# Course: 5 Steps to A5 • AP Calculus BC 2019

• Step 1 Set Up Your Study Plan
• Step 3 Develop Strategies for Success
• Step 4 Review the Knowledge You Need to Score High
• Big Idea 1 Limits
• 5 Limits and Continuity
• 5.1 The Limit of a Function
• Definition and Properties of Limits
• Evaluating Limits
• One-Sided Limits
• Squeeze Theorem
• 5.2 Limits Involving Infinities
• Infinite Limits (as 𝑥 → 𝑎)
• Limits at Infinity (as 𝑥 → ±∞)
• Horizontal and Vertical Asymptotes
• 5.3 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
• 6 Differentiation
• 6.1 Derivatives of Algebraic Functions
• Definition of the Derivative of a Function
• Power Rule
• The Sum, Difference, Product, and Quotient Rules
• The Chain Rule
• 6.2 Derivatives of Trigonometric, Inverse Trigonometric, Exponential, and Logarithmic Functions
• Derivatives of Trigonometric Functions
• Derivatives of Inverse Trigonometric Functions
• Derivatives of Exponential and Logarithmic Functions
• 6.3 Implicit Differentiation
• Procedure for Implicit Differentiation
• 6.4 Approximating a Derivative
• 6.5 Derivatives of Inverse Functions
• 6.6 Higher Order Derivatives
• L’Hôpital’s Rule for Indeterminate Forms
• 7 Graphs of Functions and Derivatives
• 7.1 Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem
• Rolle’s Theorem
• Mean Value Theorem
• Extreme Value Theorem
• 7.2 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
• 7.3 Sketching the Graphs of Functions
• Graphing without Calculators
• Graphing with Calculators
• 7.4 Graphs of Derivatives
• 7.5 Parametric, Polar, and Vector Representations
• Parametric Curves
• Polar Equations
• Types of Polar Graphs
• Symmetry of Polar Graphs
• Vectors
• Vector Arithmetic
• 8 Applications of Derivatives
• 8.1 Related Rate
• General Procedure for Solving Related Rate Problems
• Common Related Rate Problems
• Inverted Cone (Water Tank) Problem
• Angle of Elevation Problem
• 8.2 Applied Maximum and Minimum Problems
• General Procedure for Solving Applied Maximum and Minimum Problems
• Distance Problem
• Area and Volume Problem
• 9 More Applications of Derivatives
• 9.1 Tangent and Normal Lines
• Tangent Lines
• Normal Lines
• 9.2 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
• 9.3 Motion Along a Line
• Instantaneous Velocity and Acceleration
• Vertical Motion
• Horizontal Motion
• 9.4 Parametric, Polar, and Vector Derivatives
• Derivatives of Parametric Equations
• Position, Speed, and Acceleration
• Derivatives of Polar Equations
• Velocity and Acceleration of Vector Functions
• Big Idea 3 Integrals and the Fundamental Theorems of Calculus
• 10 Integration
• 10.1 Evaluating Basic Integrals
• Antiderivatives and Integration Formulas
• Evaluating Integrals
• 10.2 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
• 10.3 Techniques of Integration
• Integration by Parts
• Integration by Partial Fractions
• 11 Definite Integrals
• 11.1 Riemann Sums and Definite Integrals
• Sigma Notation or Summation Notation
• Definition of a Riemann Sum
• Definition of a Definite Integral
• Properties of Definite Integrals
• 11.2 Fundamental Theorems of Calculus
• First Fundamental Theorem of Calculus
• Second Fundamental Theorem of Calculus
• 11.3 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
• 11.4 Improper Integrals
• Infinite Intervals of Integration
• Infinite Discontinuities
• 12 Areas, Volumes, and Arc Lengths
• 12.1 The Function 𝐹(𝑥) = ∫^𝑥_𝑎 𝑓(𝑡)𝑑𝑡
• 12.2 Approximating the Area Under a Curve
• Rectangular Approximations
• Trapezoidal Approximations
• 12.3 Area and Definite Integrals
• Area under a Curve
• Area between Two Curves
• 12.4 Volumes and Definite Integrals
• Solids with Known Cross Sections
• The Disc Method
• The Washer Method
• 12.5 Integration of Parametric, Polar, and Vector Curves
• Area, Arc Length, and Surface Area for Parametric Curves
• Area and Arc Length for Polar Curves
• Integration of a Vector-Valued Function
• 13 More Applications of Definite Integrals
• 13.1 Average Value of a Function
• Mean Value Theorem for Integrals
• Average Value of a Function on [𝑎, 𝑏]
• 13.2 Distance Traveled Problems
• 13.3 Definite Integral as Accumulated Change
• Temperature Problem
• Leakage Problem
• Growth Problem
• 13.4 Differential Equations
• Exponential Growth/Decay Problems
• Separable Differential Equations
• 13.5 Slope Fields
• 13.6 Logistic Differential Equations
• 13.7 Euler’s Method
• Approximating Solutions of Differential Equations by Euler’s Method
• Big Idea 4 Series
• 14 Series
• 14.1 Sequences and Series
• Convergence
• 14.2 Types of Series
• 𝑝-Series
• Harmonic Series
• Geometric Series
• Decimal Expansion
• 14.3 Convergence Tests
• Divergence Test
• Integral Test
• Ratio Test
• Comparison Test
• Limit Comparison Test
• Informal Principle
• 14.4 Alternating Series
• Error Bound
• Absolute and Conditional Convergence
• 14.5 Power Series
• Radius and Interval of Convergence
• 14.6 Taylor Series
• Taylor Series and MacLaurin Series
• Common MacLaurin Series
• 14.7 Operations on Series
• Substitution
• Differentiation and Integration
• Error Bounds
• Step 5 Build Your Test-Taking Confidence

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