Unit 1: Limits and Continuity (10-12% of exam)
Introducing calculus
Change in tandem
Rates of change and the tangent line idea

Limits
Defining limits and limit notation
Estimating limits from graphs and tables
One-sided limits
Properties of limits

Algebraic techniques
Direct substitution
Factoring, rationalizing, and simplifying
Squeeze Theorem

Infinite limits
Vertical asymptotes
Limits at infinity and horizontal asymptotes

Continuity
Types of discontinuities (removable, jump, infinite)
Removing discontinuities
Connecting limits and continuity
Intermediate Value Theorem
Definition of the derivative
Average vs. instantaneous rate of change
Limit definition of the derivative
Derivative notation
Differentiability and continuity

Basic rules
Power, constant, sum, and difference rules
Derivatives of sin x, cos x, e^x, and ln xProduct rule
Quotient rule
Derivatives of tan x, cot x, sec x, csc x

Tangent lines
Equations of tangent and normal lines
Chain rule

Implicit differentiation

Inverse functions
Derivatives of inverse functions
Derivatives of inverse trigonometric functions

Higher-order derivatives

Selecting procedures for calculating derivatives

Logarithmic differentiation (optional technique)
Interpreting the derivative in context
Meaning of f'(x) and units
Straight-line motion: position, velocity, acceleration

Rates of change
Rates of change in applied contexts
Related rates

Approximation
Local linearity and linearization
L’Hôpital’s Rule for indeterminate forms
Mean Value Theorem and Extreme Value Theorem

Critical points
Increasing and decreasing intervals
First Derivative Test

Concavity
Points of inflection
Second Derivative Test

Extrema
Local and absolute extrema
Candidates Test on closed intervals

Curve sketching
Connecting f, f’, and f’’
Graphs of derivatives

Optimization problems

Implicit relations
Behavior of implicit relations

Domain and range analysis using calculus
Accumulation
Exploring accumulation of change
Riemann sums (left, right, midpoint, trapezoidal)
Sigma notation and definite integral as a limit

Fundamental Theorem of Calculus
Part 1: accumulation functions
Part 2: evaluating definite integrals

Properties of definite integrals

Antiderivatives
Indefinite integrals and basic rules
Integrals of trig, exponential, and 1/x functions
Integrals resulting in inverse trig functions

Integration techniques
u-substitution (indefinite and definite)Integration by long division and completing the square

Initial value problems

Improper integrals (not tested in AB; BC only)
Modeling situations with differential equations

Slope fields
Sketching and reading slope fields
Reasoning with slope fields

Separation of variables
General and particular solutions
Domain restrictions

Exponential models
Growth and decay
Rate proportional to the amount present
Unit 8: Applications of Integration (10-15%)
Average value of a function

Motion
Position, velocity, and acceleration using integrals
Displacement vs. total distance traveled

Accumulation in context
Net change and rate in/out problems

Area
Between curves (with respect to x and y)

Volume
Cross sections of known shape
Disc method
Washer method