AP Calculus ✏ In a Nutshell

1. Functions

A summary of Functions: Basic Definitions, Domain and Range, Function Operations, Function Composition, Even and Odd Functions, One-to-One and Inverses, Transformations of Functions, Polynomial and Rational Functions, Trigonometric Functions, Exponential and Logarithmic Functions, Parametric Functions (BC), Polar Functions (BC)

2. Limits and Continuity

A summary of Limits and Continuity: Understanding Limits, Types of Limits, Limit Examples, Asymptotes, Theorems on Limits, Types of Discontinuities, Algebraic Limits, Continuity, Intermediate Value Theorem (IVT), Extreme Value Theorem (EVT)

3. Differentiation

A summary of Differentiation: Definition of Derivative, Second Derivative, Basic Derivative Formulas, Chain Rule, Differentiability and Continuity, Estimating Derivatives, Differentiation of Parametric Equations (BC), Implicit Differentiation, Derivative of Inverse Functions, Derivatives of Inverse Trig Functions, Mean Value Theorem (MVT), Rolle’s Theorem, L’Hospital’s Rule, Recognizing Derivatives in Limits

4. Applications of Differential Calculus

A summary of Applications of Differential Calculus: Critical Points and Slope, Average vs. Instantaneous Rate, Tangent Lines, Increasing/Decreasing Intervals, Local/Global Extrema, Concavity & Inflection Points, Candidates Test, Curve Sketching Strategy, Optimization, Motion, Related Rates, Linearization/Tangent Approximation, Parametric & Polar Derivatives (BC), Vector/Particle interpretation

5. Antidifferentiation

A summary of Antidifferentiation: Antiderivatives & Constant of Integration, Basic Integration Formulas (power, log, exponential, trig), Substitution (u-substitution), Rational Functions & Long Division, Trig/Exponential Combinations with Identities, Chain Rule Reversal, Integration by Parts (LIATE, tabular shortcut, BC), Partial Fractions (BC), Motion Problems from v(t) and a(t), Using Initial Conditions to Solve for Constants

6. Definite Integrals

A summary of Definite Integrals: Fundamental Theorem of Calculus (FTC), Properties of Integrals (linearity, reversal, additivity), Mean Value Theorem for Integrals, Riemann Sums (left, right, midpoint), Trapezoidal Approximation, Average Value of a Function, Applications of FTC (accumulation, variable limits), Area Under/Between Curves, ln(x) as an Integral, Graphical Interpretation of Integrals, Motion/Distance and Total Accumulation, Integrals of Parametric Curves (BC)

7. Applications of Integration to Geometry

A summary of Applications of Integration to Geometry: Area Under a Curve, Area Between Curves, Using Symmetry, Polar Area, Solids with Known Cross Sections, Solids of Revolution (Disks and Washers), Shell Method, Examples of Solids, Arc Length, Parametric Arc Length (BC), Surface Area of Revolution, Improper Integrals, Comparison Test for Improper Integrals

8. Further Applications of Integration

A summary of Further Applications of Integration: Motion Along a Straight Line, Motion Along a Plane Curve (BC), Riemann Sums in Context, Net Change and Accumulated Change, Applications of Accumulated Change

9. Differential Equations

A summary of Differential Equations: Basic Definitions, Slope Fields, Euler’s Method (BC), Separable Differential Equations, Exponential Growth and Decay, Newton’s Law of Cooling, Logistic Growth (BC)

10. Sequences and Series (BC)

A summary of Sequences and Series (BC): Sequences of Real Numbers, Infinite Series, Power Series, Taylor and Maclaurin Series, Error Estimation (Lagrange Form), Using Series to Approximate Functions, Term-by-Term Differentiation and Integration