Calculus
Course description:
Calculus is an advanced mathematics course that uses meaningful
problems and appropriate technology to develop concepts and applications related
to continuity and discontinuity of functions and differentiation, and integration.
Standard 1.0: Functions
Students will expand the concept of functions to include the
analysis and interpretation of both continuous and discontinuous functions
in problem situations and the development of the concept of limit.
Learning Expectations:
Students will:
- 1.1 demonstrate
an understanding of the concepts and applications related to a variety of
continuous functions;
- 1.2 calculate
and estimate limits;
- 1.3 represent
a variety of functions graphically;
- 1.4 use graphical
representations to demonstrate an understanding of asymptotes;
- 1.5 use a variety
of methods to analyze and interpret functions;
- 1.6 apply functions
in problem situations.
Student Performance Indicators:
- analyze
the graphs of polynomial, rational, radical, and transcendental functions using
appropriate technology;
- predict
and explain the observed local and global behavior of a function;
- calculate
limits using algebra;
- estimate
limits from graphs or tables of data.
- demonstrate
an understanding of asymptotes in terms of graphical behavior;
- describe
asymptotic behavior in terms of infinite limits and limits at infinity;
- compare
relative magnitudes of functions and their rates of change.
- demonstrate
an understanding continuity in terms of limits;
- demonstrate
a geometric understanding of graphs of continuous functions.
Standard 2.0: Derivatives
Students will extend the concept of slope of a line to develop
the concept of derivative.
Learning Expectations:
The student will:
- 2.1 define, represent and interpret the concept of derivative;
- 2.2 use the derivative of a function to characterize
the function and vice versa;
- 2.1 connect
the relationships among a function and its first and second derivative;
- 2.2 apply basic
rules for differentiation;
- 2.3 apply derivatives
in problem situations.
Student Performance Indicators:
- represent
the concept of the derivative geometrically, numerically, and analytically;
- interpret
the derivative as an instantaneous rate of change;
- define
the derivative as the limit of the difference quotient;
- articulate
the relationship between differentiability and continuity.
- articulate
corresponding characteristics of graphs of f and f ´ ;
- communicate
the relationship between the increasing and decreasing behavior f and
the sign of f ´ ;
- demonstrate
an understanding of the Mean Value Theorem and its geometric consequence;
- translate
verbal descriptions into equations involving derivatives and vice versa.
- articulate
corresponding characteristics of the graphs of f , f ´ ,
and f ´´ ;
- communicate
the relationship between the concavity of f and the sign of f ´´ ;
- identify
points of inflection;
- analyze
curves using the notions of monotonicity and concavity;
- optimization,
both absolute (global) and relative (local) extrema;
- model
rates of change, including related rates problems;
- use
implicit differentiation to find the derivative of an inverse function;
- interpret
the derivative as a rate of change in varied applied contexts;
- apply
basic rules for the derivative of basic functions and their sum, product,
and quotient;
- use
the chain rule and implicit differentiation.
Standard 3.0: Integrals
- 3.1 define and
apply basic properties of definite integrals;
- 3.2 evaluate
or approximate define integrals;
- 3.3 apply techniques
of antidifferentiation.
- communicate
the relationship between a Riemann sum and a definite integral;
- apply
basic properties of definite integrals;
- evaluate
definite integrals using the Fundamental Theorem;
- apply
techniques of antidifferentiation;
- find
specific antiderivatives using initial conditions, including applications
to motion along a line;
- use
separable differential equations in modeling;
- use
Riemann sums and the Trapezoidal Rule to approximate definite integrals
of functions represented algebraically, geometrically, and by tables of values.
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