AP Calculus BC Study Guide

Unit 1: Limits and Continuity

5-10% of the multiple-choice section

A limit describes the value a function approaches as x approaches a point, which is different from evaluating the function at that point. You estimate and determine limits from graphs, tables, and algebra, including with the squeeze theorem, then use limits to define continuity, classify discontinuities, and connect limits to vertical and horizontal asymptotes. The unit ends with the Intermediate Value Theorem.

Topics in this unit

• Introducing Calculus: Can Change Occur at an Instant?
• Defining Limits and Using Limit Notation
• Estimating Limit Values from Graphs
• Estimating Limit Values from Tables
• Determining Limits Using Algebraic Properties of Limits
• Determining Limits Using Algebraic Manipulation
• Selecting Procedures for Determining Limits
• Determining Limits Using the Squeeze Theorem
• 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
• Working with the Intermediate Value Theorem (IVT)

Shaky on any of these? Paste your class notes for Unit 1 into the free flashcard maker or the free quiz maker.

Unit 2: Differentiation: Definition and Fundamental Properties

5-10% of the multiple-choice section

The derivative gives an instantaneous rate of change, defined by applying a limit to average rates of change over smaller and smaller intervals. You learn derivative notation, estimate derivatives at a point, and see when a derivative does and does not exist. You then practice the power, product, and quotient rules and the derivatives of trigonometric, exponential, and logarithmic functions.

Topics in this unit

• Defining Average and Instantaneous Rates of Change at a Point
• Defining the Derivative of a Function and Using Derivative Notation
• Estimating Derivatives of a Function at a Point
• Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not Exist
• Applying the Power Rule
• 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/or Cosecant Functions

Shaky on any of these? Paste your class notes for Unit 2 into the free flashcard maker or the free quiz maker.

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

5-10% of the multiple-choice section

This unit teaches the chain rule for composite functions and uses it to find derivatives of implicit functions, inverse functions, and inverse trigonometric functions. You also practice choosing the right differentiation procedure and calculating higher-order derivatives. College Board calls mastering the chain rule essential to success in every later unit.

Topics in this unit

• The Chain Rule
• Implicit Differentiation
• Differentiating Inverse Functions
• Differentiating Inverse Trigonometric Functions
• Selecting Procedures for Calculating Derivatives
• Calculating Higher-Order Derivatives

Shaky on any of these? Paste your class notes for Unit 3 into the free flashcard maker or the free quiz maker.

Unit 4: Contextual Applications of Differentiation

5-10% of the multiple-choice section

Here you apply derivatives to real situations, interpreting a derivative in context with correct units and connecting position, velocity, and acceleration for straight-line motion. You solve related rates problems and approximate function values with a tangent line. The unit also uses L'Hospital's rule to find limits of indeterminate forms, where you must show that the rule applies.

Topics in this unit

• Interpreting the Meaning of the Derivative in Context
• Straight-Line Motion: Connecting Position, Velocity, and Acceleration
• Rates of Change in Applied Contexts Other Than Motion
• Introduction to Related Rates
• Solving Related Rates Problems
• Approximating Values of a Function Using Local Linearity and Linearization
• Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms

Shaky on any of these? Paste your class notes for Unit 4 into the free flashcard maker or the free quiz maker.

Unit 5: Analytical Applications of Differentiation

10-15% of the multiple-choice section

This unit uses derivatives and theorems such as the Mean Value Theorem and the Extreme Value Theorem to justify conclusions about how a function behaves. You find critical points, intervals of increase and decrease, extrema, concavity, and points of inflection, and you connect a function to its first and second derivatives. It ends with optimization problems, where you justify your solution, and with the behavior of implicit relations.

Topics in this unit

• Using the Mean Value Theorem
• Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
• Determining Intervals on Which a Function Is Increasing or Decreasing
• Using the First Derivative Test to Determine Relative (Local) Extrema
• Using the Candidates Test to Determine Absolute (Global) Extrema
• Determining Concavity of Functions over Their Domains
• 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
• Introduction to Optimization Problems
• Solving Optimization Problems
• Exploring Behaviors of Implicit Relations

Shaky on any of these? Paste your class notes for Unit 5 into the free flashcard maker or the free quiz maker.

Unit 6: Integration and Accumulation of Change

15-20% of the multiple-choice section

Integration finds the accumulation of change over an interval, just as differentiation finds the rate of change at a point. You approximate areas with Riemann sums, connect them to definite integrals, and use the Fundamental Theorem of Calculus to link differentiation and integration. You also find antiderivatives with basic rules, substitution, long division, and completing the square, and BC students add integration by parts, linear partial fractions, and improper integrals.

Topics in this unit

• Exploring Accumulations of Change
• Approximating Areas with Riemann Sums
• Riemann Sums, Summation Notation, and Definite Integral Notation
• The Fundamental Theorem of Calculus and Accumulation Functions
• Interpreting the Behavior of Accumulation Functions Involving Area
• Applying Properties of Definite Integrals
• The Fundamental Theorem of Calculus and Definite Integrals
• Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
• Integrating Using Substitution
• Integrating Functions Using Long Division and Completing the Square
• Integrating Using Integration by PartsBC only
• Integrating Using Linear Partial FractionsBC only
• Evaluating Improper IntegralsBC only
• Selecting Techniques for Antidifferentiation

Shaky on any of these? Paste your class notes for Unit 6 into the free flashcard maker or the free quiz maker.

Unit 7: Differential Equations

5-10% of the multiple-choice section

You write differential equations to model situations, verify solutions, and use slope fields to see that a differential equation has many solution curves. You solve separable differential equations, using an initial condition to find a particular solution, and build exponential growth and decay models. BC students also approximate solutions with Euler's method and work with logistic models.

Topics in this unit

• Modeling Situations with Differential Equations
• Verifying Solutions for Differential Equations
• Sketching Slope Fields
• Reasoning Using Slope Fields
• Approximating Solutions Using Euler’s MethodBC only
• Finding General Solutions Using Separation of Variables
• Finding Particular Solutions Using Initial Conditions and Separation of Variables
• Exponential Models with Differential Equations
• Logistic Models with Differential EquationsBC only

Shaky on any of these? Paste your class notes for Unit 7 into the free flashcard maker or the free quiz maker.

Unit 8: Applications of Integration

5-10% of the multiple-choice section

This unit applies definite integrals to find the average value of a function, model particle motion, and measure net change in applied contexts. You find areas between curves and volumes of solids with cross sections and with the disc and washer methods, and BC students also find arc length. Seeing each of these as a limit of a Riemann sum lets you understand the formulas instead of memorizing a long list.

Topics in this unit

• Finding the Average Value of a Function on an Interval
• Connecting Position, Velocity, and Acceleration of Functions Using Integrals
• Using Accumulation Functions and Definite Integrals in Applied Contexts
• 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 Disc Method: Revolving Around the x- or y-Axis
• Volume with Disc Method: Revolving Around Other Axes
• Volume with Washer Method: Revolving Around the x- or y-Axis
• Volume with Washer Method: Revolving Around Other Axes
• The Arc Length of a Smooth, Planar Curve and Distance TraveledBC only

Shaky on any of these? Paste your class notes for Unit 8 into the free flashcard maker or the free quiz maker.

Unit 9: Parametric Equations, Polar Coordinates, and Vector-Valued Functions

BC only

10-15% of the multiple-choice section

This BC-only unit extends straight-line motion to particles moving along curves in the plane. You differentiate and integrate parametric equations and vector-valued functions, find arc lengths, and solve motion problems. You also learn that polar equations are a special case of parametric equations, differentiate in polar form, and find areas of polar regions.

Topics in this unit

• Defining and Differentiating Parametric Equations
• Second Derivatives of Parametric Equations
• Finding Arc Lengths of Curves Given by Parametric Equations
• Defining and Differentiating Vector-Valued Functions
• Integrating Vector-Valued Functions
• Solving Motion Problems Using Parametric and Vector-Valued Functions
• Defining Polar Coordinates and Differentiating in Polar Form
• Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
• Finding the Area of the Region Bounded by Two Polar Curves

Shaky on any of these? Paste your class notes for Unit 9 into the free flashcard maker or the free quiz maker.

Unit 10: Infinite Sequences and Series

BC only

15-20% of the multiple-choice section

This BC-only unit shows how a sum of infinitely many terms can converge to a finite value. You decide whether series converge or diverge using the nth term, integral, comparison, alternating series, and ratio tests, and you tell absolute from conditional convergence. You then build Taylor polynomials, bound their error, and find the radius and interval of convergence of power series that represent functions.

Topics in this unit

• Defining Convergent and Divergent Infinite Series
• Working with Geometric Series
• The nth Term Test for Divergence
• Integral Test for Convergence
• Harmonic Series and p-Series
• Comparison Tests for Convergence
• Alternating Series Test for Convergence
• Ratio Test for Convergence
• Determining Absolute or Conditional Convergence
• Alternating Series Error Bound
• Finding Taylor Polynomial Approximations of Functions
• Lagrange Error Bound
• Radius and Interval of Convergence of Power Series
• Finding Taylor or Maclaurin Series for a Function
• Representing Functions as Power Series

Shaky on any of these? Paste your class notes for Unit 10 into the free flashcard maker or the free quiz maker.