Mathematics Curriculum
  Calculus
 
The purpose of this course (Calculus AB/BC) is to prepare students to take the Advanced Placement examination given each spring, for which placement and/or credit may be awarded at the college level if a qualifying score is obtained. Content of this college-level course is defined by the Advanced Placement Course Description for Calculus AB/BC as stated by the College Board. Teachers should update course content as changes occur in future College Board publications. Content includes concepts and applications of differential and integral calculus. This course carries a weighted grade.
As mandated by the College Board, graphing calculators will be required for this course. Computers should also be used where feasible by the student and by the teacher. Any technology that will enhance student learning should be used if available. Instructional activities that engage students in solving application problems of varying complexities are encouraged.
Note: Extended refers to topics for Calculus BC
Last Updated: 08/24/12 11:03 AM
 
 
MTH.CALC
Standard 1
  DEFINE AND APPLY THE PROPERTIES OF LIMITS
 

Benchmark 1.a
Define and Apply the Properties of Limits
 
Indicator 1.a.1
Find the value of a limit from a graph of a function
 
Indicator 1.a.2
Estimate the value of a limit numerically from a table of values
 
Indicator 1.a.3
Calculate limits algebraically using various techniques
 
Indicator 1.a.4
Calculate limits using the limit laws
 
Indicator 1.a.5
Determine one-sided limits algebraically, graphically, and numerically
 
Indicator 1.a.6
Determine limits at infinity, infinite limits, and non-existent limits
 
Indicator 1.a.7
Use l’Hopital’s Rule to find the limit of functions
 
Indicator 1.a.8
Explore the precise definition of the limit
 
MTH.CALC
Standard 2
  USE LIMITS TO DEFINE AND DETERMINE CONTINUITY
 

Benchmark 2.a
Use Limits to Define and Determine Continuity
 
Indicator 2.a.1
Determine continuity from a graph or equation
 
Indicator 2.a.2
Identify a function's discontinuities and the type of discontinuity
 
Indicator 2.a.3
Determine whether a function is continuous using the formal definition
 
Indicator 2.a.4
Apply the Intermediate Value Theorem and the Extreme Value Theorem
 
MTH.CALC
Standard 3
  INVESTIGATE ASYMPTOTIC AND UNBOUNDED BEHAVIOR
 

Benchmark 3.a
Investigate Asymptotic and Unbounded Behavior
 
Indicator 3.a.1
Determine equations of asymptotes using limits at infinity
 
Indicator 3.a.2
Sketch graphs of functions given conditions involving limits
 
Indicator 3.a.3
Evaluate limits at infinity using the properties of limits
 
Indicator 3.a.4
Compare relative magnitudes to determine limits of functions
 
MTH.CALC
Standard 4
  INVESTIGATE DERIVATIVES GRAPHICALLY, NUMERICALLY, AND ANALYTICALLY
 

Benchmark 4.a
Investigate Derivatives Graphically, Numerically, and Analytically
 
Indicator 4.a.1
Define a tangent to a curve at a point
 
Indicator 4.a.2
Investigate and compare the slopes of secant lines and tangent lines
 
Indicator 4.a.3
Estimate derivatives geometrically, numerically, and analytically
 
Indicator 4.a.4
Define the derivative and use the definition to calculate slopes
 
Indicator 4.a.5
Determine the differentiability of a function at a point
 
Indicator 4.a.6
Compare and contrast the concepts of differentiability and continuity
 
Indicator 4.a.7
Identify where a function fails to be differentiable and the reason
 
MTH.CALC
Standard 5
  INVESTIGATE THE DERIVATIVE AT A POINT
 

Benchmark 5.a
Investigate the Derivative at a Point
 
Indicator 5.a.1
Find the slope at a point using the definition of the derivative
 
Indicator 5.a.2
Use the concept of local linearity to approximate the slope of a curve
 
Indicator 5.a.3
Define instantaneous rate of change
 
Indicator 5.a.4
Approximate rates of change from graphs and tables
 
Indicator 5.a.5
Write equations of lines tangent to a curve at a point
 
MTH.CALC
Standard 6
  ANALYZE THE DERIVATIVE OF A FUNCTION AS ITS OWN FUNCTION
 

Benchmark 6.a
Analyze the Derivative of a Function as its Own Function
 
Indicator 6.a.1
Compare corresponding characteristics of the graphs of f, f ' and f "
 
Indicator 6.a.2
Determine intervals of increasing and decreasing behavior
 
Indicator 6.a.3
Translate verbal descriptions into equations involving derivatives
 
Indicator 6.a.4
Apply the Mean Value Theorem and interpret its geometric properties
 
Indicator 6.a.5
Determine the concavity of functions over intervals
 
Indicator 6.a.6
Identify points of inflection as places where concavity changes
 
Indicator 6.a.7
Sketch the graph of f ' given the graph of f
 
Indicator 6.a.8
Find eqn for derivative fns algebraically using the limit definition
 
Indicator 6.a.9
Include alternative notations for the derivative
 
MTH.CALC
Standard 7
  APPLY THE DERIVATIVE TO SOLVE PROBLEMS AND TO JUSTIFY SOLUTIONS
 

Benchmark 7.a
Apply the Derivative to Solve Problems and to Justify Solutions
 
Indicator 7.a.1
Use first and second derivatives to find monotonicity and concavity
 
Indicator 7.a.2
Use the first and second derivative tests to determine extrema
 
Indicator 7.a.3
Solve optimization prob involving abs (global) & rel (local) extrema
 
Indicator 7.a.4
Find tangent lines, slope, extrema and concavity of implicit functions
 
Indicator 7.a.5
Model rates of change and solve related rates problems
 
Indicator 7.a.6
Interpret the derivative as a rate of change in applied contexts
 
Indicator 7.a.7
Differentiate functions using logarithmic differentiation
 
Indicator 7.a.8
Analyze curves given in parametric form, polar form, and vector form
 
Indicator 7.a.9
Test the convergence of improper integrals using l’Hopital’s Rule
 
MTH.CALC
Standard 8
  APPLY RULES TO FIND DERIVATIVES ALGEBRAICALLY
 

Benchmark 8.a
Apply Rules to Find Derivatives Algebraically
 
Indicator 8.a.1
Find derivatives of various functions algebraically
 
Indicator 8.a.2
Find derivatives of sums, products, quotients, inverses and composites
 
Indicator 8.a.3
Find first and second derivatives of implicitly defined relations
 
Indicator 8.a.4
Determine higher order derivatives of various functions algebraically
 
Indicator 8.a.5
Find derivatives of parametric, polar, and vectors algebraically
 
MTH.CALC
Standard 9
  ESTIMATE DEFINITE INTEGRALS USING RIEMANN SUMS AND TRAPEZOIDAL RULE
 

Benchmark 9.a
Estimate Definite Integrals Using Riemann Sums and Trapezoidal Rule
 
Indicator 9.a.1
Estimate a definite integral using left, right, and midpoint rules
 
Indicator 9.a.2
Estimate a definite integral using Trapezoidal rule
 
Indicator 9.a.3
Determine if a Riemann sum will underestimate or overestimate
 
Indicator 9.a.4
Evaluate definite integrals geometrically using formulas from Geometry
 
Indicator 9.a.5
Write the limit of the Riemann sum as a definite integral
 
Indicator 9.a.6
Write the limit of the Riemann sum as a definite integral & vice versa
 
Indicator 9.a.7
Evaluate the limit of the Riemann sum algebraically
 
MTH.CALC
Standard 10
  FIND ANTIDERIVATIVES OF BASIC FUNCTIONS
 

Benchmark 10.a
Find Antiderivatives of Basic Functions
 
Indicator 10.a.1
Understand and correctly apply the constant of integration
 
Indicator 10.a.2
Find antiderivatives of basic functions and inverse trig functions
 
Indicator 10.a.3
Apply the properties of the integrals to evaluate integrals
 
Indicator 10.a.4
Evaluate integrals using the method of integration by parts
 
Indicator 10.a.5
Evaluate integrals of rational functions by simple partial fractions
 
Indicator 10.a.6
Find improper integrals as limits of definite integrals
 
Indicator 10.a.7
Evaluate integrals using trigonometric substitution
 

Benchmark 10.b
Use Fundamental Thm of Calc & Substitution to Eval Definite Integrals
 
Indicator 10.b.1
Compare and contrast the definite and indefinite integrals
 
Indicator 10.b.2
Evaluate definite integrals of fns using Fundamental Theorem of Calc
 
Indicator 10.b.3
Evaluate integrals using substitution including change of limits
 
Indicator 10.b.4
Use a graph of f ' to find values of f using Fundamental Thm of Calc
 
Indicator 10.b.5
Apply the Fundamental Theorem of Calculus to real-world applications
 
Indicator 10.b.6
Examine a particular function f, given a graph of its derivative f '
 
MTH.CALC
Standard 11
  EXPLORE DEFINITE INTEGRAL AS AREA USING FUNDAMENTAL THEOREM OF CALC
 

Benchmark 11.a
Explore Definite Integral as an Area Using The Fundamental Thm of Calc
 
Indicator 11.a.1
Apply properties of definite integrals from multiple representations
 
Indicator 11.a.2
For each x of an accumulation function F, find f(x) or f '(x)
 
Indicator 11.a.3
Examine an accumulation function F, given a graph of its derivative f
 
Indicator 11.a.4
Find the derivatives of expressions containing integrals
 
MTH.CALC
Standard 12
  SOLVE SEPARABLE DIFFERENTIAL EQUATIONS AND THEIR APPLICATIONS
 

Benchmark 12.a
Solve Separable Differential Equations and their Applications
 
Indicator 12.a.1
Create a slope field from a differential equation
 
Indicator 12.a.2
Match a set of differential equations with corresponding slope fields
 
Indicator 12.a.3
Define, identify, and verify solutions of differential equations
 
Indicator 12.a.4
Solve separable differential equations
 
Indicator 12.a.5
Solve real-world motion problems using differential equations
 
Indicator 12.a.6
Solve growth and decay problems using differential equations
 
Indicator 12.a.7
Solve logistic differential equations and use them in modeling
 
Indicator 12.a.8
Solve differential eqn using Euler’s method given various step sizes
 
MTH.CALC
Standard 13
  USE INTEGRATION TECHNIQUES TO SOLVE REAL-WORLD PROBLEMS
 

Benchmark 13.a
Use Integration Techniques To Solve Real-World Problems
 
Indicator 13.a.1
Determine the area of a region defined by two or more functions
 
Indicator 13.a.2
Find the volume of a solid with a known cross-section
 
Indicator 13.a.3
Find the volume of a solid of revolution by the disk or washer method
 
Indicator 13.a.4
Find the average value of a function over a given interval
 
Indicator 13.a.5
Find the displacement and distance traveled by a particle along a line
 
Indicator 13.a.6
Find the arc length of curves
 
Indicator 13.a.7
Find the equation of the tangent line on a parametric curve
 
Indicator 13.a.8
Find the position, speed, velocity, and acceleration of a particle
 
Indicator 13.a.9
Find the area of a region and slope of a graph defined by polar curves
 
MTH.CALC
Standard 14
  DEFINE A SEQUENCE AND A SERIES AND TEST FOR CONVERGENCE
 

Benchmark 14.a
Define a Sequence and a Series and Test for Convergence
 
Indicator 14.a.1
Determine whether a sequence converges and find the limit
 
Indicator 14.a.2
Test a series for divergence using the Nth Term test for Divergence
 
Indicator 14.a.3
Determine the partial sum of a series
 
Indicator 14.a.4
Determine whether a geometric series converges and if so, find its sum
 
Indicator 14.a.5
Determine whether a series converges using the integral test
 
Indicator 14.a.6
Determine whether a p-series converges including the harmonic series
 
Indicator 14.a.7
Test the convergence of an alternating series and find the error bound
 
Indicator 14.a.8
Determine the convergence of a series using comparison tests
 
Indicator 14.a.9
Determine absolute or conditional convergence of a series
 
Indicator 14.a.10
Determine convergence by applying the root and ratio tests
 
MTH.CALC
Standard 15
  DEFINE, RESTATE, AND APPLY TAYLOR SERIES
 

Benchmark 15.a
Define, Restate, And Apply Taylor Series
 
Indicator 15.a.1
Determine Taylor polynomial approximations
 
Indicator 15.a.2
Find Maclaurin series and the general Taylor series
 
Indicator 15.a.3
Determine a Maclaurin series for certain functions
 
Indicator 15.a.4
Use shortcuts to create a new series from known series
 
Indicator 15.a.5
Determine the radius and interval of convergence for a power series
 
Indicator 15.a.6
Determine the Lagrange error bound of a Taylor polynomial


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