The visual guide to
AP® Calculus AB
8 units, 113 topics. Each topic is explained in writing, drawn as a lesson you can watch, and checked with practice questions.
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Each unit is as wide as its share of the exam's multiple-choice section.Filled: topics you've opened.
Unit 1Limits and Continuity
- Introducing Calculus: Change at an Instant
- Defining Limits and Using Limit Notation
- One-Sided Limits and When a Limit Fails to Exist
- Estimating Limit Values from Graphs
- Estimating Limit Values from Tables
- Determining Limits Using Algebraic Properties
- Determining Limits Using Algebraic Manipulation
- Rationalizing and Limits with Radicals
- Selecting Procedures for Determining Limits
- Determining Limits Using the Squeeze Theorem
- The Special Trigonometric Limits
- Connecting Multiple Representations of Limits
- Exploring Types of Discontinuities
- Defining Continuity at a Point
- Confirming Continuity over an Interval
- Removing Discontinuities
- Connecting Infinite Limits and Vertical Asymptotes
- Connecting Limits at Infinity and Horizontal Asymptotes
- End Behavior of Rational Functions
- Working with the Intermediate Value Theorem
Unit 2Differentiation: Definition and Fundamental Properties
- Defining Average and Instantaneous Rates of Change
- Secant Lines Becoming the Tangent Line
- Defining the Derivative and Using Derivative Notation
- The Derivative as a Function
- Estimating Derivatives of a Function at a Point
- Connecting Differentiability and Continuity
- Where Derivatives Fail to Exist
- Applying the Power Rule
- The Power Rule with Negative and Fractional Exponents
- Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- Derivatives of cos x, sin x, e^x, and ln x
- The Product Rule
- The Quotient Rule
- Finding the Derivatives of Tangent, Cotangent, Secant, and Cosecant
Unit 3Differentiation: Composite, Implicit, and Inverse Functions
- The Chain Rule
- The Chain Rule with Repeated Composition
- Implicit Differentiation
- Tangent Lines to Implicit Curves
- Differentiating Inverse Functions
- Differentiating Inverse Trigonometric Functions
- Selecting Procedures for Calculating Derivatives
- Derivatives of Exponentials and Logarithms with Other Bases
- Calculating Higher-Order Derivatives
- What the Second Derivative Means for Motion
Unit 4Contextual Applications of Differentiation
- Interpreting the Meaning of the Derivative in Context
- Units of a Derivative
- Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- Speeding Up and Slowing Down
- Rates of Change in Applied Contexts Other Than Motion
- Introduction to Related Rates
- Solving Related Rates Problems
- Related Rates with Shadows and Similar Triangles
- Approximating Values Using Local Linearity and Linearization
- Over- and Under-Estimates from Concavity
- Using L'Hospital's Rule for Indeterminate Forms
Unit 5Analytical Applications of Differentiation
- Using the Mean Value Theorem
- Rolle's Theorem and the Hypotheses That Matter
- Extreme Value Theorem, Extrema, and Critical Points
- Determining Intervals on Which a Function Is Increasing or Decreasing
- Using the First Derivative Test to Determine Relative Extrema
- Using the Candidates Test to Determine Absolute Extrema
- Determining Concavity of Functions over Their Domains
- Points of Inflection
- Using the Second Derivative Test to Determine Extrema
- Sketching Graphs of Functions and Their Derivatives
- Connecting a Function, Its First Derivative, and Its Second Derivative
- Reading a Graph of the Derivative
- Introduction to Optimization Problems
- Solving Optimization Problems
- Optimization with a Constraint Equation
- Exploring Behaviors of Implicit Relations
Unit 6Integration and Accumulation of Change
- Exploring Accumulations of Change
- Approximating Areas with Riemann Sums
- Left, Right, and Midpoint Sums
- Trapezoidal Sums
- Riemann Sums, Summation Notation, and Definite Integral Notation
- The Fundamental Theorem of Calculus and Accumulation Functions
- Differentiating an Accumulation Function
- Interpreting the Behavior of Accumulation Functions
- Applying Properties of Definite Integrals
- The Fundamental Theorem of Calculus and Definite Integrals
- Finding Antiderivatives and Indefinite Integrals
- Antiderivatives of Trigonometric and Exponential Functions
- Integrating Using Substitution
- Substitution in Definite Integrals
- Integrating Functions Using Long Division and Completing the Square
- Selecting Techniques for Antidifferentiation
Unit 7Differential Equations
- Modeling Situations with Differential Equations
- Verifying Solutions for Differential Equations
- Sketching Slope Fields
- Reasoning Using Slope Fields
- Matching a Slope Field to Its Differential Equation
- Finding General Solutions Using Separation of Variables
- Finding Particular Solutions Using Initial Conditions
- The Domain of a Particular Solution
- Exponential Models with Differential Equations
- Newton's Law of Cooling and Related Contexts
Unit 8Applications of Integration
- Finding the Average Value of a Function on an Interval
- The Mean Value Theorem for Integrals
- Connecting Position, Velocity, and Acceleration Using Integrals
- Recovering Position from Velocity
- Using Accumulation Functions and Definite Integrals in Applied Contexts
- Net Change and In-Out Rate Problems
- Finding the Area Between Curves Expressed as Functions of x
- Finding the Area Between Curves Expressed as Functions of y
- Finding the Area Between Curves That Intersect at More Than Two Points
- Volumes with Cross Sections: Squares and Rectangles
- Volumes with Cross Sections: Triangles and Semicircles
- Volume with the Disc Method: Revolving Around the x- or y-Axis
- Volume with the Disc Method: Revolving Around Other Axes
- Volume with the Washer Method: Revolving Around the x- or y-Axis
- Volume with the Washer Method: Revolving Around Other Axes
- Choosing dx or dy for a Volume Integral