Elementary Calculus: Course Notes

Title: Elementary Calculus: Course Notes

Author: Reuben L. Rosenberg

Typeset using the \TeX system by Lawrence May

Year of Publication 1984

Published by: Holt, Rinehart and Winston of Canada, Ltd

ISBN: 0039242064


Chapter 1: Review of the Differential Calculus

1.1 The Graph of a Function
1.2 Continuity
1.3 Properties of a Continuous Function
1.4 The Derivative
1.5 Rules for Finding the Derivative
1.6 Critical Points
1.7 The Nature of the Local Extrema
1.8 Sketching the Graph
1.9 Global Maximum and Minimum
1.10 The Mean Value Theorem
1.11 Applications of the Mean Value Theorem
1.12 Composite Functions
1.13 The Absolute Value Functions

Chapter 2: Anti-Derivatives

2.1 Definition and Properties
2.2 Further Anti-Derivatives
2.3 A Matter of Notation
2.4 An Important Application of Anti-Derivatives: Areas
2.5 Some Properties of the Definite Integral

Chapter 3: Rational Functions

3.1 Definition
3.2 Derivatives and Anti-Derivatives
3.3 The Graphs of the Rational Function

Chapter 4: Algebraic Functions

4.1 Implicit Functions
4.2 Algebraic Functions
4.3 The Anti-Derivative of a Function which Involves the Square Root of a Linear Function
4.4 The Graphs of Some Algebraic Functions

Chapter 5: The Definite Integral

5.1 Riemann Sums
5.2 Evaluation of a Particular Definite Integral
5.3 Properties of the Definite Integral
5.4 Change of Variable and Change of Limit
5.5 An Extension of the Mean Value Theorem for Integrals
5.6 Numerical Integration
5.7 The Trapezoidal Rule

Chapter 6: The Trigonometric Functions

6.1 Definition and Properties
6.2 The Graphs of the Function
6.3 Differentiation and Integration of the Trigonometric Functions
6.4 The Function \tan {x}
6.5 Graphs of Functions Involving the Trigonometric Functions
6.6 Trigonometric Substitutions in Integration

Chapter 7: Inverse Functions

7.1 Introduction and Properties
7.2 The Derivative of the Inverse Function
7.3 The Inverse Trigonometric Functions
7.4 Trigonometric Substitution

Chapter 8: The Logarithm and Exponential Functions

8.1 Definition
8.2 Properties of \ln {x}
8.3 Differentiating Composite Functions
8.4 Integrals Leading to the Logarithm
8.5 The Graph of \ln {x}
8.6 Inequalities for \ln {x}
8.7 The Exponential Function
8.8 Properties of e^x
8.9 Differentiation and Integration of the Exponential
8.10 Some Limits
8.11 The Function a^x
8.12 Growth and Decay

Chapter 9: Areas and Volumes

9.1 Areas
9.2 Volumes of Revolution
9.3 Volumes of Cylindrical Shells
9.4 Volumes by Slices
9.5 The Arc-Length of a Curve

Chapter 10: Further Methods of Integration

10.1 Integration by Parts
10.2 Partial Fractions
10.3 The Integration of Rational Functions
10.4 The Integration of a \displaystyle \frac {linear}{\sqrt{quadratic}}
10.5 A Summary of the Methods of Integration

Chapter 11: Improper Integrals

11.1 Improper Integrals of the First Kind
11.2 Improper Integrals of the Second Kind

Chapter 12: Expansion

12.1 The Taylor Expansion
12.2 Expansions of Standard Functions
12.3 The Evaluation of Limits

Chapter 13: Differential Equations

13.1 Problems Leading to Differential Equations
13.2 First Order Differential Equations
13.3 Separable Equations
13.4 Linear Equations

Additional Exercises in Differentiation and Integration

A Sample Examination

Listing of Exercises (R.L. Rosenberg)


Author: ascklee

Dr. Lee teaches at the Wee Kim Wee School of Communication and Information at the Nanyang Technological University in Singapore. He founded The Mathematics Digital Library in 2013.

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