	(14)
Vertical Shifts	74	(1)
Horizontal Shifts	75	(1)
Reflection about the x-Axis	76	(1)
Reflection about the y-Axis	77	(1)
Scaling: Stretching and Compressing	77	(5)
Chapter 2 Exercises	82	(6)
Companion to Limits	88	(47)
Combinations of Functions	88	(9)
Algebraic Combinations of Functions	88	(5)
Composition of Functions	93	(4)
Algebraic Simplification of Functions	97	(14)
Quotients of Polynomials (Rational Functions)	100	(4)
Quotients with Radicals	104	(2)
Complex Fractions	106	(2)
Quotients with Absolute Values	108	(3)
Inequalities	111	(13)
Linear Inequalities	113	(3)
Absolute Value: Equations and Inequalities	116	(8)
If--Then Statements	124	(11)
Chapter 3 Exercises	131	(4)
Companion to Continuous Functions	135	(27)
Polynomials	140	(2)
Zeros of a Function	142	(9)
Finding Zeros of a Polynomial	142	(6)
The Approximation of Zeros of Continuous Functions	148	(3)
More on Domains of Functions	151	(11)
Domains of Composite Functions	153	(4)
Exercises	157	(5)
The Role of Infinity	162	(25)
Graphical Interpretation	165	(8)
Horizontal Asymptotes	165	(4)
Vertical Asymptotes	169	(4)
Algebraic Manipulations	173	(14)
Finding Horizontal Asymptotes	173	(7)
Finding Vertical Asymptotes	180	(4)
Exercises	184	(3)
Probelm-Solving and Rates of Change	187	(27)
Problem-Solving	187	(7)
Applications	194	(13)
Secant and Tangent Lines	207	(7)
Exercises	210	(4)
Companion to Rules of Differentiation	214	(16)
Negative and Rational Exponents	214	(6)
Decomposition of Functions	220	(4)
Simplifying Derivatives	224	(6)
Exercises	228	(2)
Review of Trigonometric Functions	230	(53)
Angle Measures	230	(8)
Definition and Evaluation of the Trigonometric Functions	238	(11)
Definitions of the Trigonometric Functions	238	(5)
Exact Values of the Trigonometric Functions for Some Special Angles	243	(6)
Properties and Identities for the Trigonometric Functions	249	(11)
Domain, Range, and Graphs of the Trigonometric Functions	260	(6)
Combining Functions with the Trigonometric Functions	266	(17)
Exercises	277	(6)
Companion to Implicit Differentiation	283	(11)
Implicitly Defined Functions	283	(6)
Solving Equations That Contain dy/dx	289	(5)
Exercises	292	(2)
Companion to Repeated Differentiation	294	(15)
Iteration and Patterns in Higher Derivatives	294	(4)
Rate of Change of Rate of Change	298	(11)
Exercises	306	(3)
Companion to Related Rates	309	(13)
Setting Up Equations for Related Rates Problems	309	(5)
Problem-Solving Strategies for Related Rates Problems	314	(8)
Exercises	319	(3)
Linear Approximations and Differentials	322	(12)
Tangent Line Approximation	322	(4)
The Differential	326	(8)
Exercises	331	(3)
Companion to Exponential Functions	334	(20)
Rules of Exponents	335	(11)
The Natural Exponential Function	346	(8)
Exercises	351	(3)
Companion to Inverse Functions	354	(25)
One-to-One Functions	354	(5)
Properties of a Function and Its Inverse	359	(7)
Finding the Inverse of a Function	366	(13)
Finding the Inverse Function When the Function Is One-to-One	367	(1)
Restricting the Domain When the Function Is Not One-to-One	368	(9)
Exercises	377	(2)
Companion to Logarithmic Functions	379	(22)
Definition and Properties of Logarithmic Functions	380	(6)
Graphs of Logarithmic Functions	386	(6)
Solving Equations with Logarithmic and Exponential Functions	392	(9)
Exercises	397	(4)
Companion to Extreme Values of a Function	401	(18)
Extreme Values and Critical Values	401	(9)
Absolute and Relative Extreme Values	401	(5)
Solving Equations to Find Critical Values	406	(4)
Setting Up Functions to Solve Extreme Value Problems	410	(9)
Exercises	415	(4)
Companion to Curve Sketching	419	(23)
Solving Inequalities	421	(5)
Graphical Interpretation	426	(8)
Putting It All Together	434	(8)
Chapter 17 Exercises	439	(3)
Companion to Antidifferentiation	442	(21)
Antidifferentiation as the Inverse of Differentiation	442	(8)
Recognizing Antiderivatives	450	(4)
Substitution for Antiderivatives	454	(9)
Exercises	459	(4)
Companion to Area and Riemann Sums	463	(24)
Exact Areas as Sums of Basic Geometric Shapes	463	(7)
Approximations of Areas	470	(6)
Riemann Sums and Their Interpretations	476	(11)
Exercises	484	(3)
Companion to the Definite Integral	487	(25)
Area under a Curve as a Definite Integral	487	(4)
Other Interpretations of the Definite Integral	491	(9)
The Fundamental Theorem of Calculus	500	(7)
The Fundamental Theorem of Calculus, Part I	500	(3)
The Fundamental Theorem of Calculus, Part II	503	(4)
Change of Variable in Definite Integrals	507	(5)
Exercises	509	(3)
Appendix	512	(28)
A About Real Numbers	512	(5)
B About Symbols	517	(7)
C About Algebraic Manipulations	524	(8)
C-1 Use and Importance of Parentheses	524	(2)
C-2 Factoring Algebraic Expressions	526	(2)
C-3 When Can I Cancel?	528	(2)
C-4 Working with Exponents	530	(2)
D About Solving Equations	532	(5)
D-1 Completing the Square	532	(3)
D-2 Solving Pairs of Equations	535	(2)
E About Triangles	537	(3)
Answers to Exercises	540	(52)
Index	592