Calculus

by Gootman, Elliot C.

• ISBN13: 9780812098198
• ISBN10: 0812098196
• Format: Paperback
• Copyright: 1997-09-01
• Publisher: Barrons Educational Series Inc

Summary

This general review covers equations, functions, and graphs; limits, derivatives; integrals and antiderivatives; word problems; applications of integrals to geometry; and much more. Additional features make this volume especially helpful to students working on their own. They include worked-out examples, a summary of the main points of each chapter, exercises, and where needed, background material on algebra, geometry, and reading comprehension.

Table of Contents

Introduction	vii
1 Equations, Functions, and Graphs	1	(16)
1.1 Equations	1	(1)
1.2 Functions	2	(6)
1.3 Graphs	8	(5)
1.4 Summary of Main Points	13	(1)
1.5 Exercises	14	(3)
2 Change, and the Idea of the Derivative	17	(30)
2.1 Average Rates of Change	17	(6)
2.2 Instantaneous Rates of Change	23	(10)
2.3 The Derivative	33	(4)
2.4 The Geometric Meaning of Derivatives	37	(6)
2.5 Summary of Main Points	43	(1)
2.6 Exercises	44	(3)
3 The Idea of Limits	47	(22)
3.1 The Basic Definition	47	(3)
3.2 Some Complications with the Definition of Limits	50	(2)
3.3 The Problem of Division By Zero	52	(4)
3.4 The Case of (0/0)	56	(3)
3.5 Continuity and Differentiability	59	(4)
3.6 Summary of Main Points	63	(2)
3.7 Exercises	65	(4)
4 Computing Some Derivatives	69	(14)
4.1 Powers of x	70	(3)
4.2 Fractional Expressions with the Variable in the Denominator	73	(6)
4.3 Square Roots	79	(1)
4.4 Some Harder Examples	80	(2)
4.5 Summary of Main Points	82	(1)
4.6 Exercises	82	(1)
5 Formulas for Derivatives	83	(20)
5.1 Derivatives of Some Particular Functions	84	(2)
5.2 Derivatives of Combinations of Functions	86	(7)
5.3 The Chain Rule	93	(8)
5.4 Summary of Main Points	101	(1)
5.5 Exercises	101	(2)
6 Extreme Values, the Mean Value Theorem, and Curve Sketching	103	(22)
6.1 Extreme Values	104	(6)
6.2 The Mean Value Theorem	110	(4)
6.3 Curve Sketching	114	(7)
6.4 Summary of Main Points	121	(1)
6.5 Exercises	122	(3)
7 Word Problems	125	(30)
7.1 A Review of Geometry	125	(6)
7.2 Max-Min Word Problems	131	(9)
7.3 Related Rate Word Problems	140	(9)
7.4 Summary of Main Points	149	(1)
7.5 Exercises	150	(5)
8 The Idea of the Integral	155	(34)
8.1 The Basic Idea	155	(9)
8.2 Terminology and Notation	164	(16)
8.3 The Definite Integral: Definition and Notation	180	(3)
8.4 Summary of Main Points	183	(2)
8.5 Exercises	185	(4)
9 Computing Some Integrals	189	(18)
9.1 Summation Rules and Formulas	189	(10)
9.2 Computing Limits of Approximate Sums	199	(3)
9.3 Summary of Main Points	202	(2)
9.4 Exercises	204	(3)
10 Formulas for Integrals: Integrals, Antiderivatives and the Fundamental Theorem of Calculus	207	(28)
10.1 Introduction	207	(1)
10.2 The Fundamental Theorem of Calculus--The Main Idea	207	(6)
10.3 The Fundamental Theorem of Calculus--An Idea of the Proof	213	(3)
10.4 Computing Some Antiderivatives	216	(8)
10.5 Antiderivatives Involving the Chain Rule	224	(6)
10.6 Summary of Main Points	230	(2)
10.7 Exercises	232	(3)
11 Geometric Applications of the Integral	235	(58)
11.1 Horizontal vs. Vertical, x vs. y	235	(8)
11.2 Area	243	(17)
11.3 Volumes of Solids of Revolution--The Method of Cross-sectional Areas	260	(15)
11.4 Volumes of Solids of Revolution--The Method of Cylindrical Shells	275	(13)
11.5 Summary of Main Points	288	(2)
11.6 Exercises	290	(3)
12 Motion	293	(12)
12.1 Simple Initial Value Problems	293	(2)
12.2 Motion with Constant Acceleration	295	(5)
12.3 Summary of Main Points	300	(1)
12.4 Exercises	301	(4)
APPENDICES	305	(17)
A The Trigonometric Functions	305	(10)
A.1 Definitions	305	(3)
A.2 Identities	308	(1)
A.3 Derivatives	309	(2)
A.4 Antiderivatives	311	(4)
B Exponential and Logarithmic Functions	315	(7)
B.1 Definitions	315	(1)
B.2 Identities	316	(1)
B.3 Derivatives	317	(2)
B.4 Antiderivatives	319	(3)
Answers to Exercises	322	(14)
Glossary	336	(3)
Index	339