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AP Calculus
AB Correlations
BC Correlations
5th Six Weeks Assignment
Calculus BC
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All of the following came from Collegeboard's Course Discription of
AP Calculus
Topic Outline for Calculus AB
I. Functions, Graphs, and Limits
Analysis of Graphs
Limits of Functions (Including onesided limits)
An intuitive understanding of the limiting process
Calculating limits using algebra
Estimating limits from graphs or tables of data
Asymptotic and unbounded behavior
Understanding asymptotes in terms of graphical behavior
Describing asymptotic behavior in terms of limits involving infinity
Comparing relative magnitudes of functions and their rates of change
Continuity as a property of functions
An intuitive understanding of continuity.
Understanding continuity in terms of limits
Geometric understanding of graphs of continuous functions
+
Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.
II. Derivatives
Concept of the derivative
Derivative presented graphically, numerically, and analytically
Derivative interpreted as an instantaneous rate of change
Derivative defined as the limit of the difference quotient
Relationship between differentiability and continuity
Derivative at a point
Slope of a curve at a point.
Tangent line to a curve at a point and local linear approximation
Instantaneous rate of change as the limit of average rate of change
Approximate rate of change from graphs and tables of values
Derivative as a function
Corresponding characteristics of graphs of f and f'
Relationship between the increasing and decreasing behavior of f and the sign of f'
The Mean Value Theorem and its geometric consequences
Equations involving derviatives. Verbal descriptins are translated into equations involving derivatives and vice versa.
Second derivatives
Corresponding characteristics of the graphs of f, f', and f''
Relationship between the concavity of f and the sign of f''
Points of inflection as places where concavity changes
Applications of derivatives
Analysis of curves, including the notions of monotonicity and concavity
+
Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors.
Optimization, both absolute and relative extrema
Modeling rates of change, including related rates problems
Use of implicit differentiation to find the derivative of an inverse function
Interpretation of the derivative as a rate of change in varied applied contexts
Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
+
Numerical solution of differential equations using Euler's Method.
+
L'HÃ´pital's Rule, including its use in determining limits and convergence of improper integrals and series.
Computation of derivatives
Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions
Basic rules for the derivative of sums, products, and quotients of functions
Chain rule and implicit differentiation
+
Derivatives of parametric, polar, and vector functions.
III. Integrals
Interpretations and properties of definite integrals
Defined integral as a limit of Riemann sums
Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
Basic properties of definite integrals
Applications of integrals
Fundamental Theorem of Calculus
Use of the Fundamental Theorem to evaluate difinite integrals
Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined
Techniques of antidifferentiation
Antiderivatives following directly from derivatives of basic functions
+
Antiderivatives by substitution of variables(including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).
+
Improper integrals (as limits of definite integrals)
.
Applications of antidifferentiation
Finding specific antiderivatives using initial conditions, including applications to motion along a line
Solving separable differential equations and using them in modeling
+
Solving logistic differential equations and using them in modeling.
Numerical approximations to difinite integrals
Riemann Sums(using left, right, and midpoint evaluation points)
Trapezoidal sums
+IV. Polynomial Approximations and Series
Concept of a series
+
explaination of what a series is and when it converges (looking at the sequence of partial sums).
Series of constants
+
Motivating examples, including decimal expansion.
+
Geometric series with applications.
+
The harmonic series.
+
Alternating series with error bound.
+
Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its uses in testing the convergence of pseries.
+
The ratio test for convergence and divergence.
+
Comparing series to test for convergence or divergence.
Taylor series
+ Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve).
+
Maclaurin series and the general Taylor series centered at x=a.
+
Maclaurin series for the function e^x, sinx, cosx, and 1/(1x)
+ Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
+
Functions defined by power series.
+
Radius and interval of convergence of power series.
+
Lagrange error bound for Taylor Polynomials.
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Topic Outline for Calculus AB
I. Functions, Graphs, and Limits
Analysis of Graphs
Limits of Functions (Including onesided limits)
 An intuitive understanding of the limiting process
 Calculating limits using algebra
 Estimating limits from graphs or tables of data
Asymptotic and unbounded behavior Understanding asymptotes in terms of graphical behavior
 Describing asymptotic behavior in terms of limits involving infinity
 Comparing relative magnitudes of functions and their rates of change
Continuity as a property of functions An intuitive understanding of continuity.
 Understanding continuity in terms of limits
 Geometric understanding of graphs of continuous functions
+ Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form, and vector form.II. Derivatives
Concept of the derivative
 Derivative presented graphically, numerically, and analytically
 Derivative interpreted as an instantaneous rate of change
 Derivative defined as the limit of the difference quotient
 Relationship between differentiability and continuity
Derivative at a point Slope of a curve at a point.
 Tangent line to a curve at a point and local linear approximation
 Instantaneous rate of change as the limit of average rate of change
 Approximate rate of change from graphs and tables of values
Derivative as a function Corresponding characteristics of graphs of f and f'
 Relationship between the increasing and decreasing behavior of f and the sign of f'
 The Mean Value Theorem and its geometric consequences
 Equations involving derviatives. Verbal descriptins are translated into equations involving derivatives and vice versa.
Second derivatives Corresponding characteristics of the graphs of f, f', and f''
 Relationship between the concavity of f and the sign of f''
 Points of inflection as places where concavity changes
Applications of derivatives Analysis of curves, including the notions of monotonicity and concavity
+ Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration vectors. Optimization, both absolute and relative extrema
 Modeling rates of change, including related rates problems
 Use of implicit differentiation to find the derivative of an inverse function
 Interpretation of the derivative as a rate of change in varied applied contexts
 Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
+ Numerical solution of differential equations using Euler's Method.+ L'HÃ´pital's Rule, including its use in determining limits and convergence of improper integrals and series.
Computation of derivatives
 Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functionsBasic rules for the derivative of sums, products, and quotients of functions
 Chain rule and implicit differentiation
+Derivatives of parametric, polar, and vector functions.III. Integrals
Interpretations and properties of definite integrals
 Defined integral as a limit of Riemann sums
 Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
 Basic properties of definite integrals
Applications of integralsFundamental Theorem of Calculus
 Use of the Fundamental Theorem to evaluate difinite integrals
 Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined
Techniques of antidifferentiation Antiderivatives following directly from derivatives of basic functions
+ Antiderivatives by substitution of variables(including change of limits for definite integrals), parts, and simple partial fractions (nonrepeating linear factors only).+ Improper integrals (as limits of definite integrals).
Applications of antidifferentiation
 Finding specific antiderivatives using initial conditions, including applications to motion along a line
 Solving separable differential equations and using them in modeling
+ Solving logistic differential equations and using them in modeling.Numerical approximations to difinite integrals
+IV. Polynomial Approximations and Series
Concept of a series
+ explaination of what a series is and when it converges (looking at the sequence of partial sums).
Series of constants
+ Motivating examples, including decimal expansion.
+ Geometric series with applications.
+ The harmonic series.
+ Alternating series with error bound.
+Terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its uses in testing the convergence of pseries.
+ The ratio test for convergence and divergence.
+ Comparing series to test for convergence or divergence.
Taylor series
+ Taylor polynomial approximation with graphical demonstration of convergence (for example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve).
+ Maclaurin series and the general Taylor series centered at x=a.
+ Maclaurin series for the function e^x, sinx, cosx, and 1/(1x)
+ Formal manipulation of Taylor series and shortcuts to computing Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
+ Functions defined by power series.
+ Radius and interval of convergence of power series.
+ Lagrange error bound for Taylor Polynomials.