Outline for Calculus AB

• Functions, Graphs, and Limits
  • Analysis of graphs
  • Limits of functions (including one-sided limits)
    • An intuitive understanding of the limiting process
    • Calculating limits using algebra
    • Estimating limits from graphs or tablets of data
  • Asymptotic and unbounded behavior
    • Understanding asymptotes in terms of graphical behavior
    • Describing asymptotic behavior in terms of limits involving infinity
    • Comparing relative magnitudes of functions and their rates of change
  • Continuity as a property of functions
    • An intuitive understanding of continuity
    • Understanding continuity in terms of limits
    • Geometric understanding of graphs of continuous functions
• Derivatives
  • Concept of the derivative
    • Derivative presented graphically, numerically, and analytically
    • Derivative interpreted as an instantaneous rate of change
    • Derivative defined as the limit of the difference quotient
    • Relationship between differentiability and continuity
  • Derivative at a point
    • Slope of a curve at a point
    • Tangent line to a curve at a point and local linear approximation
    • Instantaneous rate of change as the limit of average rate of change
    • Approximate rate of change from graphs and tables of values
  • Derivative as a function
    • Corresponding characteristics of graphs "f" and "f'"
    • Relationship between the increasing and decreasing behavior of "f" and the sign of "f'"
    • The Mean Value Theorem and its geometric consequences
    • Equations involving derivatives
  • Second derivatives
    • Corresponding characteristics of graphs "f", "f'", and "f"
    • Relationship between the concavity of "f" and the sign of "f"
    • Points of inflection as places where concavity changes
  • Applications of derivatives
    • Analysis of curves, including the notions of monotonicity and concavity
    • Optimization, both absolute (global) and relative (local) extrema
    • Modeling rates of change, including related rates problems
    • Use of implicit differentiation to find the derivative of an inverse function
  • Computation of derivatives
    • Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions
    • Basic rules for the derivatives of sums, products, and quotients of functions
    • Chain rule and implicit differentiation
• Integrals
  • Interpretations and properties of definite integrals
    • Computation of Riemann sums using left, right, and midpoint evaluation points
    • Definite integral as a limit of Riemann sums over equal subdivisions
    • Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
    • Basic properties of definite integrals
  • Applications of integrals
  • Fundamental Theorem of Calculus
    • Use of the Fundamental Theorem to evaluate definite integrals
    • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined
  • Techniques of antidifferentiation
    • Antiderivatives following directly from derivatives of basic functions
    • Antiderivatives by substitution of variables
  • Applications of antidifferentiation
    • Finding specific antiderivatives using initial conditions, including applications to motion along a line
    • Solving separable differential equations and using them in modeling
  • Numerical approximation to definite integrals