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### Introductory Calculus Notes by Ambar N. Sengupta

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Introductory Calculus Notes written by Ambar N. Sengupta cover the following topics.

• Sets: Language and Notation
Sets and Elements, Everything from nothing, Subsets, Union, Intersections, Complements, Integers and Rationals, Cartesian Products, Mappings and Functions, Sequences

• The Extended Real Line
The Real Line, The Extended Real Line

• Suprema, Infima, Completeness
Upper Bounds and Lower Bounds, Sup and Inf: Completeness, More on Sup and Inf

• Neighborhoods, Open Sets and Closed Sets
Intervals, Neighborhoods, Types of points for a set, Interior, Exterior, and Boundary of a Set, Open Sets and Topology, Closed Sets, Open Sets and Closed Sets, Closed sets in R and in R

• Magnitude and Distance
Absolute Value, Inequalities and equalities, Distance, Neighborhoods and distance

• Limits
Limits, Sup and Inf, Limits for 1/x, A function with no limits, Limits of sequences, Lim with sups and infs

• Limits: Properties
Up and down with limits, Limits: the standard definition, Limits: working rules, Limits by comparing, Limits of composite functions

• Trigonometric Functions
Measuring angles, Geometric specification of sin, cos and tan, Reciprocals of sin, cos, and tan, Identities, Inequalities, Limits for sin and cos, Limits with sin(1/x), Graphs of trigonometric functions, Postcript on trigonometric functions, Exercises on Limits

• Continuity
Continuity at a point, Discontinuities, Continuous functions, Two examples using Q, Composites of continuous functions, Continuity on R

• The Intermediate Value Theorem
Inequalities from limits, Intermediate Value Theorem, Intermediate Value Theorem: a second formulation, Intermediate Value Theorem: an application, Locating roots

• Inverse Functions
Inverse trigonometric functions, Monotone functions: terminology, Inverse functions

• Maxima and Minima
Maxima and Minima, Maxima/minima with infinities, Closed and bounded sets

• Tangents, Slopes and Derivatives
Secants and tangents, Derivative, Notation, The derivative of x^2, Derivative of x^3, Derivative of x^n, Derivative of x^−1 = 1/x, Derivative of x^−k = 1/x^k, Derivative of x^(1/2) =√x, Derivatives of powers of x, Derivatives with infinities

• Derivatives of Trigonometric Functions
Derivative of sin is cos, Derivative of cos is − sin, Derivative of tan is sec^2

• Differentiability and Continuity
Differentiability implies continuity

• Using the Algebra of Derivatives
Using the sum rule, Using the product rule, Using the quotient rule

• Using the Chain Rule
Initiating examples, The chain rule

• Proving the Algebra of Derivatives
Sums, Products, Quotients

• Proving the Chain Rule
Why it works, Proof the chain rule

• Using Derivatives for Extrema
Quadratics with calculus, Quadratics by algebra, Distance to a line, Other geometric examples, Exercises on Maxima and Minima

• Local Extrema and Derivatives
Local Maxima and Minima, Review Exercises

• Mean Value Theorem
Rolle’s Theorem, Mean Value Theorem, Rolle’s theorem on R

• The Sign of the Derivative
Positive derivative and increasing nature, Negative derivative and decreasing nature, Zero slope and constant functions

• Differentiating Inverse Functions
Inverses and Derivatives

• Analyzing local extrema with higher derivatives
Local extrema and slope behavior, The second derivative test

• Exp and Log
Exp summarized, Log summarized, Real Powers, Example Calculations, Proofs for Exp and Log

• Convexity
Convex and concave functions, Convexity and slope, Checking convexity/concavity, Inequalities from convexity/concavity, Convexity and derivatives, Supporting Lines, Convex combinations, Exercises on Maxima/Minima , Mean Value Theorem, Convexity

• L’Hospital’s Rule
Examples, Proving l’Hospital’s rule, Exercises on l’Hosptal’s rule

• Integration
From areas to integrals, The Riemann integral, Refining partitions, Estimating approximation error, Continuous functions are integrable, A function for which the integral does not exist, Basic properties of the integral

• The Fundamental Theorem of Calculus
Fundamental theorem of calculus, Differentials and integrals, Using the fundamental theorem, Indefinite integrals, Revisiting the exponential function

• Riemann Sum Examples
Riemann sums for N-1 dx/x^2, Riemann sums for 1/x, Riemann sums for x, Riemann sums for x^2, Power sums

• Integration Techniques
Substitutions, Some trigonometric integrals, Summary of basic trigonometric integrals, Using trigonometric substitutions, Integration by parts, Exercises on the Substitution Method

• Paths and Length
Paths, Lengths of paths, Paths and Curves, Lengths for graphs,

• Selected Solutions, Bibliography

• Preliminaries
A Short Note on Proofs, Sets and Equivalence Relations

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