In 1999, the State of California adopted and mandated standards that affect all public high school students and their teachers.  The Standards set forth the concepts, skills and topics appropriate for students in Calculus.  Items in orange are not in the current AP syllabus.  Items in blue are in the BC syllabus only.

1. Demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions (one-sided, infinite, and infinity, convergence and divergence).

2. Demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.

3. Demonstrate and apply the Intermediate Value Theorem for Derivatives and the Extreme Value Theorem.

4. Demonstrate understanding of the formal definition of the derivative of a function at a point and the notion of differentiability (slope of tangent line, instantaneous rate of change, algebraic derivative shortcuts).

5. Know and apply the chain rule to calculate the derivative of a variety of composite functions.

6. Find the derivatives of parametrically defined functions and apply implicit differentiation to a wide variety of problems.

7. Students compute higher order derivatives.

8. Student’s know and apply Rolle’s theorem, the Mean Value Theorem, and L’Hôpital’s rule.

9. Use differentiation to sketch by hand the graphs of functions identifying extrema, points of inflection, and intervals of increase, decrease, and concavity.

10. Know Newton’s method for approximating the zeros of a function.

11. Use differentiation to solve optimization problems.

12. Use differentiation to solve related rate problems.

13. Know the definition of the definite integral by using Riemann sums and use Riemann sums to approximate definite integrals.

14. Apply the definition of the integral to model problems obtaining results in integral form.

15. Demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.

16. Use definite integrals to solve problems of area, velocity, acceleration, volume, area of surface, arc length, and work.

17. Compute by hand integrals of functions using substitution, integration by parts, and trigonometric substitution.

18. Know and use the properties of inverse trigonometric functions for finding indefinite integrals.

19. Compute by hand integrals using the techniques of partial fractions and completing the square.

20. Compute the integrals of trigonometric functions.

21. Understand and apply the algorithms of Simpson’s rule and Newton’s method.

22. Use improper integrals as limits of definite integrals.

23. Demonstrate understanding of the definitions of convergence and divergence of sequences and series of real numbers and apply comparison test, ration test, and alternative series test to determine convergence of a series.

24. Compute the radius (interval) of the convergence of a power series.

25. Differentiate and integrate the terms of a power series in order to form new series from know series.

26. Calculate Taylor polynomials and Taylor series of basic functions.

27. Know how to solve selected elementary differential equations and apply them to a variety of problems.

AP Syllabus Items not addressed by the California Standards:

1. The Midpoint Rule as a numerical approximation for a definite integral.

2. Average value of a function (the Mean Value Theorem for Integrals).

3. Slope fields and Euler approximations to the solutions of differential equations.

4. Population models and the Logistic Function.

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