AP Calculus BC

• Estimating Limits from Graphs and Tables
• Limits from Both Sides
• One-Sided Limits
• Limit Laws
• Limits by Factoring and Canceling
• Limits by Rationalizing
• The Squeeze Theorem
• Special Trigonometric Limits
• Infinite Limits and Vertical Asymptotes
• Limits at Infinity and Horizontal Asymptotes
• Continuity
• The Intermediate Value Theorem

• From Average to Instantaneous Rate
• The Derivative at a Point
• Tangent Lines
• The Derivative as a Function
• Where Derivatives Fail to Exist
• Constant, Power and Sum Rules
• The Product Rule
• The Quotient Rule
• Derivatives of Trig Functions
• Derivatives of Exponential and Log Functions

• The Chain Rule
• Chain Rule with Several Layers
• Implicit Differentiation
• Derivatives of Inverse Functions and Inverse Trig
• Higher Derivatives

• Leibniz Notation and Units
• Position, Velocity and Acceleration
• Related Rates
• Linear Approximation and Differentials
• L'Hôpital's Rule

• The Mean Value Theorem
• Critical Points and Absolute Extrema
• Increasing, Decreasing and the First Derivative Test
• Concavity and the Second Derivative Test
• Curve Sketching
• Optimization

The behavior of implicit relations is practiced only through Implicit Differentiation for now.

• Approximating Area with Rectangles
• Riemann Sums
• The Definite Integral
• Properties of Definite Integrals
• Accumulation Functions and FTC Part 1
• Evaluating Integrals with FTC Part 2
• Antiderivatives
• Integration by Substitution
• Substitution in Definite Integrals
• Integrals Giving Exponentials and Logarithms
• Integrals Giving Inverse Trig Functions
• Integration by Parts
• Integration by Partial Fractions
• Improper Integrals

BC adds integration by parts, linear partial fractions and improper integrals. Long division and completing the square before integrating have no topic of their own yet.

• What a Differential Equation Is
• Direction Fields and Euler's Method
• Separable Equations
• Antiderivatives with Initial Conditions
• Growth, Decay and Newton's Law of Cooling
• The Logistic Equation

• Average Value of a Function
• Net Change and Total Distance
• Area Between Curves
• Volume by Slicing
• Disks and Washers
• Arc Length

• Tangents and Arc Length for Parametric Curves
• Vector-Valued Functions and Space Curves
• Derivatives and Integrals of Vector Functions
• Velocity and Acceleration in Space
• Area in Polar Coordinates
• Polar Tangents and Arc Length

Vector-valued functions come from Calculus 3, which works in space; motion in the plane is the special case AP uses.

• Sequences and Their Limits
• Series and Partial Sums
• Geometric and Telescoping Series
• The Divergence Test
• The Integral Test and p-Series
• Comparison and Limit Comparison Tests
• Alternating Series
• Absolute and Conditional Convergence
• Ratio and Root Tests
• Estimating the Sum of a Series
• Power Series and Intervals of Convergence
• Representing Functions as Power Series
• Differentiating and Integrating Power Series
• Taylor and Maclaurin Series
• Taylor's Inequality
• Working with Known Series