Unit 1: Limits & Continuity

Limits

Definition of a Limit

Infinite Limits & Limits at Infinity

Properties of Limits

Evaluating Limits Analytically

Evaluating Limits Numerically & Graphically

Squeeze Theorem & Trigonometric Limits

Summary of Limits

Selecting Procedures for Determining Limits

Continuity

Continuity

Removable Discontinuities

Non-removable Discontinuities

Continuous Functions

Intermediate Value Theorem

Unit 2: Differentiation Definition & Fundamental Properties

Definition of Differentiation

Average Rate of Change

Instantaneous Rate of Change

Derivatives & Tangents

Estimating the Derivative at a Point

Differentiability & Continuity

Fundamental Properties of Differentiation

Derivative Rules

Derivatives of Exponentials and Logarithms

Derivatives of Sine and Cosine Functions

The Product Rule

The Quotient Rule

Derivatives of Tangent and Reciprocal Trig Functions

Unit 3: Differentiation of Composite, Implicit & Inverse Functions

Differentiation of Composite & Inverse Functions

The Chain Rule

The Inverse Function Theorem

Derivatives of Inverse Trigonometric Functions

Higher-Order Derivatives

Implicit Differentiation

Implicit Differentiation

Summary of Differentiation

Selecting Procedures for Calculating Derivatives

Unit 4: Contextual Applications of Differentiation

Rates of Change & Related Rates

Meaning of a Derivative in Context

Motion in a Straight Line

Related Rates

Linearization

Approximating Values of a Function

L'Hospital's Rule

L'Hospital's Rule

Unit 5: Analytical Applications of Differentiation

Graphs of Functions & Their Derivatives

Mean Value Theorem

Extreme Value Theorem

Critical Points

Increasing & Decreasing Functions

First Derivative Test for Local Extrema

Candidates Test for Global Extrema

Concavity of Functions

Second Derivative Test for Local Extrema

Graphs of f, f' & f''

Optimization Problems

Behaviors of Implicit Relations

Second Derivatives of Implicit Functions

Critical Points of Implicit Relations

Unit 6: Integration & Accumulation of Change

Integration & Antiderivatives

Indefinite Integrals

Derivatives & Antiderivatives

Indefinite Integral Rules

Constant of Integration

Riemann Sums & Definite Integrals

Accumulation of Change

Riemann Sums

Trapezoidal Sums

Accumulation Functions

Fundamental Theorem of Calculus

Properties of Definite Integrals

Evaluating Definite Integrals

Methods of Integration

Integrals of Composite Functions

Integration Using Substitution

Integration Using Completing the Square

Integration Using Long Division

Summary of Integration

Selecting Techniques for Integration

Unit 7: Differential Equations

First-Order Differential Equations

Introduction to Differential Equations

Slope Fields

Separation of Variables

Finding Particular Solutions

Exponential Models

Unit 8: Applications of Integration

Definite Integrals in Context

Average Value of a Function

Definite Integrals as Accumulated Change

Position, Velocity & Acceleration

Distance & Speed

Areas

Area Between a Curve & x-Axis

Area Between a Curve & y-Axis

Area Between Two Curves

Multiple Areas

Volumes with Cross Sections

Volumes from Areas of Known Cross Sections

Squares as Cross Sections

Rectangles as Cross Sections

Triangles as Cross Sections

Semicircles as Cross Sections

Volumes of Revolution

Disc Method Around the x-Axis

Disc Method Around the y-Axis

Disc Method Around Other Axes

Washer Method Around the x-Axis

Washer Method Around the y-Axis

Washer Method Around Other Axes