# Course: 5 Steps to A5 • AP Calculus AB 2019 • Elite Student Edition

• Step 1 Set up Your Study Plan
• 1 What You Need to Know about the AP Calculus AB Exam
• 2 How to Plan Your Time
• 3 Take a Diagnostic Exam
• Step 3 Develop Strategies for Success
• 4 How to Approach Each Question Type
• Step 4 Review the Knowledge You Need to Score High
• 5 Review of Precalculus
• 5.1 Lines
• Slope of a Line
• Equations of a Line
• Parallel and Perpendicular Lines
• 5.2 Absolute Values and Inequalities
• Absolute Values
• Inequalities and the Real Number Line
• Solving Absolute Value Inequalities
• Solving Polynomial Inequalities
• Solving Rational Inequalities
• 5.3 Functions
• Definition of a Function
• Operations on Functions
• Inverse Functions
• Trigonometric and Inverse Trigonometric Functions
• Exponential and Logarithmic Functions
• 5.4 Graphs of Functions
• Increasing and Decreasing Functions
• Intercepts and Zeros
• Odd and Even Functions
• Shifting, Reflecting, and Stretching Graphs
• Big Idea 1: Limits
• 6 Limits and Continuity
• 6.1 The Limit of a Function
• Definition and Properties of Limits
• Evaluating Limits
• One-Sided Limits
• Squeeze Theorem
• 6.2 Limits Involving Infinities
• Infinite Limits (as 𝑥 → 𝑎)
• Limits at Infinity (as 𝑥 → ±∞)
• Horizontal and Vertical Asymptotes
• 6.3 Continuity of a Function
• Continuity of a Function at a Number
• Continuity of a Function over an Interval
• Theorems on Continuity
• Big Idea 2: Derivatives
• 7 Differentiation
• 7.1 Derivatives of Algebraic Functions
• Definition of the Derivative of a Function
• Power Rule
• The Sum, Difference, Product, and Quotient Rules
• The Chain Rule
• 7.2 Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions
• Derivatives of Trigonometric Functions
• Derivatives of Inverse Trigonometric Functions
• Derivatives of Exponential and Logarithmic Functions
• 7.3 Implicit Differentiation
• Procedure for Implicit Differentiation
• 7.4 Approximating a Derivative
• 7.5 Derivatives of Inverse Functions
• 7.6 Higher Order Derivatives
• 7.7 L’Hôpital’s Rule for Indeterminate Forms
• 8 Graphs of Functions and Derivatives
• 8.1 Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem
• Rolle’s Theorem
• Mean Value Theorem
• Extreme Value Theorem
• 8.2 Determining the Behavior of Functions
• Test for Increasing and Decreasing Functions
• First Derivative Test and Second Derivative Test for Relative Extrema
• Test for Concavity and Points of Inflection
• 8.3 Sketching the Graphs of Functions
• Graphing without Calculators
• Graphing with Calculators
• 8.4 Graphs of Derivatives
• 9 Applications of Derivatives
• 9.1 Related Rate
• General Procedure for Solving Related Rate Problems
• Common Related Rate Problems
• Inverted Cone (Water Tank) Problem
• Angle of Elevation Problem
• 9.2 Applied Maximum and Minimum Problems
• General Procedure for Solving Applied Maximum and Minimum Problems
• Distance Problem
• Area and Volume Problems
• 10 More Applications of Derivatives
• 10.1 Tangent and Normal Lines
• Tangent Lines
• Normal Lines
• 10.2 Linear Approximations
• Tangent Line Approximation (or Linear Approximation)
• Estimating the 𝑛th Root of a Number
• Estimating the Value of a Trigonometric Function of an Angle
• 10.3 Motion Along a Line
• Instantaneous Velocity and Acceleration
• Vertical Motion
• Horizontal Motion
• Big Idea 3: Integrals and the Fundamental Theorems of Calculus
• 11 Integration
• 11.1 Evaluating Basic Integrals
• Antiderivatives and Integration Formulas
• Evaluating Integrals
• 11.2 Integration by U-Substitution
• The U-Substitution Method
• U-Substitution and Algebraic Functions
• U-Substitution and Trigonometric Functions
• U-Substitution and Inverse Trigonometric Functions
• U-Substitution and Logarithmic and Exponential Functions
• 12 Definite Integrals
• 12.1 Riemann Sums and Definite Integrals
• Sigma Notation or Summation Notation
• Definition of a Riemann Sum
• Definition of a Definite Integral
• Properties of Definite Integrals
• 12.2 Fundamental Theorems of Calculus
• First Fundamental Theorem of Calculus
• Second Fundamental Theorem of Calculus
• 12.3 Evaluating Definite Integrals
• Definite Integrals Involving Algebraic Functions
• Definite Integrals Involving Absolute Value
• Definite Integrals Involving Trigonometric, Logarithmic, and Exponential Functions
• Definite Integrals Involving Odd and Even Functions
• 13 Areas and Volumes
• 13.1 The Function 𝐹(𝑥) = ∫_𝑎^𝑥 𝑓(𝑡)d𝑡
• 13.2 Approximating the Area Under a Curve
• Rectangular Approximations
• Trapezoidal Approximations
• 13.3 Area and Definite Integrals
• Area Under a Curve
• Area between Two Curves
• 13.4 Volumes and Definite Integrals
• Solids with Known Cross Sections
• The Disc Method
• The Washer Method
• 14 More Applications of Definite Integrals
• 14.1 Average Value of a Function
• Mean Value Theorem for Integrals
• Average Value of a Function on [𝑎, 𝑏]
• 14.2 Distance Traveled Problems
• 14.3 Definite Integral as Accumulated Change