• Course Outline (topic order and time allowed is approximate)

     

    Unit 1       Limits & Continuity

    1)    Informal Definition of Limit

    a)    An Intuitive Understanding of the Limiting Process

    b)   One and Two-sided Limits

    c)    Limits that Fail to Exist

    2)   Properties of Limits

    3)   Finding Limits Graphically, Numerically, and Analytically

                   4)    Infinite Limits and Limits at Infinity 

    5)   Determining Continuity from Graphs – properties of continuity 

    6)   Determining Continuity Analytically using Limits 

    4)   Intermediate Value Theorem (IVT) 

    7)   Points of Discontinuity 

    a)    Removable and Non-removable

    b)   Asymptotic Behavior

    6)   Local Linearity 

     

     

    Unit 2       Differentiation

    1)    Exploration of the concept of a Derivative 

    a)    Instantaneous Rate of Change

    b)   Average Rate of Change

    c)    Slope of a Tangent Line

    2)   Formal Definition of a Derivative

    3)  Alternative Form of a Derivative

    4)  Local Linearity

     

    5)   Techniques of Differentiation 

    a)    power rule

    b)   product rule

    c)    quotient rule

    d)   chain rule

    5)   Derivatives of the Transcendental Functions 

    a)    power

    b)   exponential

    c)    logarithmic

    d)   trigonometric

    e)   inverse trigonometric

    6)   Higher Order Derivatives

    7)  Equations of Tangent Lines

     

     

    Unit 3       Existence Theorems

                  1) Intermediate Value Theorem

                   2)  Extreme Value Theorem

                   3) Rolle's Theorem 

                   4)  Mean Value Theorem

    Unit 4       Curve Sketching 

    1)    Relative and Absolute Extrema; Critical Values of a Function 

    2)  Increasing and Decreasing Functions

    3)    Finding Relative Extrema using the First and Second Derivative Tests 

    4.    Increasing and Decreasing Intervals – including Monotonic Functions 

    5.    Concavity and Points of Inflection

    6.    Corresponding characteristics of the graphs of f, f', and f"- graphical and numerical applications Additional Resources

     

    Unit 5       Applications of Differentiation

    1.     Optimization 

    2.    Implicit Differentiation 

    3.    Derivatives of Inverses of Functions 

    4.    Related Rates 

    5.  Motion Along a Line 

    a.    Position

    b.    Velocity

    c.    Acceleration

    6.    Introduction to Slope Fields

    7.    Tangent Line Approximation – including finding the error using concavity 

     

    Unit 6       Introduction to Integral Calculus 

    1.     Approximating Sums – left, right, midpoint, trapezoidal Additional Resources

    2.    Graphical and Numerical Application with Approximating Sums – area, distance, accumulated change. Additional Resources

    3.    Riemann Sums 

    4.    Definition of Definite Integrals as the limit of a summing process 

    5.    Properties of Definite Integrals 

    6.    Antidifferentiation from known derivatives

    7.  Constant of Integration and Particular Solutions. 

     

    Unit 7       Techniques of Integration 

    1.     First and Second Fundamental Theorems of Calculus 

    2.    Mean Value Theorem for Integrals – Average Value of a Function

    3.    Graphical Analysis of Functions using Fundamental and Mean Value Theorems

    4.    “U” Substitution – with change of variables and change of limits of definite integrals 

    5.    Integration of Transcendental Functions 

    a.    Power

    b.    Exponential

    c.    Trigonometric

    d.    Inverse Trigonometric

     

    Unit 8       Differential Equations and Applications of Integration 

    1.     Solving Separable Differential Equations 

    2.    Application with Differential Equations 

    a.    Slope Fields

    b.    Growth and Decay

    3.    Differential Equation that Model Logistic Growth and Their Applications 

    4.    Area Under a Curve and Between Curves

    5.    Volumes of Solids of Revolution About a Horizontal or Vertical Axis 

    a.    Disk

    b.    Washer

    6.    Volumes of Solids with Known Cross Sections

    7.    Using Integrals of Rates of Change to Find Accumulated Change – motion revisited

     

    *This schedule leaves 2-3 weeks for flexibility with the topics and integrating activities into the units.

     

    Course Review (31 days)

    The remainder of the time before the AP Exam (exam is usually during the 1st or 2nd week in May) is used for review with a focus on the relationships between the various concepts and on calculus as a cohesive whole.  The review includes review presentation of topics by students, individual and group work including discussion of past AP problems, review quizzes, and timed practice tests.  Occasionally after the exam, additional calculus topics are discussed as time allows.