Calculus
  • Home
  • List of Lessons
  • Version #1
    • Unit 0 >
      • 0.1 Summer Packet
      • 0.2 Calculator Skillz
    • Unit 1 >
      • 1.1 Can Change Occur at an instant
      • 1.2 Defining Limits and Using Limit Notation
      • 1.3 Limit Values from Graphs
      • 1.4 Limit Values from Tables
      • 1.5 Determining Limits Using Algebraic Properties
      • 1.6 Determining Limits Using Algebraic Manipulation
      • 1.7 Selecting Procedures for Determining Limits
      • 1.8 Determining Limits Using the Squeeze Theorem
      • 1.9 Connecting Multiple Representations of Limits
      • mid-Unit 1 Review
      • 1.10 Exploring Types of Discontinuities
      • 1.11 Defining Continuity at a Point
      • 1.12 Confirming Continuity Over an Interval
      • 1.13 Removing Discontinuities
      • 1.14 Infinite Limits and Vertical Asymptotes
      • 1.15 Limits at Infinity and Horizontal Asymptotes
      • 1.16 Intermediate Value Theorem
      • End of Unit 1 Review
    • Unit 2 >
      • 2.1 Defining Average and Instantaneous Rate of Change at a Point
      • 2.2 Defining the Derivative of a Function and Using Derivative Notation
      • 2.3 Estimating Derivatives of a Function at a Point
      • 2.4 Connecting Differentiability and Continuity
      • 2.5 Applying the Power Rule
      • 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple
      • 2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
      • 2.8 The Product Rule
      • 2.9 The Quotient Rule
      • 2.10 Derivatives of tan(x), cot(x), sec(x), csc(x)
      • Unit 2 Review
    • Unit 3 >
      • 3.1 The Chain Rule
      • 3.2 Implicit Differentiation
      • 3.3 Differentiating Inverse Functions
      • 3.4 Differentiating Inverse Trigonometric Functions
      • 3.5 Selecting Procedures for Calculating Derivatives
      • 3.6 Calculating Higher-Order Derivatives
      • Unit 3 Review
    • Unit 4 >
      • 4.1 Interpreting the Meaning of the Derivative in Context
      • 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
      • 4.3 Rates of Change in Applied Contexts Other Than Motion
      • 4.4 Introduction to Related Rates
      • 4.5 Solving Related Rates Problems
      • 4.6 Approximating Values of a Function Using Local Linearity and Linearization
      • 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms
      • Unit 4 Review
    • Unit 5 >
      • 5.1 Using the Mean Value Theorem
      • 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
      • 5.3 Determining Intervals on Which a Function is Increasing or Decreasing.
      • 5.4 Using the First Derivative Test to Determine Relative Local Extrema
      • 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema
      • 5.6 Determining Concavity of Functions over Their Domains
      • 5.7 Using the Second Derivative Test to Determine Extrema
      • mid-Unit 5 Review
      • 5.8 Sketching Graphs of Functions and Their Derivatives
      • 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
      • 5.10 Introduction to Optimization Problems
      • 5.11 Solving Optimization Problems
      • 5.12 Exploring Behaviors of Implicit Relations
      • End of Unit 5 Review
    • Unit 6 >
      • 6.1 Exploring Accumulation of Change
      • 6.2 Approximating Areas with Riemann Sums
      • 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation
      • 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
      • 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
      • Mid-Unit 6 Review
      • 6.6 Applying Properties of Definite Integrals
      • 6.7 The Fundamental Theorem of Calculus and Definite Integrals
      • 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
      • 6.9 Integrating Using Substitution
      • 6.10 Integrating Functions Using Long Division and Completing the Square
      • 6.11 Integration by Parts
      • 6.12 Integrating Using Linear Partial Fractions
      • 6.13 Evaluating Improper Integrals
      • 6.14 Selecting Techniques for Antidifferentiation
      • End of Unit 6 Review
    • Unit 7 >
      • 7.1 Modeling Situations with Differential Equations
      • 7.2 Verifying Solutions for Differential Equations
      • 7.3 Sketching Slope Fields
      • 7.4 Reasoning Using Slope Fields
      • 7.5 Approximating Solutions Using Euler’s Method
      • 7.6 General Solutions Using Separation of Variables
      • 7.7 Particular Solutions using Initial Conditions and Separation of Variables
      • 7.8 Exponential Models with Differential Equations
      • 7.9 Logistic Models with Differential Equations
      • Unit 7 Review
    • Unit 8 >
      • 8.1 Average Value of a Function on an Interval
      • 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
      • 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
      • 8.4 Finding the Area Between Curves Expressed as Functions of x
      • 8.5 Finding Area Between Curves Expressed as Functions of y
      • 8.6 Finding the Area Between Curves That Intersect at More Than Two Points
      • Mid-Unit 8 Review
      • 8.7 Volumes with Cross Sections: Squares and Rectangles
      • 8.8 Volumes with Cross Sections: Triangles and Semicircles
      • 8.9 Volume with Disc Method: Revolving Around the x- or y-Axis
      • 8.10 Volume with Disc Method: Revolving Around Other Axes
      • 8.11 Volume with Washer Method: Revolving Around the x- or y-axis
      • 8.12 Volume with Washer Method: Revolving Around Other Axes
      • 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
      • End of Unit 8 Review
    • Unit 9 >
      • 9.1 Defining and Differentiating Parametric Equations
      • 9.2 Second Derivatives of Parametric Equations
      • 9.3 Finding Arc Lengths of Curves Given by Parametric Equations
      • 9.4 Defining and Differentiating Vector-Valued Functions
      • 9.5 Integrating Vector-Valued Functions
      • 9.6 Solving Motion Problems using Parametric and Vector-Valued Functions
      • 9.7 Defining Polar Coordinates and Differentiating in Polar Form
      • 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve
      • 9.9 Finding the Area of the Region bounded by Two Polar Curves
      • Unit 9 Review
    • Unit 10 >
      • 10.1 Defining Convergent and Divergent Infinite Series
      • 10.2 Working with Geometric Series
      • 10.3 The nth Term Test for Divergence
      • 10.4 Integral Test for Convergence
      • 10.5 Harmonic Series and p-series
      • 10.6 Comparison Tests for Convergence
      • 10.7 Alternating Series Test for Convergence
      • 10.8 Ratio Test for Convergence
      • 10.9 Determining Absolute or Conditional Convergence
      • Mid-Unit 10 Review
      • 10.10 Alternating Series Error Bound
      • 10.11 Finding Taylor Polynomial Approximations of Functions
      • 10.12 Lagrange Error Bound
      • 10.13 Radius and Interval of Convergence of Power Series
      • 10.14 Finding Taylor or Maclaurin Series for a Function
      • 10.15 Representing Functions as a Power Series
      • End of Unit 10 Review
  • Version #2
    • Derivatives >
      • Unit 0 - Calc Prerequisites >
        • 0.1 Things to Know for Calc
        • 0.2 Summer Packet
        • 0.3 Calculator Skillz
      • Unit 1 - Limits >
        • 1.1 Limits Graphically
        • 1.2 Limits Analytically
        • 1.3 Asymptotes
        • 1.4 Continuity
        • Review - Unit 1
      • Unit 2 - The Derivative >
        • 2.1 Average Rate of Change
        • 2.2 Definition of the Derivative
        • 2.3 Differentiability
        • Review - Unit 2
      • Unit 3 - Basic Differentiation >
        • 3.1 Power Rule
        • 3.2 Product & Quotient Rule
        • 3.3 Velocity/Rates of Change
        • 3.4 Chain Rule
        • 3.5 Trig Derivatives
        • Review - Unit 3
      • Unit 4 - More Derivatives >
        • 4.1 Exp and Log Derivatives
        • 4.2 Inverse Derivatives
        • 4.3 L'Hopitals Rule
        • Review - Unit 4
      • Unit 5 - Curve Sketching >
        • 5.1 Extreme Values
        • 5.2 First Derivative Test
        • 5.3 Second Derivative Test
        • Review - Unit 5
      • Unit 6 - Implicit Differentiation >
        • 6.1 Implicit Differentiation
        • 6.2 Related Rates
        • 6.3 Optimization
        • Review - Unit 6
      • SEMESTER REVIEW
    • Integrals >
      • Unit 7 - Approximation Methods >
        • 7.1 Rectangular Approximation
        • 7.2 Trapezoidal Approximation
      • Unit 8 - Integration >
        • 8.1 Definite Integral
        • 8.2 First Fundamental Theorem of Calculus
        • 8.3 Antiderivatives
        • Review - Unit 8
      • Unit 9 - FTC Part 2 >
        • 9.1 The 2nd FTC
        • 9.2 Trig Integrals
        • 9.3 Average Value
        • 9.4 Net Change
        • Review - Unit 9
      • Unit 10 - More Integrals >
        • 10.1 Slope Fields
        • 10.2 u Sub Indefinite Integral
        • 10.3 u Sub Definite Integral
        • 10.4 Separation of Variables
        • Review - Unit 10
      • Unit 11 - Area and Volume >
        • 11.1 Area Between Curves
        • 11.2 Solids of Revolution (Discs)
        • 11.3 Solids of Revolution (Washers)
        • 11.4 Perpendicular Cross Sections
        • Review - Unit 11
  • FRQ
    • 2022 AB
    • 2022 BC
  • Teacher Resources

Two versions of calculus lessons are listed below.  Both cover all of *AP Calculus AB

Version #1
​The course below follows CollegeBoard's Course and Exam Description.

Unit 0 - Calc Prerequisites (Summer Work)
        0.1 Summer Packet
Unit 1 - Limits and Continuity
        1.1 Can Change Occur at an Instant?
        1.2 Defining Limits and Using Limit Notation
        1.3 Estimating Limit Values from Graphs
        1.4 Estimating Limit Values from Tables
        1.5 Determining Limits Using Algebraic Properties
              (1.5 includes piecewise functions involving limits)
        1.6 Determining Limits Using Algebraic Manipulation
        1.7 Selecting Procedures for Determining Limits
              (1.7 includes rationalization, complex fractions, and                    absolute value)
        1.8 Determining Limits Using the Squeeze Theorem
        1.9 Connecting Multiple Representations of Limits
        Mid-Unit Review - Unit 1
        1.10 Exploring Types of Discontinuities
        1.11 Defining Continuity at a Point
        1.12 Confirming Continuity Over an Interval

        1.13 Removing Discontinuities
        1.14 Infinite Limits and Vertical Asymptotes
        1.15 Limits at Infinity and Horizontal Asymptotes

        1.16 Intermediate Value Theorem (IVT)
        Review - Unit 1
Unit 2 - Differentiation: Definition and Fundamental Properties
        2.1 Defining Average and Instantaneous Rate of
              Change at a Point

        2.2 Defining the Derivative of a Function and Using
              Derivative Notation

              (2.2 includes equation of the tangent line)
        2.3 Estimating Derivatives of a Function at a Point
        2.4 Connecting Differentiability and Continuity
        2.5 Applying the Power Rule
        2.6 Derivative Rules: Constant, Sum, Difference, and
              Constant Multiple

              (2.6 includes horizontal tangent lines, equation of the
                normal line, and differentiability of piecewise
)
        2.7 Derivatives of cos(x), sin(x), e^x, and ln(x)
        2.8 The Product Rule
        2.9 The Quotient Rule
        2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x) 

        Review - Unit 2
Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions
        3.1 The Chain Rule
        3.2 Implicit Differentiation
        3.3 Differentiating Inverse Functions
        3.4 Differentiating Inverse Trigonometric Functions
        3.5 Selecting Procedures for Calculating Derivatives
        3.6 Calculating Higher-Order Derivatives
        Review - Unit 3
Unit 4 - Contextual Applications of Differentiation
        4.1 Interpreting the Meaning of the Derivative in Context
        4.2 Straight-Line Motion: Connecting Position, Velocity,
              and Acceleration

        4.3 Rates of Change in Applied Contexts Other Than
              Motion

        4.4 Introduction to Related Rates
        4.5 Solving Related Rates Problems
        4.6 Approximating Values of a Function Using Local
              Linearity and Linearization

        4.7 Using L'Hopital's Rule for Determining Limits of
              Indeterminate Forms

       Review - Unit 4
Unit 5 - Analytical Applications of Differentiation
        5.1 Using the Mean Value Theorem
        5.2 Extreme Value Theorem, Global Versus Local
              Extrema, and Critical Points

        5.3 Determining Intervals on Which a Function is
              Increasing or Decreasing

        5.4 Using the First Derivative Test to Determine Relative
              Local Extrema

        5.5 Using the Candidates Test to Determine Absolute
              (Global) Extrema

        5.6 Determining Concavity of Functions over Their
              Domains

         5.7 Using the Second Derivative Test to Determine
               Extrema

        Mid-Unit Review - Unit 5
        5.8 Sketching Graphs of Functions and Their Derivatives
        5.9 Connecting a Function, Its First Derivative, and Its
              Second Derivative

              (5.9 includes a revisit of particle motion and
               determining if a particle is speeding up/down.)
        5.10 Introduction to Optimization Problems
        5.11 Solving Optimization Problems
        5.12 Exploring Behaviors of Implicit Relations

        Review - Unit 5
Unit 6 - Integration and Accumulation of Change
        6.1 Exploring Accumulation of Change
        6.2 Approximating Areas with Riemann Sums
        6.3 Riemann Sums, Summation Notation, and Definite
              Integral Notation

        6.4 The Fundamental Theorem of Calculus and
              Accumulation Functions
        6.5 Interpreting the Behavior of Accumulation Functions
​              Involving Area

        Mid-Unit Review - Unit 6
        6.6 Applying Properties of Definite Integrals
        6.7 The Fundamental Theorem of Calculus and Definite
              Integrals

        6.8 Finding Antiderivatives and Indefinite Integrals:
              Basic Rules and Notation

        6.9 Integrating Using Substitution
        6.10 Integrating Functions Using Long Division
​                and 
Completing the Square
        6.11 Integrating Using Integration by Parts (BC topic)
        6.12 Integrating Using Linear Partial Fractions (BC topic)
        6.13  Evaluating Improper Integrals (BC topic)
        6.14 Selecting Techniques for Antidifferentiation
        Review - Unit 6
Unit 7 - Differential Equations
        7.1 Modeling Situations with Differential Equations
        7.2 Verifying Solutions for Differential Equations
        7.3 Sketching Slope Fields
        7.4 Reasoning Using Slope Fields
        7.5 Euler's Method (BC topic)
        7.6 General Solutions Using Separation of Variables

        7.7 Particular Solutions using Initial Conditions and
             Separation of Variables

        7.8 Exponential Models with Differential Equations
        7.9 Logistic Models with Differential Equations (BC topic)

        Review - Unit 7
Unit 8 - Applications of Integration
        8.1 Average Value of a Function on an Interval
        8.2 Position, Velocity, and Acceleration Using Integrals
        8.3 Using Accumulation Functions and Definite Integrals
              in Applied Contexts

        8.4 Area Between Curves (with respect to x)

        8.5 Area Between Curves (with respect to y)
        8.6 Area Between Curves - More than Two Intersections
        Mid-Unit Review - Unit 8
        8.7 Cross Sections: Squares and Rectangles
        8.8 Cross Sections: Triangles and Semicircles
        8.9 Disc Method: Revolving Around the x- or y- Axis
        8.10 Disc Method: Revolving Around Other Axes
        8.11 Washer Method: Revolving Around the x- or y-Axis
        8.12 Washer Method: Revolving Around Other Axes
        8.13 The Arc Length of a Smooth, Planar Curve and
                Distance Traveled
(BC topic)

        Review - Unit 8
Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions  (BC topics)
        9.1 Defining and Differentiating Parametric Equations
        9.2 Second Derivatives of Parametric Equations
        9.3 Arc Lengths of Curves (Parametric Equations)
        9.4 Defining and Differentiating Vector-Valued Functions

        9.5 Integrating Vector-Valued Functions
        9.6 Solving Motion Problems Using Parametric and
              Vector-Valued Functions
            

        9.7 Defining Polar Coordinates and Differentiating in
              Polar Form

        9.8 Find the Area of a Polar Region or the Area Bounded
              by a Single Polar Curve

        9.9 Finding the Area of the Region Bounded by Two
              Polar Curves

        Review - Unit 9
Unit 10 - Infinite Sequences and Series (BC topics)
        10.1 Defining Convergent and Divergent Infinite Series
        10.2 Working with Geometric Series
        10.3 The nth Term Test for Divergence
        10.4 Integral Test for Convergence

        10.5 Harmonic Series and p-Series
        10.6 Comparison Tests for Convergence
        10.7 Alternating Series Test for Convergence
        10.8 Ratio Test for Convergence

        10.9 Determining Absolute or Conditional Convergence
        Mid-Unit Review - Unit 10
​        10.10 Alternating Series Error Bound
        10.11 Finding Taylor Polynomial Approximations of
                  Functions

        10.12 Lagrange Error Bound
        10.13 Radius and Interval of Convergence of Power
                  Series

        10.14 Finding Taylor Maclaurin Series for a Function
        10.15 Representing Functions as a Power Series

        Review - Unit 10

Version #2
​The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day.

Lessons and packets are longer because they cover more material.

Unit 0 - Calc Prerequisites (Summer Work)
        0.1 Things to Know for Calc
        0.2 Summer Packet
        0.3 Calculator Skillz
Unit 1 - Limits
        1.1 Limits Graphically
        1.2 Limits Analytically
        1.3 Asymptotes
        1.4 Continuity
        Review - Unit 1
 Unit 2 - The Derivative
        2.1 Average Rate of Change
        2.2 Definition of the Derivative
        2.3 Differentiability [Calculator Required]
​        Review - Unit 2
 Unit 3 - Basic Differentiation
        3.1 Power Rule
        3.2 Product and Quotient Rules
        3.3 Velocity and other Rates of Change
        3.4 Chain Rule
        3.5 Trig Derivatives
        Review - Unit 3   
Unit 4 - More Deriviatvies
        4.1 Derivatives of Exp. and Logs
        4.2 Inverse Trig Derivatives
        4.3 L'Hopital's Rule
        Review - Unit 4
Unit 5 - Curve Sketching
        5.1 Extrema on an Interval
        5.2 First Derivative Test
        5.3 Second Derivative Test
        Review - Unit 5
Unit 6 - Implicit Differentiation
        6.1 Implicit Differentiation
        6.2 Related Rates
        6.3 Optimization
​        Review - Unit 6
Unit 7 - Approximation Methods
        7.1 Rectangular Approximation Method
        7.2 Trapezoidal Approximation Method
        Review - Unit 7   
Unit 8 - Integration
        8.1 Definite Integral
        8.2 Fundamental Theorem of Calculus (part 1)
        8.3 Antiderivatives (and specific solutions)
        Review - Unit 8   
Unit 9 - The 2nd Fundamental Theorem of Calculus
        9.1 The 2nd FTC
        9.2 Trig Integrals
        9.3 Average Value (of a function)
        9.4 Net Change
        Review - Unit 9
Unit 10 - More Integrals
        10.1 Slope Fields
        10.2 u-Substitution (indefinite integrals)
        10.3 u-Substitution (definite integrals)
        10.4 Separation of Variables
        Review - Unit 10
Unit 11 - Area and Volume
        11.1 Area Between Two Curves
        11.2 Volume - Disc Method
        11.3 Volume - Washer Method
​        11.4 Perpendicular Cross Sections
        Review - Unit 11
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