## Lecture Notes

### Calculus

Much part of these lecture notes came from calculus classes I taught.

#### Lecture Notes

#### Differential Calculus

#### Integral Calculus

#### Sequences, Series, and Vector-Valued Functions

#### Multi-Variable Calculus

#### Free Online Textbooks

#### References and Further Reading

Click on linked topics to view lecture notes.

Some Important Formulas from College Algebra and Trigonometry for Freshmen Calculus

Limits of Functions

Examples of Non-Existing Limits

The Precise Definition of a Limit

How to Calculate Limits I

How to Calculate Limits II

How to Calculate Limits III

Continuity

Limits Involving Infinity and Asymptotes

Finding a Line Tangent to a Curve

Derivatives

A Physical Meaning of Derivative: Velocity and Acceleration

Basic Differentiation Formulas

Continuity versus Differentiability

The Product and Quotient Rule

Derivatives of Trigonometric Functions

The Chain Rule

The Proof of the Chain Rule

Implicit Differentiation

Related Rates

Derivatives of Logarithmic and Exponential Functions

Inverse Trigonometric Functions

Linear Approximations and Differentials

Maximum and Minimum

Mean Value Theorem

The First and Second Derivative Test

Optimization Problems

L'Hôpital's Rule

Antiderivatives

Areas under Curves

Definite Integral

The Fundamental Theorem of Calculus

Substitution Rule

Areas Between Curves

Applications of Definite Integrals: Volumes by Slicing

Applications of Definite Integrals: Solids of Revolution (The Disk Method)

Applications of Definite Integrals: Solids of Revolution (The Washer Method)

Applications of Definite Integrals: Volumes by Cylindrical Shells

Applications of Definite Integrals: Arc Length

Applications of Definite Integrals:Surfaces of Revolution

Integration by Parts

Trigonometric Integrals

Trigonometric Substitution

Inverse Trigonometric Functions

Integration of Rational Functions by Partial Fractions

Improper Integrals

Sequences

Infinite Series

Cauchy-Maclaurin Integral Test

The Ratio, Root and Comparison Tests

Alternating Series, Absolute and Conditional Convergence

Approximating Functions with Polynomials

Properties of Power Series

Taylor Series

Vectors

The Dot Product

The Cross Product

Lines and Planes in Space

Derivatives and Integrals of Vector-Valued Functions

Arc Length and Reparametrization

Calculus

Benjamin Crowell, Calculus

Michael Spivak, Calculus

Gilbert Strang, Calculus

Ellery Williams Davis and William Charles Brenke, The Calculus

Advanced Calculus

George A. Gibson, Advanced Calculus. An alternative copy is available here.

Francis B. Hildebrand, Advanced Calculus for Applications

David V. Widder, Advanced Calculus. An alternative copy is available here. (Much better scanned copy!)

Robert Wrede and Murray R. Spegel, Advanced Calculus, Schaum's Outlines

Ivan S. Sokolnikoff, Advanced Calculus

Tom M. Apostol, Calculus, Volum I. One-Variable Calculus, with an Introduction to Linear Algebra, 2nd Edition, John Wiley & Sons, 1967

Tom M. Apostol, Calculus, Volume II. Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability, 2nd Edition, John Wiley & Sons, 1969

Jerrold Marsden and Anthony Tromba, Vector Calculus, 5th Edition, W. H. Freeman and Company, 2003

Jerrold Marsden and Alan Weinstein, Calculus I, II, III, Second Edition, Springer, 1984

Jerrold Marsden and Alan Weinstein, Calculus Unlimited, The Benjamin/Cummings Publishing Company, Inc., 1981

Ivan S. Sokolnikoff, Advanced Calculus, McGraw-Hill Book Company, Inc., 1939