## AP Calculus Content & Skills

#### Functions, Graphs, and Limits Unit

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

#### Derivatives Unit

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

#### Applications of the Derivatives Unit

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

#### Integration Unit

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

#### Application of the Integral Unit

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

#### Sequences, Series, and Function Approximation Unit (only for BC Calculus)

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