AP Calculus AB: Course Outline
Unit 1: Functions and GraphsA. Functions and Function Notation
a. Topic 1: Definition of a function
b. Topic 2: Vertical line test
c. Topic 3: Function notation
d. Topic 4: Finding input & output values
e. Topic 5: Domain & range
B. Absolute Value and Piecewise Defined Functions
a. Topic 1: Piecewise defined functions
b. Topic 2: The absolute value function
c. Topic 3: Solving equations involving absolute values
C. Inequalities
a. Topic 1: Solving inequalities
b. Topic 2: Solving inequalities involving absolute values
D. Composition and Combination of Functions
a. Topic 1: Composition of functions
b. Topic 2: Inverse functions
c. Topic 3: Non-invertible functions
E. Exponential and Logarithmic Functions
a. Topic 1: The family of exponential functions
b. Topic 2: The number e
c. Topic 3: Logarithmic functions
d. Topic 4: Solving exponential and logarithmic functions
e. Topic 5: Changing the base of logarithmic functions
F. Transformation of Functions
a. Topic 1: Horizontal & vertical shifts
b. Topic 2: Reflections & symmetry
c. Topic 3: Vertical stretches & compressions
d. Topic 4: Horizontal stretches & compressions
G. Trigonometric Functions
a. Topic 1: Radians and arc length
b. Topic 2: The sin and cosine functions
c. Topic 3: Other trigonometric functions
d. Topic 4: Inverse trigonometric functions
e. Topic 5: Trigonometric identities
H. Power, Polynomial and Rational Functions
a. Topic 1: Variations
b. Topic 2: Power functions
c. Topic 3: Polynomial functions
d. Topic 4: Rational functions
Unit 2: Limits and Continuity
A. Intuitive Definition of a Limit
a. Topic 1: What is a limit?
b. Topic 2: Using tables & graphs to find limits
c. Topic 3: Application: Using limits to find velocity
B. Algebraic Techniques for Finding Limits
a. Topic 1: Calculating limits using limit laws
b. Topic 2: Direct substitution property
c. Topic 3: Indeterminate forms / C. One-Sided Limits
a. Topic 1: Definition of one-sided limits
b. Topic 2: Finding one-sided limits
c. Topic 3: Existence of limits
D. Infinite Limits
a. Topic 1: Definition of infinite limits
b. Topic 2: Vertical asymptotes
E. Limits at Infinity
a. Topic 1: End-behavior of functions & horizontal asymptotes
b. Topic 2: Limit laws of infinite limits
c. Topic 3: Oblique asymptotes
F. Limits of Special Trigonometric Functions
a. Topic 1: Special limits involving the sine function
b. Topic 2: Special limits involving the cosine function
c. Topic 3: Special limits involving the tangent function
G. Continuity
a. Topic 1: Continuity at a point
b. Topic 2: Continuity on a closed interval
c. Topic 3: Continuity on an open interval
d. Topic 4: Intermediate Value Theorem
Unit 3: Derivatives
A. Definition of the Derivative
a. Topic 1: The derivative as the slope of a tangent
b. Topic 2: The derivative as the rate of change
c. Topic 3: The derivative as a function
B. Differentiation Rules
a. Topic 1: Constant rule, constant multiple rule
b. Topic 2: Power rule
c. Topic 3: Sum rule, difference rule
d. Topic 4: Product rule
e. Topic 5: Quotient rule
f. Topic 6: Trigonometric rules
C. The Chain Rule
a. Topic 1: Chain rule definition
b. Topic 2: The chain rule with other rules
c. Topic 3: Higher-order derivatives
D. Derivatives of Exponential Functions
a. Topic 1: Exponential rule
b. Topic 2: Base-a exponentials rule
E. Derivative of Logarithmic Functions
a. Topic 1: Natural logarithmic rule
b. Topic 2: General logarithmic rule
F. Derivatives of Inverse Functions
a. Topic 1: Inverse trig rule
b. Topic 2: Derivatives of Inverses
G. Differentiability and Continuity
a. Topic 1: Differentiability implies continuity
b. Topic 2: Non-differentiable functions
H. Implicit Differentiation
a. Topic 1: Derivatives of implicitly defined functions
I. Logarithmic Differentiation
a. Topic 1: derivatives of complicated expressions
Unit 4: Antiderivatives and Definite Integrals
A. Differential Equations and Slope Fields
a. Topic 1: Differential Equations
b. Topic 2: Slope Fields
B. Antiderivatives
a. Topic 1: Introduction to antiderivatives
b. Topic 2: Basic antidifferentiation rules
c. Topic 3: Trigonometric antidifferentiation rules
C. The Chain Rule for Antiderivatives
a. Topic 1: Simple substitutions
b. Topic 2: Trigonometric Integrals
D. Antiderivatives of Exponentials
a. Topic 1: Natural exponential functions
b. Topic 2: General exponential functions
E. Antiderivatives and Logarithms
a. Topic 1: Natural logarithmic functions
b. Topic 2: General logarithmic functions
F. Antiderivatives and Inverse Trigonometric Functions
a. Topic 1: Rules yielding inverse trig functions
b. Topic 2: Rules yielding inverse trig cofunctions
G. Trigonometric Substitutions
a. Topic 1: Substitution technique
b. Topic 2: Types of expressions
H. The Definite Integral
a. Topic 1: Riemann sums
b. Topic 2: Definition of the definite integral
c. Topic 3: Area between curves
d. Topic 4: Approximate integration
I. Fundamental Theorem of Calculus
a. Topic 1: Properties of the definite integral
b. Topic 2: The Fundamental Theorem of Calculus
c. Topic 3: Integral defined functions
d. Topic 4: The Mean Value Theorem
Unit 5: Application of the Derivative
A. Tangent and Normal Lines
a. Topic 1: Tangent lines
b. Topic 2: Normal lines
B. Position, Velocity, and Acceleration (PVA)
a. Topic 1: Position & velocity
b. Topic 2: Acceleration
C. Related Rates
a. Topic 1: Defining the problem
b. Topic 2: Example problems
D. Relative Extrema and the First Derivative Test
a. Topic 1: Relative extrema & critical numbers
b. Topic 2: Increasing, decreasing functions & the first derivative
E. Concavity and the Second Derivative Test
a. Topic 1: Concavity
b. Topic 2: The second derivative test / F. Absolute Extrema and Optimization
a. Topic 1: Extreme value theorem
b. Topic 2: Sample problem
G. Rolle's Rule and the Mean Value Theorem
a. Topic 1: Rolle’s rule
b. Topic 2: Mean value theorem
H. Differentials
a. Topic 1: Linear approximation
b. Topic 2: Differentials
Unit 6: Application of Antiderivatives and Definite Integrals
A. Net Change and Displacement
a. Topic 1: Net Change Theorem
b. Topic 2: Displacement vs. total distance
B. Volume
a. Topic 1: Volume of a solid of revolution: circular disks & rings
b. Topic 2: Volume of a solid of revolution: cylindrical shells
c. Topic 3: Volume of a solid of known cross-section
C. Separable Differential Equations
a. Topic 1: Separable equations
b. Topic 2: Exponential, bounded growth & decay
D. Work
a. Topic 1: Work done moving an object
b. Topic 2: Work done on a fluid
E. Other Applications of Definite Integrals
a. Topic 1: Center of mass & density
b. Topic 2: Costs & probability