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Introduction
Ordering Info
Copyright

© 2006 by Elkhorn Public Schools and Scantron Corporation.
All Rights Reserved.

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Curriculum Designer by
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Elkhorn Public Schools
Math
Mathematics - AP Calculus BC

Functions, Graphs, and Limits
Analysis of graphs
The learner will be able to produce graphs of functions with and without the aid of technology.
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Calculus Master Elkhorn Public Schools(a)
  
Analysis of graphs 2
The learner will be able to use calculus both to predict and to explain the observed local and global behavior of a function.
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Calculus Master Elkhorn Public Schools(a)
  
Limits of functions 1
The learner will be able to intuitively understand the limiting process.
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Calculus Master Elkhorn Public Schools(a)
  
Limits of functions 2
The learner will be able to calculate limits using algebra.
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Calculus Master Elkhorn Public Schools(a)
  
Limits of functions 3
The learner will be able to estimate limits from graphs or tables of data.
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Calculus Master Elkhorn Public Schools(a)
  
Asymptotic and unbounded behavior 1
The learner will be able to understand asymptotes in terms of graphical behavior.
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Calculus Master Elkhorn Public Schools(a)
  
Asymptotic and unbounded behavior 2
The learner will be able to describe asymptotic behavior in terms of limits involving infinity.
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Calculus Master Elkhorn Public Schools(a)
  
Asymptotic and unbounded behavior 3
The learner will be able to compare relative magnitudes of functions and their rates of change. (For example, contrasting exponential growth, polynomial growth, and logarithmic growth.).
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Calculus Master Elkhorn Public Schools(a)
  
Continuity as a property of functions 1
The learner will be able to intuitively understand continuity. (Close values of the domain lead to close values of the range.}.
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Calculus Master Elkhorn Public Schools(a)
  
Continuity as a property of functions 2
The learner will be able to understand continuity in terms of limits.
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Calculus Master Elkhorn Public Schools(a)
  
Continuity as a proeerty of functions 3
The learner will be able to understand geometrically the graphs of continuous functions. (Intermediate Value Theorem and Extreme Value Theorem.
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Calculus Master Elkhorn Public Schools(a)
  
Parametric, polar, and vector functions
The learner will be able to analyze planar curves including those given in parametric form, polar form, and vector form.
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Calculus Master Elkhorn Public Schools(a)
  
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Derivatives
Concept of the derivative
The learner will be able to present the derivative graphically, numerically, and analytically.
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Calculus Master Elkhorn Public Schools(a)
  
Concept of the derivative 2
The learner will be able to interpret the derivative as an instantaneous rate of change.
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Calculus Master Elkhorn Public Schools(a)
  
Concept of the derivative 3
The learner will be able to define the derivative as the limit of the difference quotient.
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Calculus Master Elkhorn Public Schools(a)
  
Concept of the derivative 4
The learner will be able to understand and apply the relationship between differentiablity and continuity.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative at a point 1
The learner will be able to determine the slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative at a point 2
The learner will be able to determine a tangent line to a curve at a point and local linear approximation.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative at a point 3
The learner will be able to determine instantaneous rate of change as the limit of average rate of change.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative at a point 4
The learner will be able to approximate rate of change from graphs and tables of values.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative as a function 1
The learner will be able to analyze corresponding characteristics of graphs of f and f '.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative as a function 2
The learner will be able to understand and apply the relationship between the increasing and decreasing behavior of f and the sign of f '.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative as a function 3
The learner will be able to understand and apply the Mean Value Theorem and its geometric consequences.
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Calculus Master Elkhorn Public Schools(a)
  
Derivative as a function 4
The learner will be able to write equations involving derivatives. Verbal descriptions are translated into equations involving derivatives andvice versa.
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Calculus Master Elkhorn Public Schools(a)
  
Second derivatives 1
The learner will be able to apply and use the corresponding characteristics of the graphs of f, f ', f ".
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Calculus Master Elkhorn Public Schools(a)
  
Second derivatives 2
The learner will be able to apply and analyze the relationship between the concavity of f and the sign of f ".
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Calculus Master Elkhorn Public Schools(a)
  
Second derivatives 3
The learner will be able to understand that points of inflection are places where concavity changes.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 1
The learner will be able to analyze curves, including the notions of monotonicity and concavity.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 2
The learner will be able to analyze planar curves given in parametric form polar form, and vector form, including velocity and acceleration vectors.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 3
The learner will be able to understand and apply optimization, both absolute (global) and relative (local) extrema.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 4
The learner will be able to model rates of change, including related rates problems.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 5
The learner will be able to use implicit differentiation to find the derivative of an inverse function.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 6
The learner will be able to interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 7
The learner will be able to interpret geometrically differential equations via slope fields and the relationshop between slope fields and solution curves for differential equations.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 8
The learner will be able to find numerical solutions of differential equations using Euler's method.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of derivatives 9
The learner will be able to apply L'Hopital's Rule, including its use in determining limits and convergence of improper integrals and series.
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Calculus Master Elkhorn Public Schools(a)
  
Computation of derivatives 1
The learner will be able to find derivatives of basic functions, including power, exponential, logarithmic, trigonometric functions.
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Calculus Master Elkhorn Public Schools(a)
  
Computation of derivatives 2
The learner will be able to use the basic rules for the derivative of sums, products, and quotients of functions.
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Calculus Master Elkhorn Public Schools(a)
  
Computation of derivatives 3
The learner will be able to use the chain rule and implicit differentiation.
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Calculus Master Elkhorn Public Schools(a)
  
Computation of derivatives 4
The learner will be able to determine derivatives of parametric, polar, and vector functions.
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Calculus Master Elkhorn Public Schools(a)
  
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Integrals
Properties of definite integrals 1
The learner will be able to compute Riemann sums using left, right, and midpoint evaluation points.
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Calculus Master Elkhorn Public Schools(a)
  
Properties of definite integrals 2
The learner will be able to understand and apply basic properties of definite integrals. (Examples include additivity and linearity.).
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Calculus Master Elkhorn Public Schools(a)
  
Properties of definite integrals 3
The learner will be able to understand that the definite integral is a limit of Riemann sums over equal subdivisions.
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Calculus Master Elkhorn Public Schools(a)
  
Properties of definite integrals 4
The learner will be able to understand and use the definite integral as the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of integrals
The learner will be able to adapt their knowledge and techniques to solve application problems. Specific applications should include finding the area of a region (including a region bunded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled, and the length of a curve (including a curve given in parametric form).
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Calculus Master Elkhorn Public Schools(a)
  
Fundamental Theorem of Calculus 1
The learner will be able to use the Fundamental Theorem to evaluate definite integrals.
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Calculus Master Elkhorn Public Schools(a)
  
Fundamental Theorem of Calculus 2
The learner will be able to use the Fundamental Theorem to represent a particular anti-derivative, and the analytical and graphical analysis of functions so defined.
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Calculus Master Elkhorn Public Schools(a)
  
Techniques of antidifferentiation 1
The learner will be able to find antiderivaties following directly from derivatives of basic functions.
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Calculus Master Elkhorn Public Schools(a)
  
Techniques of antidifferentiation 2
The learner will be able to determine antiderivatives by substitution of variables (including change of limits for definite integrals), parts, and simle partial fractions (nonrepeating linear factors only).
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Calculus Master Elkhorn Public Schools(a)
  
Techniques of antidifferentiation 3
The learner will be able to determine improper integrals (as limits of definte integrals.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of antidifferentiation 1
The learner will be able to find specific antiderivatives using intial conditions, including applications to motion along a line.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of antidifferentiation 2
The learner will be able to solve separable differential equations and use them in modeling. In particular, studying the equation y ' = ky and exponential growth.
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Calculus Master Elkhorn Public Schools(a)
  
Applications of antidifferentiation 3
The learner will be able to solve logistic differential equations and use them in modeling.
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Calculus Master Elkhorn Public Schools(a)
  
Numerical approx to definte integrals
The learner will be able to use Riemann and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values.
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Calculus Master Elkhorn Public Schools(a)
  
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Polynomial Approximations and Series
Concept of series
The learner will be able to define a series as a sequence of partial sums, and convergence in terms of the limit of the sequence of partial sums. Technology can be used to explore convergence or divergence.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 1
The learner will be able to use decimal expansion.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 2
The learner will be able to apply geometric series.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 3
The learner will be able to use and apply harmonic series.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 4
The learner will be able to use alternating series with an error bound.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 5
The learner will be able to determine terms of series as areas of rectangles and their relationship to improper integrals, including the integral test and its use in testing the convergence of p-series.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 6
The learner will be able to apply the ratio test for convergence and divergence.
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Calculus Master Elkhorn Public Schools(a)
  
Series of constants 7
The learner will be able to compare series to test for convergence and divergence.
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Calculus Master Elkhorn Public Schools(a)
  
Taylor series 1
The learner will be able to use Taylor polynomial approximation with graphical demonstration of convergence. (For example, viewing graphs of various Taylor polynomials of the sine function approximating the sine curve.
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Calculus Master Elkhorn Public Schools(a)
  
Taylor series 2
The learner will be able to use Maclaurin series and the general Tayor series centered at x = a, and use Maclaurin series for the exponential function, sin x, cos x, and 1/(1-x).
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Calculus Master Elkhorn Public Schools(a)
  
Taylor series 3
The learner will be able to formally manipulate the Taylor series and use shortcuts to compute the Taylor series, including substitution, differentiation, antidifferentiation, and the formation of new series from known series.
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Calculus Master Elkhorn Public Schools(a)
  
Taylor series 4
The learner will be able to define functions using power series.
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Calculus Master Elkhorn Public Schools(a)
  
Taylor series 5
The learner will be able to find the radius and interval of convergence of power series.
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Calculus Master Elkhorn Public Schools(a)
  
Taylor series 6
The learner will be able to determine Lagrange error bund for Taylor polynomials.
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Calculus Master Elkhorn Public Schools(a)
  
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