Calculus

Concept Map

Concepts
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Limit

Concept of Limit

Evaluate Limits using Graphs

Evaluate Limits using Tables

Limits of Composite Functions

Evaluate Limits Using Substitution

Limits of Polynomial and Rational Functions

Basic Trigonometric Limits

Limits of Exponential and Logarithmic Functions

Limits Involving Radical Functions

Continuity

Continuity of a Function

Continuity at a Point, Continuity Test, Types of Discontinuity

Differentiability at a Point

Intermediate Value Theorem, Existence of Solutions

Find Imaginary Solutions

Differentiation

Derivatives

Slope Models using Derivatives

Average and Instantaneous Rates of Change

Slope of Tangent Line

Tangent Line Approximation

Differentiation Rules and Formulas

Constant, Identity, and Power Rules

Sum and Difference Differentiation Rules

Products and Quotients Differentiation Rules

Derivatives of Trigonometric Functions

Chain Rule

Implicit Differentiation

Multi-Step Differentiation

Higher Order Derivatives

Related Rates

Derivatives of Exponential Functions

Increasing and Decreasing Values and Extrema

Derivatives with Mean Value Theorem and Rolle's Theorem

Absolute versus Local Extrema

First Derivative Test

Second Derivative Test

Concavity and Inflection

Derivative Applications

Absolute Extrema and Optimization

Newton's Method

Analyzing the Graphs of Functions

Integral

Area Under the Curve

Antiderivative

Integration of Standard Functions

Area Computations

Area Sums

Probability and Probability Density Functions

Trapezoidal and Midpoint Approximations

Riemann Sum

Definite Integrals and Integration Techniques

Definite Integrals

U Substitution

Properties of Definite Integrals

Fundamental Theorem of Calculus

Integration with Mean Value Theorem

Simpson's Rule

Area Between Curves

Length of a Plane Curve

Solid of Revolution

Introduction to Solids of Revolution

Volume by Cross Section

Volume by Disks

Volume by Washers

Method of Cylindrical Shells

Basic Formula of Areas of Surfaces of Revolution

Definite Integral Application

Fluid Pressure

Work and Force

Integration by Parts

One Pass Integration by Parts

Multiple Pass Integration by Parts

Recursive Integration by Parts

Trigonometric Integrals

Powers of Sines and Cosines

Powers of Secants and Tangents

Products of Sines and Cosines of Different Angles

Trigonometric Substitution

Other Integral Concepts

Partial Fraction Decomposition

Convergence and Divergence of Integrals

Improper Integrals and Infinite Discontinuities

Differential Equations

General and Particular Solutions

Introduction to Differential Equations

Formation of Differential Equations

Euler's Method

Slope Fields and Isoclines

Differential Equations and Integration

Solving Separable First-Order Differential Equations

Logistic Functions

Differential Equations Representing Growth and Decay

Functions

Graphs and Functions

Identify Functions and the Vertical Line Test

Even and Odd Functions and Function Symmetry

Domain and Range of a Function

Restricted Domain and Range

Domain, Range, and Signs of Trigonometric Functions

Zeroes and Intercepts of Polynomials

Quadratic and Exponential Equations and Functions

Linear, Quadratic, and Cubic Models

Exponential Properties in Variable Expressions

Exponential Properties Involving Products

Exponential Properties Involving Quotients

Exponential Terms Raised to an Exponent

Solving Equations with Exponents

Exponential Growth and Decay

Exponential Growth

Exponential Decay

Geometric Sequences and Exponential Functions

Graphs of Exponential Functions

Applications of Exponential Functions

Compound Interest

The Number e

Compound Interest per Year

Compound Interest per Period

Definition of Inverse Functions

Existence (One-to-One, Monotonic)

Finding Inverse Functions

Continuity and Differentiability

Derivative Rule for Inverses

Properties of Logarithms

Logarithm Properties

Change of Base

Product and Quotient Properties of Logarithms

Power Property of Logarithms

Inverse Properties of Logarithms

Logarithmic Function

Logarithmic Functions

Common and Natural Logarithms

Analysis of Logarithmic Graphs

Solving Logarithmic Equations

Logarithmic Models

Logarithmic Differentiation

Logarithmic Integration

Inverse Trigonometric Functions

Inverse Trig Functions

Properties of Inverse Trigonometric Functions

Analyzing Graphs of Inverse Trigonometric Functions

Derivatives of Six Inverse Trigonometric Properties

Integration of Inverse Trigonometric Functions by Substitution

Hyperbolic Functions

Hyperbolic Function

Derivatives and Integrals of Hyperbolic Functions

Inverse Hyperbolic Functions

Derivatives and Integrals of Inverse Hyperbolic Functions

Sequences and Series

Sequences and Series in Calculus

Recursive Formulas

Explicit Formulas

Sum Notation and Properties of Sigma

Mathematical Induction

Inductive Reasoning from Patterns

Proof by Induction

Induction and Factors

Induction and Inequalities

Sequences

Sequence

Arithmetic Progression

Arithmetic Sequences

Finding the nth Term Given the Common Difference and a Term

Finding the nth Term Given Two Terms for an Arithmetic Sequence

Geometric Sequence

Geometric Sequences

Finding the nth Term Given the Common Ratio and the First Term

Finding the nth Term Given Two Terms for a Geometric Sequence

Geometric Mean

Sequence Limits

Limit of a Sequence

Convergence and Divergence of Sequences

L'Hopital's Rule

Series

Infinite Series and Series Sums

Infinite Polynomials

Sequence of Partial Sums

Series Sums and Gauss' Formula

Problem Solving with Series Sums

Alternating Series

Arithmetic Series

Arithmetic Series Sums

Sums of Finite Arithmetic Series

Geometric Series

Sums of Geometric Series

Sums of Finite Geometric Series

Sums of Infinite Geometric Series

Special Series

Convergence and Divergence

Limit Comparison Test, Simplified Limit Comparison Test

Alternating Series Test

Absolute and Conditional Convergence

Summary of Procedures for Determining Convergence

Power Series and Convergence

Series Multiplication of Power Series

Taylor and Maclaurin Polynomials

Maclaurin Series Truncation Error

Calculations with Series

Binomial Series for Powers

Choosing Centers

Conic Section

Conic Sections

Conic Sections and Dandelin Spheres

Ellipse

Ellipses

Ellipses Centered at the Origin

Equation of an Ellipse

Ellipses Not Centered at the Origin

Focal Property of Ellipses

Parabola

Parabolas

Parabolas with Vertex at the Origin

Parabolic Functions in Focus Form

Parabolas and the Distance Formula

Parabolas with any Vertex

Parabolas and Analytic Geometry

Applications of Parabolas

Hyperbola

Hyperbolas

Graphs of Hyperbolas Centered at the Origin

Equations of Hyperbolas Centered at the Origins

Hyperbolas with any Center

Hyperbola Equations and the Focal Property

Hyperbolas and Asymptotes

Classify Conic Sections

General Forms of Conic Sections

Classifying Conic Sections

Circle

Equations of Circles

Circles Centered at the Origin

Circles Not Centered at the Origin