Table of Contents
 Step 1 Set Up Your Study Plan
 Step 2 Determine Your Test Readiness
 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
 OneSided 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

5.1 The Limit of a Function

5 Limits and 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

6.1 Derivatives of Algebraic Functions

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

7.1 Rolle’s Theorem, Mean Value Theorem, and Extreme Value Theorem

8 Applications of Derivatives

8.1 Related Rate
 General Procedure for Solving Related Rate Problems
 Common Related Rate Problems
 Inverted Cone (Water Tank) Problem
 Shadow 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
 Business Problems

8.1 Related Rate

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

9.1 Tangent and Normal Lines

6 Differentiation

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 USubstitution
 The USubstitution Method
 USubstitution and Algebraic Functions
 USubstitution and Trigonometric Functions
 USubstitution and Inverse Trigonometric Functions
 USubstitution and Logarithmic and Exponential Functions

10.3 Techniques of Integration
 Integration by Parts
 Integration by Partial Fractions

10.1 Evaluating Basic Integrals

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

11.1 Riemann Sums and Definite Integrals

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 VectorValued 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
 Business Problems
 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

13.1 Average Value of a Function

10 Integration

Big Idea 4 Series

14 Series

14.1 Sequences and Series
 Convergence

14.2 Types of Series
 𝑝Series
 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

14.1 Sequences and Series

14 Series

Big Idea 1 Limits
 Step 5 Build Your TestTaking Confidence