# Calculus

Contents# Topics in Calculus

### Differentiation of Functions

- Definition of the Derivative
- Basic Differentiation Rules
- Derivatives of Power Functions
- The Product and Quotient Rules
- Chain Rule
- Derivatives of Inverse Functions
- Derivatives of Trigonometric Functions
- Derivatives of Exponential and Logarithmic Functions
- Derivatives of Hyperbolic Functions
- Logarithmic Differentiation
- Derivatives of Parametric Functions
- Derivatives of Polar Functions
- Implicit Differentiation
- Higher-Order Derivatives
- Leibniz Formula
- Table of Derivatives
- Differential of a Function
- Higher-Order Differentials

### Applications of the Derivative

- Approximation by Differentials
- Monotonic Functions
- Local Extrema of Functions
- Global Extrema of Functions
- Convex Functions
- Inflection Points
- Asymptotes
- Curve Sketching
- Rolle’s Theorem
- Lagrange’s Mean Value Theorem
- Cauchy’s Mean Value Theorem
- Classical Inequalities
- Proving of Inequalities
- Tangent and Normal Lines
- Curvature and Radius of Curvature
- Evolute and Involute
- Envelope of a Family of Curves
- Osculating Curves
- Optimization Problems in Geometry
- Optimization Problems in Physics
- Optimization Problems in Economics
- Van der Waals Equation

### Integration of Functions

- The Indefinite Integral and Basic Formulas of Integration. Table of Integrals
- Change of Variable
- Integration by Parts
- Integration of Rational Functions
- Integration of Irrational Functions
- Integration of Rational Expressions of Trigonometric Functions
- Integration of Some Classes of Trigonometric Functions
- Integration of Hyperbolic Functions
- Using Trigonometric and Hyperbolic Substitutions
- The Definite Integral and Fundamental Theorem of Calculus
- Improper Integrals

### Double Integrals

- Definition and Properties of Double Integrals
- Iterated Integrals
- Double Integrals over Rectangular Regions
- Double Integrals over General Regions
- Change of Variables in Double Integrals
- Double Integrals in Polar Coordinates
- Geometric Applications of Double Integrals
- Physical Applications of Double Integrals

### Triple Integrals

- Definition and Properties of Triple Integrals
- Triple Integrals in Cartesian Coordinates
- Change of Variables in Triple Integrals
- Triple Integrals in Cylindrical Coordinates
- Triple Integrals in Spherical Coordinates
- Calculation of Volumes Using Triple Integrals
- Physical Applications of Triple Integrals

### Fourier Series

- Definition of Fourier Series and Typical Examples
- Fourier Series of Functions with an Arbitrary Period
- Even and Odd Extensions
- Complex Form of Fourier Series
- Convergence of Fourier Series
- Bessel’s Inequality and Parseval’s Theorem
- Differentiation and Integration of Fourier Series
- Applications of Fourier Series to Differential Equations
- Orthogonal Polynomials and Generalized Fourier Series