Mathematics Standards

Calculus

 
 
When taught in high school, calculus should be presented with the same level of depth and rigor as entry level college and university calculus courses. These standards outline a complete college curriculum in one variable calculus. It is recognized that many high school programs may have insufficient time to cover all of the following content in a typical academic year. For example some districts may treat differential equations lightly and spend substantial time on infinite sequences and series. Others may do the opposite. Consideration of the College Board syllabi for the AB and BC Advanced Placement exams may be helpful in making curricular decisions. Calculus is a widely applied area of mathematics, and also involves a beautiful intrinsic theory. Students mastering this content will be exposed to both these important aspects of the subject.
 
  1. Students demonstrate knowledge of both the formal definition and the graphical interpretation of limit of values of functions. This includes one-sided limits, infinite limits, and limits at infinity. Students know the definition of convergence and divergence of a function as the domain variable either approaches a number or infinity.

    1. Students prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.

    2. Students verify and estimate limits using graphical calculators.

    3. Students prove and use special limits such as the limits of (sin(x))/x and (1 - cos(x))/x as x tends to 0.

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

  3. Students demonstrate understanding and application of the Intermediate Value Theorem and the Extreme Value Theorem.

  4. Students demonstrate understanding of the formal definition of the derivative of a function at a point, and the notion of differentiability.

    1. Students demonstrate understanding of the derivative of a function as the slope of the tangent line to the graph of the function.

    2. Students demonstrate understanding of the interpretation of the derivative as instantaneous rate of change. Students can use derivatives to solve a variety of problems coming from physics, chemistry, economics, etc, that involve the rate of change of a function.

    3. Students understand the relation between differentiability and continuity.

    4. Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

  5. Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions.

  6. Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems coming from physics, chemistry, economics, etc.

  7. Students compute derivatives of higher orders.

  8. Students know and can apply Rolle's theorem, the Mean Value Theorem, and L'Hopital's rule.

  9. Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals where the function is increasing and decreasing.

  10. Students know Newton's method for approximating the zeros of a function.

  11. Students use differentiation to solve optimization (maximum - minimum problems) in a variety of pure and applied contexts.

  12. Students use differentiation to solve related rate problems in a variety of pure and applied contexts.

  13. Students know the definition of the definite integral using Riemann sums. They use this definition to approximate integrals.

  14. Students apply the definition of the integral to model problems in physics, economics, etc, obtaining results in terms of integrals.

  15. Students demonstrate knowledge of and proof of the Fundamental Theorem of Calculus, and use it to interpret integrals as anti-derivatives.

  16. Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

  17. Students compute, by hand, the integrals of a wide variety of functions using techniques of integration such as:

    1. Substitution

    2. Integration by parts

    3. Trigonometric substitution

    They can also combine these techniques when appropriate.

  18. Students know the definitions and properties of inverse trigonometric functions, and their appearance as indefinite integrals.

  19. Students compute, by hand, the integrals of rational functions by combining the above techniques with the algebraic techniques of partial fractions and completing the square.

  20. Students compute the integrals of trigonometric functions using the above techniques.

  21. Students understand the algorithms involved in Simpson's rule and Newton's method. They use calculators and/or computers to approximate integrals numerically.

  22. Students understand improper integrals as limits of definite integrals.

  23. Students demonstrate understanding of the definitions of convergence and divergence of sequences and series of real numbers. They can determine whether a series converges using such tests as the comparison test, ratio test, and alternate series test.

  24. Students understand and can compute the radius (interval) of convergence of power series.

  25. Students differentiate and integrate the terms of a power series in order to form new series from known ones.

  26. Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.

  27. Students know the techniques of solution of selected elementary differential equations, and their applications to a wide variety of situations, including growth and decay problems.
 
 
 

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