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AP Calculus BC study notes
Short notes for every topic in AP Calculus BC, unit by unit, each with key terms and a sample AP-style question.
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Study AP Calculus BC free in APrepUnit 1: Limits & Continuity
- 1.1Introducing Calculus: Can Change Occur at an Instant?
- 1.2Defining Limits and Using Limit Notation
- 1.3Estimating Limit Values from Graphs
- 1.4Estimating Limit Values from Tables
- 1.5Determining Limits Using Algebraic Properties of Limits
- 1.6Determining Limits Using Algebraic Manipulation
- 1.7Selecting Procedures for Determining Limits
- 1.8Determining Limits Using the Squeeze Theorem
- 1.9Connecting Multiple Representations of Limits
- 1.10Exploring Types of Discontinuities
- 1.11Defining Continuity at a Point
- 1.12Confirming Continuity over an Interval
- 1.13Removing Discontinuities
- 1.14Connecting Infinite Limits and Vertical Asymptotes
- 1.15Connecting Limits at Infinity and Horizontal Asymptotes
- 1.16Working with the Intermediate Value Theorem (IVT)
Unit 2: Differentiation: Definition & Rules
- 2.1Defining Average and Instantaneous Rate of Change at a Point
- 2.2Defining the Derivative of a Function and Using Derivative Notation
- 2.3Estimating Derivatives of a Function at a Point
- 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
- 2.5Applying the Power Rule
- 2.6Derivative Rules: Constant, Sum, Difference, and Constant Multiple
- 2.7Derivatives of cos x, sin x, e^x, and ln x
- 2.8The Product Rule
- 2.9The Quotient Rule
- 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
Unit 3: Composite, Implicit & Inverse Derivatives
Unit 4: Contextual Applications of Differentiation
- 4.1Interpreting the Meaning of the Derivative in Context
- 4.2Straight-Line Motion: Connecting Position, Velocity, and Acceleration
- 4.3Rates of Change in Applied Contexts Other Than Motion
- 4.4Introduction to Related Rates
- 4.5Solving Related Rates Problems
- 4.6Approximating Values of a Function Using Local Linearity and Linearization
- 4.7Using L'Hopital's Rule for Determining Limits of Indeterminate Forms
Unit 5: Analytical Applications of Differentiation
- 5.1Using the Mean Value Theorem
- 5.2Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
- 5.3Determining Intervals on Which a Function is Increasing or Decreasing
- 5.4Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.5Using the Candidates Test to Determine Absolute (Global) Extrema
- 5.6Determining Concavity of Functions over Their Domains
- 5.7Using the Second Derivative Test to Determine Extrema
- 5.8Sketching Graphs of Functions and Their Derivatives
- 5.9Connecting a Function, Its First Derivative, and Its Second Derivative
- 5.10Introduction to Optimization Problems
- 5.11Solving Optimization Problems
- 5.12Exploring Behaviors of Implicit Relations
Unit 6: Integration & Accumulation of Change
- 6.1Exploring Accumulations of Change
- 6.2Approximating Areas with Riemann Sums
- 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6Applying Properties of Definite Integrals
- 6.7The Fundamental Theorem of Calculus and Definite Integrals
- 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9Integrating Using Substitution
- 6.10Integrating Functions Using Long Division and Completing the Square
- 6.11Integrating Using Integration by Parts
- 6.12Integrating Using Linear Partial Fractions
- 6.13Evaluating Improper Integrals
- 6.14Selecting Techniques for Antidifferentiation
Unit 7: Differential Equations
- 7.1Modeling Situations with Differential Equations
- 7.2Verifying Solutions for Differential Equations
- 7.3Sketching Slope Fields
- 7.4Reasoning Using Slope Fields
- 7.5Approximating Solutions Using Euler's Method
- 7.6Finding General Solutions Using Separation of Variables
- 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
- 7.8Exponential Models with Differential Equations
- 7.9Logistic Models with Differential Equations
Unit 8: Applications of Integration
- 8.1Finding the Average Value of a Function on an Interval
- 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4Finding the Area Between Curves Expressed as Functions of x
- 8.5Finding the Area Between Curves Expressed as Functions of y
- 8.6Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7Volumes with Cross Sections: Squares and Rectangles
- 8.8Volumes with Cross Sections: Triangles and Semicircles
- 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10Volume with Disc Method: Revolving Around Other Axes
- 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12Volume with Washer Method: Revolving Around Other Axes
- 8.13The Arc Length of a Smooth, Planar Curve and Distance Traveled
Unit 9: Parametric, Polar & Vector Functions
- 9.1Defining and Differentiating Parametric Equations
- 9.2Second Derivatives of Parametric Equations
- 9.3Finding Arc Lengths of Curves Given by Parametric Equations
- 9.4Defining and Differentiating Vector-Valued Functions
- 9.5Integrating Vector-Valued Functions
- 9.6Solving Motion Problems Using Parametric and Vector-Valued Functions
- 9.7Defining Polar Coordinates and Differentiating in Polar Form
- 9.8Finding the Area of a Polar Region or the Area Bounded by a Single Polar Curve
- 9.9Finding the Area of the Region Bounded by Two Polar Curves
Unit 10: Infinite Sequences & Series
- 10.1Defining Convergent and Divergent Infinite Series
- 10.2Working with Geometric Series
- 10.3The nth Term Test for Divergence
- 10.4Integral Test for Convergence
- 10.5Harmonic Series and p-Series
- 10.6Comparison Tests for Convergence
- 10.7Alternating Series Test for Convergence
- 10.8Ratio Test for Convergence
- 10.9Determining Absolute or Conditional Convergence
- 10.10Alternating Series Error Bound
- 10.11Finding Taylor Polynomial Approximations of Functions
- 10.12Lagrange Error Bound
- 10.13Radius and Interval of Convergence of Power Series
- 10.14Finding Taylor or Maclaurin Series for a Function
- 10.15Representing Functions as Power Series