AP Calculus BC - Study Guides, Flashcards, AP-style Practice & Mock Exams

This unit establishes the concept of limits to precisely describe function behavior, laying the groundwork for analyzing continuity and identifying vertical and horizontal asymptotes.

• 1.0Unit Overview
• 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)
• 1.17Unit Exam

We will establish the formal definition of derivatives and develop a versatile toolkit of rules for differentiating algebraic, trigonometric, and transcendental functions.

• 2.0Unit Overview
• 2.1Defining Average and Instantaneous Rates 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
• 2.11Unit Exam

We will develop advanced techniques for finding derivatives of composite, inverse, and implicitly defined functions, including the calculation of successive rates of change.

• 3.0Unit Overview
• 3.1The Chain Rule
• 3.2Implicit Differentiation
• 3.3Differentiating Inverse Functions
• 3.4Differentiating Inverse Trigonometric Functions
• 3.5Selecting Procedures for Calculating Derivatives
• 3.6Calculating Higher-Order Derivatives
• 3.7Unit Exam

We will apply derivatives to analyze motion, solve problems involving connected rates, approximate function values, and evaluate certain indeterminate forms.

• 4.0Unit Overview
• 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'Hospital's Rule for Determining Limits of Indeterminate Forms
• 4.8Unit Exam

We will use derivatives to analyze function behavior, locating extreme values and points of inflection to connect graphical features with applied optimization problems.

• 5.0Unit Overview
• 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
• 5.13Unit Exam

We will connect the concept of accumulation to antidifferentiation, developing various analytic techniques to evaluate definite, indefinite, and even improper integrals.

• 6.0Unit Overview
• 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 (BC ONLY)
• 6.12Integrating Using Linear Partial Fractions (BC ONLY)
• 6.13Evaluating Improper Integrals (BC ONLY)
• 6.14Selecting Techniques for Antidifferentiation
• 6.15Unit Exam

This unit explores modeling rates of change, visualizing solutions with slope fields, and applying analytical and numerical methods to solve various differential equations.

• 7.0Unit Overview
• 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 (BC ONLY)
• 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 (BC ONLY)
• 7.10Unit Exam

We will apply definite integrals to model and solve problems involving accumulation, from calculating areas and volumes to determining arc length and total distance traveled.

• 8.0Unit Overview
• 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 (BC ONLY)
• 8.14Unit Exam

We will use derivatives and integrals to analyze motion, arc length, and area for curves defined by parameters, vectors, or in the polar plane.

• 9.0Unit Overview
• 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
• 9.8Find 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
• 9.10Unit Exam

We will apply various convergence tests to infinite series, then construct polynomial approximations for functions and analyze the bounds of their associated error.

• 10.0Unit Overview
• 10.1Defining Convergent and Divergent Infinite Series
• 10.2Working with Geometric Series
• 10.3The n^{th} 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
• 10.16Unit Exam