﻿ FCPS Program of Studies
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

Essential - Standard, benchmark, or indicator from the VDOE Standards of Learning document. In the absence of VDOE standards for a given course, content subject to testing such as AP and IB can be labeled Essential.
Expected - Standard, benchmark, or indicator added by the FCPS Program of Studies to provide a context, a bridge, or an enhancement to the Essential SBIs.
Extended - Standard, benchmark, or indicator added by the FCPS Program of Studies generally used to differentiate instruction for advanced learners (Honors/GT)