AP Calculus Content & Skills

Learning Expectations
  • Know the formal definition of a limit.
  • Interpret graphs to determine the value of a limit.
  • Use theorems of limits to evaluate sums, products, quotients, and composition of functions.
  • Use graphing calculators to verify and estimate limits
  • Interpret graphical representations of continuity
  • Apply the intermediate value theorem
  Functions, graphs, and limits tutorials
Learning Expectations
  • Demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
  • Understand that the derivative is the instantaneous rate of change.
  • Use derivatives to solve a variety of problems (e.g., physics, etc.)
  • Understand the relation between differentiability and continuity.
  • Derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
  • Apply the chain rule to the calculation of the derivative of a variety of composite functions.
  • Compute derivatives of higher order.
  Derivatives tutorials
Learning Expectations
  • Find derivatives of parametrically defined functions
  • Use implicit differentiation in a wide variety of problems (e.g., physics, etc.)
  • Know and apply Rolle’s theorem, the mean value theorem, and L’Hôpital’s rule
  • Use differentiation to sketch, by hand, graphs of functions.
  • Identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.
  • Know Newton’s method for approximating zeros of a function.
  • Use differentiation to solve optimization problems in a variety of pure and applied contexts.
  • Use differentiation to solve related rate problems in a variety of pure and applied contexts.
  Applications of the derivative tutorials
Learning Expectations
  • Know the definition of a definite integral by using Riemann sums and use this definition to approximate integrals.
  • Apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
  • Use the fundamental theorem of calculus to interpret integrals as anti-derivatives.
  Integration tutorials
Learning Expectations
  • Use definite integrals in problems involving area, velocity, acceleration, volume of a solid, and length of a curve.
  • Compute, by hand, the integrals of a wide variety of functions using techniques of integration, such as substitution, integration by parts, and trigonometric substitution.
  • Know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
  • Compute, by hand, the integrals of rational functions by combining the techniques such as substitution, integration by parts, and trigonometric substitution, with algebraic techniques of partial fractions and completing the square.
  • Understand improper integrals as limits of definite integrals.
  Application of the integral tutorials
Learning Expectations
  • Understand the definition of convergence and divergence of sequences and series of real numbers.
  • Use tests such as the comparison test, ratio test, and alternate series test to determine if a series converges.
  • Compute the radius (interval) of the convergence of power series.
  • Differentiate and integrate the terms of a power series in order to form new series from known ones.
  • Calculate Taylor approximations and Taylor series of basic functions, including the remainder term.
  Sequences, series, and function approximation tutorials