AP Calculus AB: Course Outline

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