# Calculus Topics

## 1. Single-Variable Functions, Limits and Continuity

- Different Types of Functions
- Symmetric Functions
- Increasing and Decreasing Functions
- Introduction to Limits
- Limit Laws
- One-Sided Limits
- Limits to Infinity
- Limits to Determine Vertical Asymptotes
- Limits at Infinity
- Limits to Determine Horizontal Asymptotes
- Squeeze Theorem
- Continuity of a Function
- Types of Discontinuities

## 2. Derivatives of Single-Variable Functions

- Definition of a Derivative
- Derivative Notation
- Differentiability of a Function
- Basic Laws for Differentiation
- Derivatives of Polynomials
- Derivatives of Trigonometric Functions
- Derivatives of Exponential Functions
- Derivatives of Logarithmic Functions
- Derivative Product Rule
- Derivative Quotient Rule
- Derivative Chain Rule
- Derivatives of Even and Odd Functions
- Higher Order Differentiation
- Implicit Differentiation
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Reciprocal Trigonometric Functions
- Derivatives of Inverse Reciprocal Trigonometric Functions

## 3. Extreme Values and Applications of Derivatives

- Local Maxima and Minima, and Absolute Maxima and Minima
- Fermat's Theorem for Extrema
- Concavity and Inflection Points of a Function
- The First Derivative Test
- The Second Derivative Test
- Linear Approximation of Single Variable Functions
- The Intermediate Value Theorem
- The Existence of Roots Theorem
- The Extreme Value Theorem
- Rolle's Theorem
- The Mean Value Theorem
- Related Rates Applications
- Optimization with Derivatives
- Biology Application - Population Growth Curves
- Physics Application - Displacement, Acceleration, Velocity, and Jerk Applications
- Economics Application - Cost Functions

## 4. Parametric Curves

- Indeterminate Forms
- L'Hospital's Rule ( Indeterminate Quotients | Indeterminate Differences | Indeterminate Products | Indeterminate Powers )
- Parametric Curves
- Eliminating Parameters in Parametric Curves
- The Cycloid
- Derivatives for Parametric Curves
- Horizontal and Vertical Tangents of Parametric Curves
- Concavity of Parametric Curves
- Polar Coordinate Systems
- Polar Curves
- Families of Polar Curves, r = asinΘ and r = acosΘ
- Families of Polar Curves, r = sin(bΘ) and r =cos(bΘ)
- Families of Polar Curves, r = atan(Θ) and r = tan(bΘ)
- Sketching Polar Curves Examples
- Symmetry in Polar Curves
- Derivatives for Polar Curves

## 5. Integration

- Antiderivatives
- Defining the Integral and Riemann Sums
- Definite Integrals
- Properties of Definite Integrals
- The Fundamental Theorem of Calculus Part 1
- The Fundamental Theorem of Calculus Part 2
- Indefinite Integrals
- Properties of Indefinite Integrals
- Indefinite Integrals of Polynomials
- Indefinite Integrals of Trigonometric Functions
- Indefinite Integrals of Exponential Functions
- Indefinite Integrals of Logarithmic Functions
- Indefinite Integrals of Inverse Trigonometric Functions
- U-Substitution of Indefinite Integrals ( Examples 1 | Examples 2 )
- U-Substitution of Definite Integrals
- Integration by Parts of Indefinite Integrals ( Examples 1 | Examples 2 )
- Integration by Parts of Definite Integrals
- Tabular Integration
- Reduction Formulas
- Advanced Trigonometric Function Integration ( Examples 1 | Examples 2 )
- The Indefinite Integrals of Secant and Cosecant
- Integration by Trigonometric Substitution ( Examples 1 | Examples 2 )
- Long Division of Improper Rational Functions
- Integration with Partial Fractions ( Examples 1 | Examples 2 )
- Strategy Summary for Integration

## 6. Computing Areas, Volumes, and Arc Length with Integration

- Calculating Areas Bounded by the x-Axis
- Calculating Areas Bounded by the y-Axis
- Areas Between Curves ( Examples 1 )
- Areas Under Parametric Curves ( Examples 1 )
- Areas Enclosed by Polar Curves
- Calculating Volumes - Washer/Disk Method ( Examples 1 | Examples 2 )
- Calculating Volumes - Cylindrical Shell Method ( Examples 1 ) )
- Volumes of Geometric Shapes ( Examples 1 )
- Improper Integrals ( Examples 1 | Examples 2 )
- Comparison Test for Improper Integral Convergence/Divergence ( Examples 1 )
- Arc Length of a Curve ( Examples 1 )
- Arc Length of a Parametric Curve

## 7. Sequences

- Proving the Existence of Limits ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 )
- Proving the Existence of Limits to Infinity
- Proving the Existence of Limits at Infinity
- Proving an Incorrect Limit of a Function
- Sequences ( Examples 1 )
- Limit of a Sequence ( Examples 1 )
- Uniqueness of a Convergent Sequence's Limit
- Proof that Convergent Sequences are Bounded
- The Monotonic Sequence Theorem for Convergence
- Limit Sum/Difference Laws for Convergent Sequences
- Limit Product/Quotient Laws for Convergent Sequences
- Limit Constant Multiple/Power Laws for Convergent Sequences
- The Squeeze Theorem for Convergent Sequences ( Examples 1
- Comparison Theorems for Sequences
- Evaluating Limits of Sequences ( Examples 1 | Examples 2 )
- Evaluating Limits of Recursive Sequences ( Examples 1 | Examples 2 )
- Sequence Convergence and Divergence Proofs
- Sequences Review

## 8. Infinite Series

- Series
- Geometric Series
- Computing the Sum of a Geometric Series ( Examples 1 )
- The Harmonic Series
- Telescoping Series ( Examples 1 | Examples 2 )
- Properties of Convergent Series
- Convergence and Divergence Theorems for Series
- Series Convergence and Divergence Proofs
- Infinite Series Review
- The Divergence Theorem for Series
- The Integral Test for Positive Series ( Examples 1 | Examples 2 )
- The p-Series Test ( Examples 1 )
- The Comparison Test for Positive Series ( Examples 1 | Examples 2 )
- The Limit Comparison Test for Positive Series ( Examples 1 | Examples 2 )
- The Ratio Test for Positive Series ( Examples 1 | Examples 2 )
- The Root Test for Positive Series ( Examples 1 | Examples 2 )
- Absolute and Conditional Convergence ( Examples 1 )
- The Alternating Series Test ( Examples 1 | Examples 2 )
- Intervals of Absolute and Conditional Convergence of a Series
- Applying Convergence and Divergence Tests for Series ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 )
- Error Estimation for Approximating Alternating Series ( Examples 1 )
- Rearrangement of Terms in Convergent Series

## 9. Power Series

- Power Series
- The Radius of Convergence of a Power Series ( Examples 1 | Examples 2 | Examples 3 )
- Algebraic Operations of Power Series ( Examples 1 )
- Cauchy Product of Power Series
- Determining Power Series Representations of Functions ( Examples 1 | Examples 2 )
- Differentiation and Integration of Power Series ( Examples 1 )
- Abel's Theorem
- Power Series for Functions in Powers of x
- Power Series for Functions in Powers Other than x ( Examples 1 )
- Determining a Function Representing a Power Series ( Examples 1 | Examples 2 )
- Computing Sums with Power Series
- Taylor and Maclaurin Polynomials
- Approximating Functions with Taylor and Maclaurin Polynomials
- Taylor and Maclaurin Series ( Examples 1 )
- Frequently Used Maclaurin Series
- Maclaurin Series for The Natural Exponential Function
- Maclaurin Series for Sine and Cosine
- Maclaurin Series of Combinations of Functions
- Taylor Series of Combinations of Functions ( Examples 1 )
- Taylor's Theorem and The Lagrange Remainder ( Examples 1 | Examples 2 )
- Estimation with Taylor Polynomials and Error Bounds
- The Integral Remainder ( Examples 1 )

# 10. Geometry in 3-Space

- The Cylindrical Coordinate System
- The Spherical Coordinate System
- Geometry in Three-Dimensional Space
- The Distance Between Two Points in Three Dimensional Space
- Sets of Points Describing Surfaces
- Introduction to Vectors
- The Dot Product of Vectors
- Vector Projections
- 2 x 2 and 3 x 3 Determinants
- The Cross Product of Vectors in Three-Dimensional Space
- The Scalar Triple Product of Vectors in Three-Dimensional Space
- Equations of Planes in Three-Dimensional Space
- Equations of Lines in Three-Dimensional Space
- Distances Between Linear Objects in Three-Dimensional Space
- Quadric Surfaces

## 11. Vector-Valued Functions

- Vector-Valued Functions
- Classification of Curves
- Sketching Space Curves in Three-Dimensional Space
- Limits of Vector-Valued Functions ( Examples 1 )
- Continuity of Vector-Valued Functions
- Plane Curves and Space Curves
- Derivatives of Vector-Valued Functions
- Derivative Rules for Vector-Valued Functions ( Examples 1 )
- Integrals of Vector-Valued Functions
- Vector-Valued Functions - Velocity, Acceleration, and Speed
- Parameterization of Curves in Three-Dimensional Space ( Examples 1 )
- Arc Length of Curves in Three-Dimensional Space ( Examples 1 | Examples 2 )
- Arc Length Parameterization of Curves in Three-Dimensional Space
- Unit Tangent Vectors to a Space Curve ( Examples 1 | Examples 2 )
- Curvature at a Point on a Curve ( Examples 1 | Examples 2 | Examples 3 )
- Curvature at a Point on a Single Variable Real-Valued Function ( Examples 1 )
- The Curvature of Straight Lines and Circles
- The Curvature of Plane Polar Curves
- The Radius of Curvature at a Point on a Curve
- Unit Normal and Unit Binormal Vectors to a Space Curve
- Method for Calculating Unit Normal and Unit Binormal Vectors ( Examples 1 )
- The Osculating Circle at a Point on a Curve
- The Evolute of a Curve ( Examples 1 )
- Normal, Rectifying, and Osculating Planes
- Determining Equations of Normal, Rectifying, and Osculating Planes ( Examples 1 )
- Torsion at a Point on a Curve ( Examples 1 )
- The Frenet-Serret Formulas
- The Fundamental Theorem of Space Curves

## 12. Multivariable Functions

- Functions of Several Variables
- The Domain of a Function of Several Variables
- Level Curves and Contour Plots
- Sets of Points in One, Two, and Three Dimensions
- Limits of Functions of Two Variables ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
- Proving Limits of Functions of Two Variables
- Limits of Functions of Three Variables
- Limit Laws for Functions of Several Variables ( Examples 1 )
- Continuity of Functions of Several Variables ( Examples 1 | Examples 2 )
- Discontinuities in Functions of Several Variables

## 13. Partial Differentiation

- Partial Derivatives ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )
- Implicit Partial Differentiation
- Higher Order Partial Derivatives
- Clairaut's Theorem on Higher Order Partial Derivatives ( Examples 1 )
- The Laplace Equations
- Tangent Planes to Surfaces
- Finding a Tangent Plane on a Surface ( Examples 1 | Examples 2 | Examples 3 )
- Normal Lines on a Surface ( Examples 1 )
- Linear Approximations of Functions of Several Variables
- Differentiability of Functions of Several Variables
- The Chain Rule Type 1 for Functions of Several Variables ( Examples 1 )
- The Chain Rule Type 2 for Functions of Several Variables
- Applying The Chain Rule to Functions of Several Variables ( Examples 1 | Examples 2 )
- Positively Homogenous Functions of Several Variables
- The Implicit Differentiation Formulas ( Examples 1 )
- Systems of Multivariable Equations
- Jacobian Determinants ( Examples 1 )
- The Implicit Function Theorem ( Examples 1 | Examples 2 )
- Directional Derivatives ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 )
- Higher Order Directional Derivatives
- The Gradient of Functions of Several Variables
- The Perpendicularity of The Gradient at a Point on a Level Curve
- The Perpendicularity of The Gradient at a Point on a Level Surface
- The Maximum Rate of Change at a Point on a Function of Several Variables ( Examples 1 )
- Tangent Planes to Level Surfaces ( Examples 1 | Examples 2 )
- Normal Lines to Level Surfaces ( Examples 1 | Examples 2 )

## 14. Extreme Values of Multivariable Functions

- Maximum and Minimum Values of Functions of Several Variables
- Determining Extreme Values of Functions of Several Variables
- The Extreme Value Theorem for Functions of Several Variables
- Analytically Determining Extreme Values for Functions of Several Variables
- Definite, Semi-Definite, and Indefinite Matrices
- Hessian Matrices ( Examples 1 )
- The Second Derivatives Test for Functions of Two Variables ( Examples 1 | Examples 2 )
- The Second Derivatives Test for Functions of Several Variables
- Lagrangian Functions
- The Method of Lagrange Multipliers
- Lagrange Multipliers with One Constraint ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 | Examples 6 )
- Lagrange Multipliers with Two Constraints ( Examples 1 | Examples 2 | Examples 3 | Examples 4 )

## 15. Multiple Integrals

- Partial Integration
- Double Integrals over Rectangles
- Double Integrals over General Domains
- Properties of The Double Integral
- Iterated Integrals
- Fubini's Theorem and Evaluating Double Integrals over Rectangles ( Examples 1 )
- Evaluating Double Integrals over General Domains ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 | Examples 6 )
- The Mean Value Theorem for Double Integrals
- Improper Double Integrals ( Examples 1 | Examples 2 )
- Evaluating Double Integrals in Polar Coordinates ( Examples 1 | Examples 2 | Examples 3 )
- Change of Variables in Double Integrals ( Examples 1 | Examples 2 )
- Computing Surface Areas with Double Integrals
- The Gamma and Beta Functions
- Triple Integrals over Boxes
- Fubini's Theorem for Evaluating Triple Integrals over Boxes ( Examples 1 )
- Triple Integrals over General Domains ( Examples 1 | Examples 2 )
- Changing The Order of Integration in Triple Integrals
- Evaluating Triple Integrals in Cylindrical Coordinates
- Evaluating Triple Integrals in Spherical Coordinates
- Change of Variables in Triple Integrals ( Examples 1 )

## 16. Vector Fields, Line Integrals, and Surface Integrals

- Vector Fields
- Gradient Fields
- Conservative Vector Fields ( Examples 1 | Examples 2 | Examples 3 )
- Equipotential Curves and Surfaces
- Line Integrals ( Examples 1 | Examples 2 | Examples 3 )
- Line Integrals on Piecewise Smooth Curves
- Line Integrals with Respect to Specific Variables ( Examples 1 )
- Line Integrals of Vector Fields ( Examples 1 )
- The Fundamental Theorem for Line Integrals
- Line Integrals of Nonconservative Vector Fields
- Line Integrals Review
- Connected and Simply Connected Domains
- Independence of Path
- Independence of Path Theorem
- Parametric Surfaces
- Parameterizing Surfaces ( Examples 1 )
- Tangent Planes to Parametric Surfaces
- Surface Integrals ( Examples 1 | Examples 2 | Examples 3 | Examples 4 | Examples 5 )
- Surface Integrals over Basic Surfaces ( Examples 1 | Examples 2 )
- Computing Surface Area with Surface Integrals ( Examples 1 | Examples 2 )
- Orientable Surfaces
- Surface Integrals of Vector Fields ( Examples 1 | Examples 2 )
- Surface Integrals Review
- The Divergence of a Vector Field ( Examples 1 | Examples 2 )
- The Curl of a Vector Field ( Examples 1 | Examples 2 )
- The Divergence and Curl of a Vector Field in Two-Dimensions
- Properties of The Divergence and Curl of a Vector Field
- The Curl of Conservative Vector Fields
- Divergence Identities
- Curl Identities
- Green's Theorem ( Examples 1 | Examples 2 )
- Stokes' Theorem ( Examples 1 | Examples 2 )
- The Divergence Theorem

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