# Calculus

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```					                                                                      Calculus, Advanced Placement

Standard 1: Limits and              Standard 2: Differential        Standard 3: Applications         Standard 4: Integral            Standard 5: Applications
Continuity                          Calculus                        of Derivatives                   Calculus                        of Integration

Students understand the             Students find derivatives of    Students find slopes and         Students define integrals       Students find velocity
concept of limit, find limits of    algebraic, trigonometric,       tangents, maximum and            using Riemann Sums, use         functions and position
functions at points and at          logarithmic, and exponential    minimum points, and points       the Fundamental Theorem of      functions from their
infinity, decide if a function is   functions. They find            of inflection. They solve        Calculus to find integrals,     derivatives, solve separable
continuous, and use                 derivatives of sums,            optimization problems and        and use basic properties of     differential equations, and
continuity theorems.                products, and quotients, and    find rates of change.            integrals. They integrate by    use definite integrals to find
composite and inverse                                            substitution and find           areas and volumes.
functions. They find                                             approximate integrals.
derivatives of higher order
and use logarithmic
differentiation and the Mean
Value Theorem.

1. Understand the                  1. Understand the concept        1. Find the slope of a           1. Use rectangle                1. Find specific
concept of limit and               of derivative                    curve at a point,                approximations to find          antiderivatives using
estimate limits from               geometrically,                   including points at              approximate values of           initial conditions,
graphs and tables of               numerically, and                 which there are                  integrals.                      including finding
values.                            analytically, and                vertical tangents and         2. Calculate the values of         velocity functions from
2. Find limits by                     interpret the derivative         no tangents.                     Riemann Sums over               acceleration functions,
substitution.                      as a rate of change.          2. Find a tangent line to           equal subdivisions              finding position
3. Find limits of sums,            2. State, understand, and           a curve at a point and           using left, right, and          functions from velocity
differences, products,             apply the definition of          a local linear                   midpoint evaluation             functions, and
and quotients.                     derivative.                      approximation.                   points.                         applications to motion
4. Find limits of rational         3. Find the derivatives of       3. Decide where                  3. Interpret a definite            along a line.
functions that are                 functions, including             functions are                    integral as a limit of       2. Solve separable
undefined at a point.              algebraic,                       decreasing and                   Riemann Sums.                   differential equations
5. Find one-sided limits.             trigonometric,                   increasing.                   4. Understand the                  and use them in
6. Find limits at infinity.           logarithmic, and                 Understand the                   Fundamental Theorem             modeling.
7. Decide when a limit is             exponential functions.           relationship between             of Calculus: Interpret       3. Solve differential
infinite and use limits         4. Find the derivatives of          the increasing and               a definite integral of          equations of the form
involving infinity to              sums, products, and              decreasing behavior of           the rate of change of a         y' = ky as applied to
describe asymptotic                quotients.                       f and the sign of f'.            quantity over an                growth and decay

Standard 1: Limits and              Standard 2: Differential       Standard 3: Applications         Standard 4: Integral             Standard 5: Applications
Continuity                          Calculus                       of Derivatives                   Calculus                         of Integration

Students understand the             Students find derivatives of   Students find slopes and         Students define integrals        Students find velocity
concept of limit, find limits of    algebraic, trigonometric,      tangents, maximum and            using Riemann Sums, use          functions and position
functions at points and at          logarithmic, and exponential   minimum points, and points       the Fundamental Theorem of       functions from their
infinity, decide if a function is   functions. They find           of inflection. They solve        Calculus to find integrals,      derivatives, solve separable
continuous, and use                 derivatives of sums,           optimization problems and        and use basic properties of      differential equations, and
continuity theorems.                products, and quotients, and   find rates of change.            integrals. They integrate by     use definite integrals to find
composite and inverse                                           substitution and find            areas and volumes.
functions. They find                                            approximate integrals.
derivatives of higher order
and use logarithmic
differentiation and the Mean
Value Theorem.

behavior.                       5. Find the derivatives of      4. Find local and absolute          interval as the change           problems.
8. Find special limits                composite functions,            maximum and                      of the quantity over          4. Use definite integrals
sin x
using the chain rule.           minimum points.                  the interval, that is            to find the area
such as            /x .         6. Find the derivatives of      5. Analyze curves,                                                   between a curve and
9. Understand continuity              implicitly-defined              including the notions                                             the x-axis, or between
in terms of limits.                                                                                     f'(x)dx = f(b) –
functions.                      of monotonicity and                                               two curves.
10.Decide if a function is                                                                             f(a).
7. Find derivatives of             concavity.                                                     5. Use definite integrals
continuous at a point.                                                                           5. Use the Fundamental
inverse functions.           6. Find points of                                                    to find the average
11.Find the types of                                                                                   Theorem of Calculus
8. Find second derivatives         inflection of functions.                                          value of a function
discontinuities of a                                                                                to evaluate definite
and derivatives of              Understand the                                                    over a closed interval.
function.                                                                                           and indefinite integrals
higher order.                   relationship between                                           6. Use definite integrals
12.Understand and use                                                                                  and to represent
9. Find derivatives using          the concavity of f and                                            to find the volume of a
the Intermediate                                                                                    particular
logarithmic                     the sign of f".                                                   solid with known
Value Theorem on a                                                                                  antiderivatives.
differentiation.                Understand points of                                              cross-sectional area.
function over a closed                                                                              Perform analytical and
10.Understand and use              inflection as places                                           7. Apply integration to
interval.                                                                                           graphical analysis of
the relationship                where concavity                                                   model and solve
13.Understand and apply                                                                                functions so defined.

Standard 1: Limits and              Standard 2: Differential       Standard 3: Applications          Standard 4: Integral           Standard 5: Applications
Continuity                          Calculus                       of Derivatives                    Calculus                       of Integration

Students understand the             Students find derivatives of   Students find slopes and          Students define integrals      Students find velocity
concept of limit, find limits of    algebraic, trigonometric,      tangents, maximum and             using Riemann Sums, use        functions and position
functions at points and at          logarithmic, and exponential   minimum points, and points        the Fundamental Theorem of     functions from their
infinity, decide if a function is   functions. They find           of inflection. They solve         Calculus to find integrals,    derivatives, solve separable
continuous, and use                 derivatives of sums,           optimization problems and         and use basic properties of    differential equations, and
continuity theorems.                products, and quotients, and   find rates of change.             integrals. They integrate by   use definite integrals to find
composite and inverse                                            substitution and find          areas and volumes.
functions. They find                                             approximate integrals.
derivatives of higher order
and use logarithmic
differentiation and the Mean
Value Theorem.

the Extreme Value                 between                         changes.                       6. Understand and use              problems in physics,
Theorem: If f(x) is               differentiability and        7. Use first and second              properties of these             biology, economics,
continuous over a                 continuity.                     derivatives to help               definite integrals:             etc., using the integral
closed interval, then f        11.Understand and apply            sketch graphs.                                                    as a rate of change to
has a maximum and a               the Mean Value                  Compare the                                                       give accumulated
[f(x) + g(x)]dx =
minimum on the                    Theorem.                        corresponding                                                     change and using the
interval.                                                         characteristics of the                                            method of setting up
graphs of f, f', and f''.             f(x)dx +                    an approximating
8. Use implicit                      g(x)dx                          Riemann Sum and
differentiation to find                                           representing its limit
the derivative of an                                              as a definite integral.
inverse function.                     k . f(x)dx = k
9. Solve optimization                f(x)dx
problems.
10.Find average and                      f(x)dx = 0
instantaneous rates of

Standard 1: Limits and              Standard 2: Differential       Standard 3: Applications         Standard 4: Integral            Standard 5: Applications
Continuity                          Calculus                       of Derivatives                   Calculus                        of Integration

Students understand the             Students find derivatives of   Students find slopes and         Students define integrals       Students find velocity
concept of limit, find limits of    algebraic, trigonometric,      tangents, maximum and            using Riemann Sums, use         functions and position
functions at points and at          logarithmic, and exponential   minimum points, and points       the Fundamental Theorem of      functions from their
infinity, decide if a function is   functions. They find           of inflection. They solve        Calculus to find integrals,     derivatives, solve separable
continuous, and use                 derivatives of sums,           optimization problems and        and use basic properties of     differential equations, and
continuity theorems.                products, and quotients, and   find rates of change.            integrals. They integrate by    use definite integrals to find
composite and inverse                                           substitution and find           areas and volumes.
functions. They find                                            approximate integrals.
derivatives of higher order
and use logarithmic
differentiation and the Mean
Value Theorem.

change. Understand
the instantaneous rate
of change as the limit               f(x)dx =
of the average rate of           f(x)dx
change. Interpret a
derivative as a rate of              f(x)dx +     f(x)dx
change in applications,
including velocity,
=      f(x)dx
speed, and
If f(x)    g(x) on
acceleration.
11.Find the velocity and
acceleration of a                [a,b], then     f(x)dx
particle moving in a
straight line.
12.Model rates of change,                  g(x)dx.
7. Understand and use
including related rates
integration by

Standard 1: Limits and              Standard 2: Differential       Standard 3: Applications         Standard 4: Integral           Standard 5: Applications
Continuity                          Calculus                       of Derivatives                   Calculus                       of Integration

Students understand the             Students find derivatives of   Students find slopes and         Students define integrals      Students find velocity
concept of limit, find limits of    algebraic, trigonometric,      tangents, maximum and            using Riemann Sums, use        functions and position
functions at points and at          logarithmic, and exponential   minimum points, and points       the Fundamental Theorem of     functions from their
infinity, decide if a function is   functions. They find           of inflection. They solve        Calculus to find integrals,    derivatives, solve separable
continuous, and use                 derivatives of sums,           optimization problems and        and use basic properties of    differential equations, and
continuity theorems.                products, and quotients, and   find rates of change.            integrals. They integrate by   use definite integrals to find
composite and inverse                                           substitution and find          areas and volumes.
functions. They find                                            approximate integrals.
derivatives of higher order
and use logarithmic
differentiation and the Mean
Value Theorem.

problems.                       substitution (or
change of variable) to
find values of
integrals.
8. Understand and use
Riemann Sums, the
Trapezoidal Rule, and
technology to
approximate definite
integrals of functions
represented
algebraically,
geometrically, and by
tables of values.

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 views: 5 posted: 12/1/2011 language: English pages: 5