## 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*

- Normal-and-tangent-line
- Motion-along-line-derivatives
- Implicit-differentiation
- Derivatives-inverse-functions
- Minima-maxima-and-critical-points
- Absolute-relative-maxima-minima
- Concavity-and-inflection-points
- Optimization
- Differentiation-applications-in-different-fields
- Related-rates
- Mean-value-theorem
- Introduction-to-L'Hôpital's-rule
- Local-linearization

*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.