- #1
- 22,183
- 3,322
- Author: Serge Lang
- Title: A First Course in Calculus
- Amazon Link: https://www.amazon.com/dp/0387962018/?tag=pfamazon01-20
- Prerequisities: High-school mathematics
- Level: Undergrad
Table of Contents:
Code:
[LIST]
[*] Review of Basic Material
[LIST]
[*] Numbers and Functions
[LIST]
[*] Integers, rational numbers, and real numbers
[*] Inequalities
[*] Functions
[*] Powers
[/LIST]
[*] Graphs and Curves
[LIST]
[*] Coordinates
[*] Graphs
[*] The straight line
[*] Distance between two points
[*] Curves and equations
[*] The circle
[*] The parabola. Changes of coordinates
[*] The hyperbola
[/LIST]
[/LIST]
[*] Differentiation and Elementary Functions
[LIST]
[*] The Derivative
[LIST]
[*] The slope of a curve
[*] The derivative
[*] Limits
[*] Powers
[*] Sums, products, and quotients
[*] The chain rule
[*] Higher derivatives
[*] A Rate of change
[/LIST]
[*] Sine and Cosine
[LIST]
[*] The sine and cosine functions
[*] The graphs
[*] Addition formula
[*] The derivatives
[*] Two basic limits
[/LIST]
[*] The Mean Value Theorem
[LIST]
[*] The maximum and minimum theorem
[*] The mean value theorem
[*] Increasing and decreasing functions
[/LIST]
[*] Sketching Curves
[LIST]
[*] Behavior as x becomes very large
[*] Curve sketching
[*] Convexity
[*] Polar coordinates
[*] Parametric curves
[/LIST]
[*] Inverse Functions
[LIST]
[*] Definition of inverse functions
[*] Derivative of inverse functions
[*] The arcsine
[*] The arctangent
[/LIST]
[*] Exponents and Logarithms
[LIST]
[*] The logarithm
[*] The exponential function
[*] The general exponential function
[*] Order of magnitude
[*] Some applications
[/LIST]
[/LIST]
[*] Integration
[LIST]
[*] Integration
[LIST]
[*] The indefinite integral
[*] Continuous functions
[*] Area
[*] Fundamental theorem
[*] Upper and lower sums
[*] The basic properties
[*] Integrable functions
[/LIST]
[*] Properties ot the Integral
[LIST]
[*] Further connection with the derivative
[*] Sums
[*] Inequalities
[*] Improper integrals
[/LIST]
[*] Techniques of Integration
[LIST]
[*] Substitution
[*] Integration by parts
[*] Trigonometric integrals
[*] Partial fractions
[/LIST]
[*] Some Substantial Exercises
[LIST]
[*] An estimate for (n!)^{l/n}
[*] Stirling's formula
[*] Wallis' product
[/LIST]
[*] Applications of Integration
[LIST]
[*] Length of curves
[*] Area in polar coordinates
[*] Volumes of revolution
[*] Work
[*] Density and mass
[*] Probability
[*] Moments
[/LIST]
[/LIST]
[*] Series
[LIST]
[*] Taylor's Formula
[LIST]
[*] Taylor's formula
[*] Estimate for the remainder
[*] Trigonometric functions
[*] Exponential function
[*] Logarithm
[*] The arctangent
[*] The binomial expansion
[*] Uniqueness theorem
[/LIST]
[*] Series
[LIST]
[*] Convergent series
[*] Series with positive terms
[*] The ratio test
[*] The integral test
[*] Absolute and alternating convergence
[*] Power series
[*] Differentiation and integration of power series
[/LIST]
[/LIST]
[*] Miscellaneous
[LIST]
[*] Complex Numbers
[LIST]
[*] Definition
[*] Polar form
[*] Complex valued functions
[/LIST]
[*] Appendix: \varepsilon and \delta
[LIST]
[*] Least upper bound
[*] Limits
[*] Points of accumulation
[*] Continuous functions
[/LIST]
[*] Appendix: Induction
[*] Appendix: Sine and Cosine
[*] Appendix: Physics and Mathematics
[/LIST]
[*] Functions of Several Variables
[LIST]
[*] Vectors
[LIST]
[*] Definition of points in n-space
[*] Located vectors
[*] Scalar product
[*] The norm of a vector
[*] Lines and planes
[/LIST]
[*] Differentiation of Vectors
[LIST]
[*] Derivative
[*] Length of curves
[/LIST]
[*] Functions of Several Variables
[LIST]
[*] Graphs and level curves
[*] Partial derivatives
[*] Differentiability and gradient
[/LIST]
[*] The Chain Rule and the Gradient
[LIST]
[*] The chain rule
[*] Tangent plane
[*] Directional derivative
[*] Conservation law
[/LIST]
[/LIST]
[*] Answers
[*] Index
[/LIST]
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