Grades
Eight Through Twelve - Mathematics Content Standards.
When
taught in high school, calculus should be presented with the same level of
depth and rigor as are entry-level college and university calculus courses.
These standards outline a complete college curriculum in one variable calculus.
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 Calculus AB and Calculus BC sections of the Advanced
Placement Examination in Mathematics may be helpful in making curricular decisions.
Calculus is a widely applied area of mathematics and involves a beautiful
intrinsic theory. Students mastering this content will be exposed to both
aspects of the subject.
1.0 Students
demonstrate knowledge of both the formal definition and the graphical
interpretation of limit of values of functions. This knowledge 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
approaches either a number or infinity:
1.1 Students prove and use theorems
evaluating the limits of sums, products, quotients, and composition of
functions.
1.2 Students use graphical calculators to verify and estimate limits.
1.3 Students prove and use special limits, such as the limits of (sin(x))/x and
(1-cos(x))/x as x tends to 0.
2.0 Students demonstrate knowledge of both the formal definition and
the graphical interpretation of continuity of a function.
3.0 Students demonstrate an understanding and the application of the
intermediate value theorem and the extreme value theorem.
4.0 Students demonstrate an understanding of the formal definition of
the derivative of a function at a point and the notion of differentiability:
4.1 Students demonstrate an
understanding of the derivative of a function as the slope of the tangent line
to the graph of the function.
4.2 Students demonstrate an understanding of the interpretation of the
derivative as an instantaneous rate of change. Students can use derivatives to
solve a variety of problems from physics, chemistry, economics, and so forth
that involve the rate of change of a function.
4.3 Students understand the relation between differentiability and continuity.
4.4 Students derive derivative formulas and use them to find the derivatives of
algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic
functions.
5.0 Students know the chain rule and its proof and applications to the
calculation of the derivative of a variety of composite functions.
6.0 Students find the derivatives of parametrically defined functions
and use implicit differentiation in a wide variety of problems in physics,
chemistry, economics, and so forth.
7.0 Students compute derivatives of higher orders.
8.0 Students know and can apply Rolle's theorem, the mean
value theorem, and L'Hôpital's rule.
9.0 Students use differentiation to sketch, by hand, graphs of
functions. They can identify maxima, minima, inflection points, and intervals
in which the function is increasing and decreasing.
10.0
Students know Newton's method for
approximating the zeros of a function.
11.0 Students use differentiation to solve optimization
(maximum-minimum problems) in a variety of pure and applied contexts.
12.0
Students use differentiation to solve
related rate problems in a variety of pure and applied contexts.
13.0
Students know the definition of the
definite integral by using Riemann sums. They use this definition to
approximate integrals.
14.0 Students apply the definition of the integral to model
problems in physics, economics, and so forth, obtaining results in terms of
integrals.
15.0
Students demonstrate knowledge and
proof of the fundamental theorem of calculus and use it to interpret integrals
as antiderivatives.
16.0
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.
17.0
Students compute, by hand, the
integrals of a wide variety of functions by using techniques of integration,
such as substitution, integration by parts, and trigonometric substitution.
They can also combine these techniques when appropriate.
18.0
Students know the definitions and
properties of inverse trigonometric functions and the expression of these
functions as indefinite integrals.
19.0
Students compute, by hand, the
integrals of rational functions by combining the techniques in standard 17.0
with the algebraic techniques of partial fractions and completing the square.
20.0
Students compute the integrals of
trigonometric functions by using the techniques noted above.
21.0
Students understand the algorithms
involved in Simpson's rule and Newton's method. They use calculators or
computers or both to approximate integrals numerically.
22.0
Students understand improper
integrals as limits of definite integrals.
23.0 Students demonstrate an understanding of the definitions
of convergence and divergence of sequences and series of real numbers. By using
such tests as the comparison test, ratio test, and alternate series test, they
can determine whether a series converges.
24.0 Students understand and can compute the radius (interval)
of the convergence of power series.
25.0
Students differentiate and integrate
the terms of a power series in order to form new series from known ones.
26.0
Students calculate Taylor
polynomials and Taylor series of basic functions, including the remainder term.
27.0
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.