
Introduction to Calculus

Limits
 Introduction to Limits
 Determining Limits Graphically
 Formal Definition of a Limit at a Point
 Determining One Sided Limits
 Determining When a Limit does not Exist
 Determining Limits Algebraically
 Infinite Limits and Vertical Asymptotes
 Limits for The Squeeze Theorem
 Limits at Infinity and Horizontal Asymptotes
 Continuous Functions
 Intemediate Value Theorem
 Definition of Continuity at a Point
 Classifying Topics of Discontinuity (removable vs. nonremovable)

Derivatives
 Rate of Change of a Function
 Average Rate of Change Over an Interval
 Instantaneous Rate of Change at a Point
 Slope of a Curve at a Point
 Tangent Line to a Curve
 Normal Line to a Tangent
 Average Velocity
 Instantaneous Velocity
 Limit Definition of Derivative
 First Principles Example 1: x²
 First Principles Example 2: x³
 First Principles Example 3: square root of x
 Standard Notation and Terminology
 Differentiable vs. Nondifferentiable Functions

Basic Differentiation Rules

Differentiating Trigonometric Functions
 Limits Involving Trigonometric Functions
 Intuitive Approach to the derivative of y=sin(x)
 Derivative Rules for y=cos(x) and y=tan(x)
 Differentiating sin(x) from First Principles
 Special Limits Involving sin(x), x, and tan(x)
 Graphical Relationship Between sin(x), x, and tan(x), using Radian Measure
 Derivatives of y=sec(x), y=cot(x), y= csc(x)
 Differentiating Inverse Trigonometric Functions

Differentiating Exponential Functions

Differentiating Logarithmic Functions

Graphing with the First Derivative
 Interpreting the Sign of the First Derivative (Increasing and Decreasing Functions)
 Identifying Stationary Points (Critical Points) for a Function
 Identifying Turning Points (Local Extrema) for a Function
 Classifying Critical Points and Extreme Values for a Function
 Mean Value Theorem for Continuous Functions

Graphing with the Second Derivative
 Relationship between First and Second Derivatives of a Function
 Analyzing Concavity of a Function
 Notation for the Second Derivative
 Application of the Second Derivative (Acceleration)
 Determining Points of Inflection for a Function
 First Derivative Test vs Second Derivative Test for Local Extrema
 The special case of x⁴
 Critical Points of Inflection
 Examples of Curve Sketching

Applications of Derivatives

Introduction to Integration
 Sigma Notation
 Integration: the Area Problem
 Formal Definition of the Definite Integral
 Definite and indefinite integrals
 Integrals of Polynomial functions
 Determining Basic Rates of Change Using Integrals
 Integrals of Trigonometric Functions
 Integrals of Exponential Functions
 Integrals of Rational Functions
 Basic Properties of Definite Integrals
 The Fundamental Theorem of Calculus

Techniques of Integration

Using Integrals to Find Areas and Volumes

Methods of Approximating Integrals

Applications of Definite Integrals
 The Average Value of a Function
 Solving Separable Differential Equations
 Slope Fields
 Exponential Growth and Decay Models
 Logistic Growth Models
 Net Change: Motion on a Line
 Determining the Surface Area of a Solid of Revolution
 Determining the Length of a Curve
 Determining the Volume of a Solid of Revolution
 Determining Work and Fluid Force

Parametric Functions

Polar Curves

Tests of Convergence / Divergence
 Indeterminate Forms and de L'hospital's Rule
 Infinite Sequences
 Infinite Series
 Partial Sums of Infinite Series
 Geometric Series
 Harmonic Series
 Nth Term Test for Divergence of an Infinite Series
 Integral Test for Convergence of an Infinite Series
 Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series
 Direct Comparison Test for Convergence of an Infinite Series
 Limit Comparison Test for Convergence of an Infinite Series
 Ratio Test for Convergence of an Infinite Series
 Root Test for for Convergence of an Infinite Series
 Strategies to Test an Infinite Series for Convergence

Power Series
 Introduction to Power Series
 Determining the Radius and Interval of Convergence for a Power Series
 Differentiating and Integrating Power Series
 Constructing a Taylor Series
 Constructing a Maclaurin Series
 Lagrange Form of the Remainder Term in a Taylor Series
 Applications of Power Series
 Power Series Representations of Functions
 Power Series and Exact Values of Numerical Series
 Power Series and Estimation of Integrals
 Power Series and Limits
 Product of Power Series
 Binomial Series
 Power Series Solutions of Differential Equations