EAST LONGMEADOW PUBLIC SCHOOLS
2006 - 2007 DISTRICT CURRICULUM GUIDES

Mathematics - Calculus and AP Calculus

Mathematics

The Massachusetts Mathematics Curriculum Framework envisions all students in the Commonwealth achieving mathematical competence through a strong mathematics program that emphasizes problem solving, communicating, reasoning and proof, making connections, and using representations. Acquiring such competence depends in large part on a clear, comprehensive, coherent, and developmentally appropriate set of standards to guide curriculum expectations.


Algebraic Concepts

    Algorithms: Create
The learner will be able to create algorithms to solve problems.
Strand Scope Source
Algorithms Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  
    Algorithms: Analyze
The learner will be able to analyze algorithms.
Strand Scope Source
Algorithms Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  
    Algorithms: Exploring/Validation
The learner will be able to explore problems which involve the validation of algorithms using calculators and computers.
Strand Scope Source
Algorithms Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  

Calculus and Pre-Calculus

    Problem Solving: Graphs of Maxima/Minima
The learner will be able to find a maxima and minima of a graph and use it to solve a problem.
Strand Scope Source
Problem Solving Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.67
  
    Limiting Processes
The learner will be able to explore the following limiting processes by analyzing them graphically: infinite sequence, infinite series, and area under curves.
Strand Scope Source
Limits Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.67
  
    Linear Programming: Problem Solving
The learner will be able to use the concept of linear programming to solve problems.
Strand Scope Source
Linear Programming Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  
    Networks: Inductive/Deductive Reasoning
The learner will be able to explore networks using both inductive and deductive reasoning.
Strand Scope Source
Graph Theory/Networks Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  
    Linear Programming: Represent Problems
The learner will be able to use concepts of linear programming to represent problems.
Strand Scope Source
Linear Programming Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  
    Matrices: Variable Quantities/Represent
The learner will be able to use matrices to represent real-world situations where the amounts involved are variable.
Strand Scope Source
Matrices Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.66
  
    Matrices: Problem Solving
The learner will be able to apply matrix algebra to obtain problem solutions that make use of finite graphs.
Strand Scope Source
Matrices Master Massachusetts Curriculum Frameworks, January 1996, Grade 12, p.48
  
    Antiderivative: Apply/Growth/Decay
The learner will be able to use the antiderivative in problems involving growth and/or decay.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Antiderivatives: Determine
The learner will be able to determine specific antiderivatives applying initial conditions including applications to motion along a line.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Antiderivatives: Distance/Velocity
The learner will be able to use antiderivatives in determining distance and velocity when acceleration and initial conditions are given.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Apply Calculus Concepts: Fundamental
The learner will be able to recognize the Fundamental Theorem of Calculus.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Apply Calculus Concepts: L'Hopitals Rule
The learner will be able to apply L'Hopital's Rule.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Apply Calculus Concepts: Limit Functions
The learner will be able to apply L'Hopital's Rule to determine the limit of functions whose limits yield the indeterminate forms: 0/0 and infinity/infinity.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Apply Calculus Concepts: Limit Functions
The learner will be able to apply L'Hopital's Rule to determine the limit of functions whose limits yield the indeterminate forms: 0 to the 0th power; 1 to the infinity power; infinity to the infinity power; and infinity minus infinity.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Apply Calculus Concepts: Max/Min
The learner will be able to determine maximum and minimum.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Applying: Fundamental Theorem
The learner will be able to illustrate particular antiderivatives both graphically and analytically applying the Fundamental Theorem.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Calculus Concepts: Antidifferentiation
The learner will be able to apply strategies of antidifferentiation.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Calculus Concepts: Differentiability
The learner will be able to describe the relationship between differentiability and continuity.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Antiderivatives: Growth Decay
The learner will be able to use antiderivatives in solving growth and decay problems using differential equations with separable variables.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Applying: Equation/Tangent Lines
The learner will be able to determine the equation of tangent lines to a curve at a point and apply it to estimate function values.
Strand Scope Source
Applying Calculus Concepts Master East Longmeadow Public School
  
    Antiderivative: Define
The learner will be able to understand the meaning of the antiderivative.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivative: Interpret/Rate of Change
The learner will be able to interpret the derivative as a rate of change in different applied contexts including velocity, speed, and acceleration.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Curves: Concavity
The learner will be able to recognize points of concavity on a curve.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Curves: Constant/Increasing/Decreasing
The learner will be able to recognize points on a curve where it is constant, increasing, or decreasing.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Curves: Equations of Tangents/Slopes
The learner will be able to find the equation of tangents to the graphs of curves using the slope formula.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Curves: Maxima/Minima/Critical Points
The learner will be able to recognize the following points on a curve: critical points, absolute maxima and minima, and relative maxima and minima.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Curves: Points of Inflection
The learner will be able to calculate points of inflection on a curve.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Curves: Slopes/Steepness
The learner will be able to apply slopes to the steepness of curves to analyze and measure changes in dimensions.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Curves: Study Graphs/Derivatives
The learner will be able to apply derivatives to study graphs of curves.
Strand Scope Source
Curve Sketching Master East Longmeadow Public School
  
    Definite Integral: Antiderivatives
The learner will be able to evaluate definite integrals using antiderivatives.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Area
The learner will be able to understand the definite integral as measure of area.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Area Between Curves
The learner will be able to use concepts of the definite integral to calculate the area between two given curves.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Area Under Curve
The learner will be able to use concepts of the definite integral to calculate the area between a curve and the x-axis.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Area/Approximation
The learner will be able to approximate area when it is described as a definite integral.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Average Value
The learner will be able to use concepts of the definite integral to calculate the average value of a function.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Definition
The learner will be able to understand that the definite integral is by definition the limit of a sum.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Fundamental Theorems
The learner will be able to evaluate definite integrals by applying the Fundamental Theorems of calculus.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Fundamental Theorems
The learner will be able to understand the fundamental theorems of integral calculus.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Limit of Riemann Sums
The learner will be able to rewrite definite integrals as the limit of Riemann Sums and vice-versa.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Problem Solving
The learner will be able to apply the definite integral in problem solving situations.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Problem Solving
The learner will be able to use the concept of the definite integral to obtain solutions to problems involving the average value of a function, area between curves, volumes of solids of revolution about the axes or lines parallel to the axes using disc/washer and shell methods, and volumes of solids with known cross-sectional area.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Properties
The learner will be able to understand the properties of the definite integral.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Rectangle Estimations
The learner will be able to understand rectangle estimations as approximations to the definite integral.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Reimann Sums
The learner will be able to apply a knowledge of the definition of the definite integral using Reimann Sums to approximate the actual value of integrals.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Riemann Sums
The learner will be able to interpret the definite integral as a limit of Riemann sums.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Riemann Sums
The learner will be able to apply Riemann Sums and the Trapezoidal Rule to calculate an approximate value for a definite integral.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Riemann Sums
The learner will be able to describe the relationship between a Riemann sum and a definite integral.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Simpson's Rule
The learner will be able to apply Simpson's Rule in approximating definite integrals.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Technology
The learner will be able to use technology to approximate definite integrals.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Trapezoidal Rule
The learner will be able to understand trapezoidal rule as an approximation of the definite integral.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Trapezoidal Rule
The learner will be able to apply the Trapezoidal Rule in approximating definite integrals.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Use
The learner will be able to use the basic properties of definite integrals.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Volume/Known Areas
The learner will be able to use concepts of the definite integral to calculate the volume of a solid of revolution where the cross-sectional area is a known value.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Volume/Solid of Rev.
The learner will be able to use concepts of the definite integral to calculate the volume of a solid of revolution (washer, disc, or shell) about the x-axis, y-axis, or a line parallel to one of the two axes.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integral: Work
The learner will be able to use concepts of the definite integral to determine work when either force or displacement is used as the variable.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integrals: Evaluate
The learner will be able to apply elementary properties to evaluate definite integrals.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Definite Integrals: Problems
The learner will be able to apply definite integrals to problems involving work, volume, velocity, and/or acceleration.
Strand Scope Source
Definite Integral Master East Longmeadow Public School
  
    Derivatives: Basic Functions
The learner will be able to calculate derivatives of elementary functions including X and inverse trigonometric.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Basic Rules
The learner will be able to use basic rules for the derivative of basic functions and their sum, product,and quotient.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Basic Theorems/Apply
The learner will be able to apply the basic theorems of derivatives.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Chain Rule
The learner will be able to apply the chain rule in finding derivatives.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Chain Rule/Limits
The learner will be able to demonstrate the ability to use second derivative test, implicit differentiation, the chain rule, and the limit of a function at infinity.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Chain Rule/Proof
The learner will be able to understand the chain rule and its proof.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Chain/Implicit
The learner will be able to apply the chain rule and implicit differentiation.
Strand Scope Source
Differentiation Master East Longmeadow Public School
  
    Derivatives: Circle/Equation Tangent
The learner will be able to determine the equation of a line tangent to a circle at a given point using the equation of the circle.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Composite/Chain Rule
The learner will be able to calculate the derivative of composite functions using the chain rule.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Concavity
The learner will be able to apply derivatives to determine whether curves are concave or convex at a given point.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Curve Sketching
The learner will be able to sketch a curve using information from the first and second derivatives.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Define
The learner will be able to describe the meaning of the derivative of a function.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Define
The learner will be able to give the definition of the derivative as the limit of the difference quotient.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Define Geometrically
The learner will be able to define derivatives geometrically .
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Definition
The learner will be able to identify the definition of derivative.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Derive Formulas
The learner will be able to derive formulas for determining derivatives.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Derived Function
The learner will be able to find a derived function given the definition.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Deriving Formulas
The learner will be able to use the definition of the derivative as the limit of a sum to derive formulas for the derivatives.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Determine
The learner will be able to apply implicit differentiation to determine derivatives of inverse functions.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Differential Approximation
The learner will be able to use derivatives in solving differential approximations.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Equation Tangent/Normal
The learner will be able to determine the equation of both tangent and normal lines to a curve using the derivative.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Estimate/Rates of Change
The learner will be able to estimate rates of change using graphs and tables of values.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Finding Zeros
The learner will be able to calculate the zeros of the derivative of a function.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Function/Implicitly Defined
The learner will be able to calculate the derivative of implicitly defined functions.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Geometric Consequences
The learner will be able to determine the geometric consequences of the Mean Value Theorem.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Higher Order
The learner will be able to calculate higher order derivatives (second derivative, third derivative, etc.).
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Implicit Differentiation
The learner will be able to apply the concept of implicit differentiation to determine the derivative of an inverse function.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Implicit Differentiation
The learner will be able to apply the concept of implicit differentiation in many different problem types originating in a variety of curriculum areas.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Initial Conditions
The learner will be able to apply initial conditions to determine a specific antiderivative.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Interpret
The learner will be able to interpret the derivative as a rate of change in many different applied contexts.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Inverse Functions
The learner will be able to calculate the derivative of the inverses of functions, including trigonometric inverses.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: l'Hopital's Rule
The learner will be able to use derivatives in applying l'Hopital's rule to evaluate limits in a variety of forms.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Logarithmic Differentiation
The learner will be able to calculate derivatives using logarithmic differentiation.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Maxima/Minima
The learner will be able to use derivatives in solving maxima and minima problems, including marginal analysis.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Maxima/Minima
The learner will be able to apply derivatives to find maxima and minima points.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Mean Value Theorem
The learner will be able to solve problems using the mean value theorem.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Mean Value Theorem
The learner will be able to illustrate an understanding of the Mean Value Theorem and its geometric consequence.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Notation
The learner will be able to understand derivative notation.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Operations
The learner will be able to calculate derivatives of sums, products, and quotients.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Optimization
The learner will be able to apply derivatives to obtain solutions to optimization problems.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Over Interval
The learner will be able to determine the first and second derivative of a given function over an interval using its definition and appropriate notation.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Points of Inflection
The learner will be able to apply derivatives to find points of inflection.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Points of Inflection
The learner will be able to determine points of inflections.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Polynomials/Limits
The learner will be able to apply the limit concept to finding the derivatives of polynomials.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Problem Solving
The learner will be able to obtain solutions to problems including tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, Newton's method, differential and linear approximations, and optimization problems.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Problem Solving
The learner will be able to solve problems using derivatives and include tangent and normal lines to a curve, curve sketching, velocity, acceleration, related rates of change, Newton's Method, differentials and linear approximations, and optimization problems.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Problem Solving
The learner will be able to apply derivatives in obtaining solutions to many different problems (from a wide variety of areas) that involve the rate of change of a function.
Strand Scope Source
Derivatives/Antiderivatives Master East Longmeadow Public School
  
    Derivatives: Problem Solving
The learner will be able to solve problems using derivatives.