Unit 1: Limits and Continuity (4-7% of exam)
Introducing calculus
Rates of change and the tangent line idea

Limits
Limit notation, estimating from graphs and tables
One-sided limits and properties of limits

Algebraic techniques
Direct substitution, factoring, rationalizing
Squeeze Theorem

Infinite limits
Vertical and horizontal asymptotes

Continuity
Types of discontinuities
Intermediate Value Theorem
Definition of the derivative
Limit definition and derivative notation
Differentiability and continuity

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

Implicit differentiation

Inverse functions
Derivatives of inverse functions
Derivatives of inverse trigonometric functions

Higher-order derivatives

Selecting procedures for finding derivatives
Interpreting the derivative in context
Meaning and units of f'(x)Straight-line motion: position, velocity, acceleration

Related ratesApproximation
Local linearity and linearization
L’Hôpital’s Rule
Mean Value Theorem and Extreme Value Theorem

Critical points
Increasing/decreasing intervals and First Derivative Test

Concavity
Points of inflection and Second Derivative Test

Extrema
Local and absolute extrema, Candidates Test

Curve sketching
Connecting f, f’, and f’’

Optimization problemsBehavior of implicit relations
Accumulation
Riemann sums (left, right, midpoint, trapezoidal)
Definite integral as a limit of a sum

Fundamental Theorem of Calculus
Accumulation functions and evaluating definite integrals

Properties of definite integrals

Antiderivatives
Indefinite integrals and basic rules
Integrals involving trig, exponential, 1/x, and inverse trig forms

Integration techniques
u-substitution
Long division and completing the square

Integration by parts (BC only)

Partial fractions with linear, non-repeating factors (BC only)

Improper integrals (BC only)

Initial value problems
Modeling situations with differential equations

Slope fields
Sketching and reasoning with slope fields

Separation of variables
General and particular solutions

Exponential models
Growth and decay

Euler’s Method (BC only)

Logistic models (BC only)
Logistic differential equation
Carrying capacity and growth rate analysis
Average value of a function

Motion
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 shapeDisc and washer methods

Arc length of a curve (BC only)
Parametric equations
Derivatives of parametric functions
Second derivatives
Arc length of a parametric curve

Vector-valued functions
Derivatives of vector-valued functions
Position, velocity, acceleration, and speed
Distance traveled in the plane

Polar coordinates
Converting between polar and rectangular
Graphing polar curves
Derivatives of polar functions and tangent lines
Area of a polar region and area between polar curves
Convergence of series
Sequences and partial sums
The nth Term Test for divergence
Infinite geometric seriesp-series and harmonic series

Convergence tests
Integral Test
Direct Comparison Test
Limit Comparison Test
Alternating Series Test
Ratio Test

Error bounds
Alternating series error bound
Lagrange error bound

Absolute vs. conditional convergence

Taylor and Maclaurin series
aylor polynomial approximations
Maclaurin series for e^x, sin x, cos x, and 1/(1-x)Constructing new series by substitution, differentiation, and integration

Power series
Radius and interval of convergence
Differentiating and integrating power series