ap calculus bc - hershey.k12.pa.us€¦ · instructional unit derivatives ap calculus bc unit...

AP Calculus BC

Instructional Unit Derivatives

AP Calculus BC

Unit Content Objective Performance Indicator Performance Task State Standards

Derivatives The students will be -Determine the derivative of -written -Derivative presented able to differentiate a composite function using documentation ofgraphically, numerically, and composite functions the Chain Rule problem solvinganalytically using the Chain -Utilize the Chain Rule two -applications to real -Derivative interpreted as an Rule, find or more times to determine life problemsinstantaneous rate of change derivatives using the derivative -AP Calculus exam -Derivative defined as the the Power Rule for -Understand the use and practice questionslimit of the difference quotient Rational Powers, notation for an implicitly -technology

and find derivatives defined function applications -Relationship between -Find the tangent anddifferentiability and continuity normal lines to a conic -Slope of a curve at a point -Find a higher order -Tangent line to a curve at a derivative implicitlypoint and local linear -Apply the Power Rule forapproximation Rational Powers of the -Instantaneous rate of change exponent as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'

-The Mean Value Theorem and

Monday, August 18, 2003 of 61


AP Calculus BC


its geometric consequences using implicit -classroom discussion -Equations involving differentiation.derivatives. Verbaldescriptions are translated intoequations involving derivatives and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems

-Use of implicit differentiation to find the derivative of aninverse function -Interpretation of thederivative as a rate of change



AP Calculus BC


in varies applied contexts,including velocity, speed, andacceleration -Geometric interpretation ofdifferential equations via slopefields and the relationshipbetween slope fields andsolution curves for differentialequations -Numerical solution ofdifferential equations usingEuler's method -L'Hopital's Rule, including itsuse in determining limits andconvergence of improperintegrals and series -Knowledge of derivatives ofbasic functions, includingpower, exponential, logarithmic, trigonometric, and inversetrigonometric functions



AP Calculus BC


Derivatives The students will be -Understand how a -written 2.11.11.C -Derivative presented able to find where a derivative fails to exist documentation ofgraphically, numerically, and function is not -Understand the relationship problem solvinganalytically differentiable and between differentiability -applications to real -Derivative interpreted as an approximate a and local linearity life problemsinstantaneous rate of change derivative -Compute a numerical -AP Calculus exam -Derivative defined as the numerically and derivative practice questionslimit of the difference quotient -Understand the relationship -technology

between differentiability applications -Relationship between and continuitydifferentiability and continuity -Apply the Intermediate -Slope of a curve at a point Value Theorem for -Tangent line to a curve at a Derivativespoint and local linearapproximation -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'

-The Mean Value Theorem and its geometric consequences -Equations involvingderivatives. Verbaldescriptions are translated into



AP Calculus BC


equations involving derivatives graphically. -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems

-Use of implicit differentiation to find the derivative of aninverse function -Interpretation of thederivative as a rate of changein varies applied contexts,including velocity, speed, andacceleration -Geometric interpretation of



AP Calculus BC


differential equations via slopefields and the relationshipbetween slope fields andsolution curves for differentialequations -Numerical solution ofdifferential equations usingEuler's method -L'Hopital's Rule, including itsuse in determining limits andconvergence of improperintegrals and series -Knowledge of derivatives ofbasic functions, includingpower, exponential, logarithmic, trigonometric, and inversetrigonometric functions



AP Calculus BC


Derivatives The students will be -Determine the first -written -Derivative presented able to find first and derivative of a parametric documentation ofgraphically, numerically, and second derivatives function problem solvinganalytically of parametrically -Determine the second -applications to real -Derivative interpreted as an defined functions, derivative of a parametric life problemsinstantaneous rate of change differentiate vector function -AP Calculus exam -Derivative defined as the valued functions, -Determine horizontal and practice questionslimit of the difference quotient and calculate the vertical tangent lines to a -technology

slope of a polar parametric function applications -Relationship between -Understand and apply thedifferentiability and continuity definition for a derivative at -Slope of a curve at a point a point of a vector valued -Tangent line to a curve at a functionpoint and local linear -Analyze the velocity,approximation speed, acceleration, and -Instantaneous rate of change direction of motion of a as the limit of average rate of vector valued functionchange -Understand and apply -Approximate rate of change differentiation rules forfrom graphs and tables of vector valued functionsvalues -Determine the slope of a -Corresponding polar curvecharacteristics of graphs of f -Determine horizontal andand f' vertical tangents for a polar -Relationship between the curveincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives curve. -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Define symbolically a -written 2.11.11.A -Derivative presented able to calculate derivative documentation ofgraphically, numerically, and slopes and -Determine the derivative of problem solvinganalytically derivatives using a function using the -applications to real -Derivative interpreted as an the definition of the definition of a derivative life problemsinstantaneous rate of change derivative, and -Graph f' given f -AP Calculus exam -Derivative defined as the graph f from the -Graph f given f' practice questionslimit of the difference quotient graph of f', graph f' -Graph the derivative for a -technology

from the graph of f, table of data applications -Relationship between and graph the -Determine one-sideddifferentiability and continuity derivative of a derivatives -Slope of a curve at a point function given -Tangent line to a curve at a numerically withpoint and local linearapproximation -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives data. -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Identify all possible -written -Derivative presented able to find limits of indeterminate forms documentation ofgraphically, numerically, and indeterminate forms -Understand and apply problem solvinganalytically using L'Hopital's L'Hopital's Rule -applications to real -Derivative interpreted as an -Convert, using algebraic life problemsinstantaneous rate of change techniques, indeterminate -AP Calculus exam -Derivative defined as the forms into forms for which practice questionslimit of the difference quotient L'Hopital's Rule applies -technology

-Compare graphical applications -Relationship between estimations of limits todifferentiability and continuity results obtained from using -Slope of a curve at a point L'Hopital's Rule -Tangent line to a curve at apoint and local linearapproximation -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives Rule. -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Use the definition of a -written 2.5.11.A, -Derivative presented able to use derivative to determine an documentation of 2.8.11.D,graphically, numerically, and derivatives to object's instantaneous rate problem solving 2.8.11.P,analytically analyze straight line of change -applications to real 2.11.11.A, -Derivative interpreted as an motion and solve -Understand the difference life problems 2.11.11.B,instantaneous rate of change other problems between speed and velocity -AP Calculus exam 2.11.11.C -Derivative defined as the involving rates of practice questionslimit of the difference quotient -Utilize the concept of a -technology

derivative to determine an applications -Relationship between object's velocity anddifferentiability and continuity acceleration given its -Slope of a curve at a point position -Tangent line to a curve at a -Use the derivative topoint and local linear model other rates of changeapproximation such as free fall, -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives change. sensitivity to change, and -classroom discussion and vice versa. economics -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Determine a linear -written 2.5.11.A, -Derivative presented able to find approximation using a documentation of 2.5.11.B,graphically, numerically, and linearizations and linearization process problem solving 2.5.11.C,analytically use Newton's -Apply the procedure for -applications to real 2.5.11.D -Derivative interpreted as an method to Newton's Method to a life problemsinstantaneous rate of change approximate the function -AP Calculus exam -Derivative defined as the zeros of a function, -Estimate the change in a practice questionslimit of the difference quotient estimate the change function using differentials -technology

in a function using -Understand the difference applications -Relationship between differentials, and between absolute, relative,differentiability and continuity solve related rate and percentage change -Slope of a curve at a point -Determine the related rate -Tangent line to a curve at a equation of a definedpoint and local linear quantityapproximation -Apply the concept of -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives problems. related rates to real life -classroom discussion and vice versa. situations -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Solve initial value -written 2.8.11.Q -Derivative presented able to construct differential problems documentation ofgraphically, numerically, and slope fields using -Construct a slope field problem solvinganalytically technology, using graphical techniques -applications to real -Derivative interpreted as an interpret slope fields and technology life problemsinstantaneous rate of change as visualizations of -Predict a functions -AP Calculus exam -Derivative defined as the differential equation given the slope practice questionslimit of the difference quotient equations, and use field of the function -technology

Euler's Method to -Use Euler's Method to find applications -Relationship between find approximate approximations for initialdifferentiability and continuity solutions to value problems -Slope of a curve at a point differential -Visualize Euler's Method -Tangent line to a curve at a equations with initial using graphical solutions topoint and local linear values. initial value problemapproximation approximations -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Determine absolute -written 2.11.11.A, -Derivative presented able to determine extreme values documentation of 2.11.11.Bgraphically, numerically, and local or global -Apply the Extreme Value problem solvinganalytically extreme values of a Theorem -applications to real -Derivative interpreted as an function, apply the -Determine all critical points life problemsinstantaneous rate of change Mean Value of a function -AP Calculus exam -Derivative defined as the Theorem, and find -Determine all local extreme practice questionslimit of the difference quotient the intervals on values -technology

which a function is -Apply the Mean Value applications -Relationship between increasing or Theorem for Derivatives -classroomdifferentiability and continuity decreasing. -Determine the intervals discussion-written -Slope of a curve at a point over which a function is documentation of -Tangent line to a curve at a increasing and decreasing problem solvingpoint and local linear -Understand the concept of -applications to realapproximation an antiderivative and the life problems -Instantaneous rate of change process of -AP Calculus exam as the limit of average rate of antidifferentiation practice questionschange -technology -Approximate rate of change applicationsfrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Apply the first derivative -written 2.5.11.A, -Derivative presented able to apply the test for local extrema documentation of 2.5.11.D,graphically, numerically, and first and second -Determine the concavity of problem solving 2.11.11.A,analytically derivative tests, a function using the second -applications to real 2.11.11.B -Derivative interpreted as an determine the derivative test life problemsinstantaneous rate of change concavity of a -Determine the points of -AP Calculus exam -Derivative defined as the function, locate inflection of a function practice questionslimit of the difference quotient points of inflection, -Utilize the second -technology

and solve problems derivative test for local applications -Relationship between involving minimum extremadifferentiability and continuity or maximum values -Develop a mathematical -Slope of a curve at a point model to optimize a given -Tangent line to a curve at a situationpoint and local linearapproximation -Instantaneous rate of change as the limit of average rate ofchange -Approximate rate of changefrom graphs and tables ofvalues -Correspondingcharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives of a function. -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC





AP Calculus BC


Derivatives The students will be -Apply differentiation rules -written -Derivative presented able to use the rules for integer powers, documentation ofgraphically, numerically, and of differentiation to multiples, sums and problem solvinganalytically calculate differences -applications to real -Derivative interpreted as an derivatives, -Determine the derivative of life problemsinstantaneous rate of change including second and a product or quotient of -AP Calculus exam -Derivative defined as the higher order functions practice questionslimit of the difference quotient derivatives of basic -Determine second and -technology

functions, including higher order derivatives applications -Relationship between trigonometric -Determine the derivativesdifferentiability and continuity functions, inverse of the six basic -Slope of a curve at a point trigonometric trigonometric functions -Tangent line to a curve at a functions, and -Determine the derivativespoint and local linear exponential and of the six basic inverseapproximation logarithmic trigonometric functions -Instantaneous rate of change -Determine the derivative of as the limit of average rate of an exponential functionchange -Determine the derivative of -Approximate rate of change a power functionfrom graphs and tables of -Determine the derivative ofvalues both a natural and common -Corresponding logarithmic functioncharacteristics of graphs of fand f' -Relationship between theincreasing and decreasingbehavior of f and the sign of f'




AP Calculus BC


equations involving derivatives functions. -classroom discussion and vice versa. -Correspondingcharacteristics of the graphs of f, f', and f" -Relationship between theconcavity of f and the sign off" -Points of inflections asplaces where concavitychanges -Analysis of curves, including the notions of monotonicityand concavity -Analysis of planar curvesgiven in parametric form, polarform, and vector form,including velocity andacceleration vectors -Optimization, both absoluteand relative extrema -Modeling rates of change,including related rates problems




AP Calculus BC




Instructional Unit Functions, Graphs, and Limits

AP Calculus BC


Functions, Graphs, and Limits The students will be -Construct graphs of -writtenable to analyze parametric, polar, and documentation of

-An intuitive understanding of planar curves given vector planar curves both problem solvingthe limiting process in parametric form, with and without technology -applications to real -Calculating limits (both one- polar form, and life problemsand two-sided) using algebra -Define a vector in -AP Calculus exam -Estimating limits from graphs component form practice questions and tables of data -Understand and apply -technology -Understanding asymptotes in vector operations applications terms of graphical behavior -Understand and apply -Describing asymptotic properties of vectorbehavior in terms of limits operationsinvolving infinity -Determine the angle -Comparing relative between two nonzeromagnitudes of functions and vectorstheir rates of change -Analyze a graphical -An intuitive understanding of representation of a functioncontinuity given in parametric, polar, -Understanding continuity in and vector formterms of limits -Convert equations between -Geometric understanding of polar and Cartesiangraphs of continuous functions coordinates

-Determine the symmetry -Analysis of curves given in of a polar equation using



AP Calculus BC


Functions, Graphs, and Limits The students will be -Understand the definition -written 2.8.11.Sable to evaluate the of a limit documentation of

-An intuitive understanding of limits of functions. -Apply properties of limits to problem solvingthe limiting process applications -applications to real -Calculating limits (both one- -Evaluate both one-sided life problemsand two-sided) using algebra and two-sided limits -AP Calculus exam -Estimating limits from graphs -Apply the Sandwich practice questions and tables of data -technology -Understanding asymptotes in applications terms of graphical behavior -Describing asymptoticbehavior in terms of limitsinvolving infinity -Comparing relativemagnitudes of functions andtheir rates of change -An intuitive understanding ofcontinuity -Understanding continuity interms of limits -Geometric understanding ofgraphs of continuous functions

-Analysis of curves given in



AP Calculus BC


Functions, Graphs, and Limits The students will be -Determine the continuity at -written 2.8.11.Oable to identify the both an endpoint and an documentation of

-An intuitive understanding of intervals upon which interior point on the domain problem solvingthe limiting process a given function is of a function -applications to real -Calculating limits (both one- continuous, -Define continuity of a life problemsand two-sided) using algebra understand the function -AP Calculus exam -Estimating limits from graphs meaning of a -Understand and apply practice questions and tables of data continuous function, properties of continuous -technology -Understanding asymptotes in and be able to functions applications terms of graphical behavior remove removable -Analyze the continuity of a -Describing asymptotic discontinuities by function through the use ofbehavior in terms of limits extending or composite functionsinvolving infinity modifying a -Apply the Intermediate -Comparing relative function. Value Theorem formagnitudes of functions and Continuous Functionstheir rates of change -An intuitive understanding ofcontinuity -Understanding continuity interms of limits -Geometric understanding ofgraphs of continuous functions




AP Calculus BC


Functions, Graphs, and Limits The students will be -Define both in words and -written 2.8.11.Q,able to find and symbolically a horizontal documentation of 2.11.11.A

-An intuitive understanding of verify end behavior asymptote problem solvingthe limiting process models for various -Apply properties of the limit -applications to real -Calculating limits (both one- functions, calculate as x approaches both life problemsand two-sided) using algebra limits as x positive and negative -AP Calculus exam -Estimating limits from graphs approaches both infinity practice questions and tables of data positive and -Define a vertical -technology -Understanding asymptotes in negative infinity, asymptote using the applications terms of graphical behavior and identify vertical concept of a limit -Describing asymptotic -Determine both a right andbehavior in terms of limitsinvolving infinity -Comparing relativemagnitudes of functions andtheir rates of change -An intuitive understanding ofcontinuity -Understanding continuity interms of limits -Geometric understanding ofgraphs of continuous functions



Instructional Unit Integrals

AP Calculus BC


Integrals The students will be -Evaluate integrals using a -written -Computation of Riemann able to compute substitution method documentation ofsums using left, right, and indefinite and -Understand and apply the problem solvingmidpoint evaluation points definite integrals substitution method to -applications to real -Definite integral as a limit of using the methods trigonometric integrands life problemsRiemann sums over equal of substitution, -Understand and apply the -AP Calculus examsubdivisions parts, tabular, partial integration by parts formula practice questions -Definite integral of the rate fractions, integral -technologyof change of a quantity over tables, and -Investigate the repeated applicationsan interval trigonometric use of integration by parts -Basic properties of definite substitution. to evaluate certain integralsintegrals -Use of the Fundamental -Use tabular integration as aTheorem of Calculus to shortcut to problems fittingevaluate definite integrals a specific pattern -Use of the Fundamental -Rewrite a rationalTheorem of Calculus to expression as the sum ofrepresent a particular partial fractionsantiderivative, and the -Use the procedure ofanalytical and graphical partial fractions to evaluateanalysis of functions so the integral of a rationaldefined expression -Antiderivatives following -Utilize integral tables todirectly from derivatives of evaluate nonroutinebasic functions integration procedures -Antiderivatives by -Utilize trigonometricsubstitution of variables, parts, substitution to evaluate and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line integrals fitting a specific -classroom discussion -Solving separable differential pattern equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Use the Trapezoidal Rule to -written 2.11.11.E -Computation of Riemann able to approximate approximate the area under documentation ofsums using left, right, and the definite integral a curve problem solvingmidpoint evaluation points by using the -Approximate an integral -applications to real -Definite integral as a limit of Trapezoidal Rule and numerically by using life problemsRiemann sums over equal by using Simpson's Simpson's Rule -AP Calculus examsubdivisions Rule, and estimate -Determine error bounds practice questions -Definite integral of the rate the error in using the when using an -technologyof change of a quantity over Trapezoidal and approximation technique applicationsan interval -Basic properties of definiteintegrals -Use of the FundamentalTheorem of Calculus toevaluate definite integrals -Use of the FundamentalTheorem of Calculus torepresent a particularantiderivative, and theanalytical and graphicalanalysis of functions sodefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line Simpson's Rules. -classroom discussion -Solving separable differential equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Determine the length of a -written -Computation of Riemann able to calculate smooth parametrized curve documentation ofsums using left, right, and lengths and surface problem solvingmidpoint evaluation points areas of -Determine the surface area -applications to real -Definite integral as a limit of parametrically from a smooth life problemsRiemann sums over equal defined curves, parametrized curve -AP Calculus examsubdivisions integrate vector -Write both the indefinite practice questions -Definite integral of the rate valued functions, and definite integral form of -technologyof change of a quantity over and calculate a vector valued function applicationsan interval lengths, areas of -Determine the area of the -Basic properties of definite regions in the plane, region between the originintegrals and surface areas and a polar curve -Use of the Fundamental determined by polar -Determine the areaTheorem of Calculus to curves. between two polar curvesevaluate definite integrals -Determine the length of a -Use of the Fundamental polar curveTheorem of Calculus to -Determine the area of arepresent a particular surface of revolution of aantiderivative, and the polar curve revolved aboutanalytical and graphical both the x and y axisanalysis of functions sodefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line -classroom discussion -Solving separable differential equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Understand and explain the -written 2.11.11.C -Computation of Riemann able to solve law of exponential change documentation ofsums using left, right, and problems involving -Apply the law of problem solvingmidpoint evaluation points exponential growth exponential change to such -applications to real -Definite integral as a limit of and decay, and areas as compounded life problemsRiemann sums over equal exponential or interest, radioactivity, -AP Calculus examsubdivisions logistic population Newton's law of cooling, and practice questions -Definite integral of the rate resistance proportional to -technologyof change of a quantity over an object's velocity applicationsan interval -Understand and apply the -Basic properties of definite exponential modelintegrals -Understand and apply the -Use of the Fundamental logistic growth modelTheorem of Calculus to -Use technology to exploreevaluate definite integrals a logistic growth model -Use of the FundamentalTheorem of Calculus torepresent a particularantiderivative, and theanalytical and graphicalanalysis of functions sodefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line growth. -classroom discussion -Solving separable differential equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Determine the area -written 2.11.11.E -Computation of Riemann able to use between two nonintersecting documentation ofsums using left, right, and integration to curves in a plane problem solvingmidpoint evaluation points calculate areas of -Determine the area -applications to real -Definite integral as a limit of regions in a plane, enclosed by the intersection life problemsRiemann sums over equal volumes of solids, of curves in a plane -AP Calculus examsubdivisions surface areas of -Determine the area practice questions -Definite integral of the rate solids of revolution, enclosed by intersecting -technologyof change of a quantity over and lengths of curves by partitioning the applicationsan interval curves in a plane. region into several -Basic properties of definite subregionsintegrals -Integrate an enclosed area -Use of the Fundamental with respect to both x and yTheorem of Calculus to evaluate definite integrals -Determine the volume of a -Use of the Fundamental solid of known integrableTheorem of Calculus to cross section areasrepresent a particular -Apply the washer andantiderivative, and the cylindrical shells methods toanalytical and graphical determine volumes ofanalysis of functions so hollow solidsdefined -Determine the surface area -Antiderivatives following of a solid of revolutiondirectly from derivatives of -Determine the arc length ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line a smooth curve -classroom discussion -Solving separable differential equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Develop a strategy for -written 2.5.11.A, -Computation of Riemann able to adapt their modeling with integrals documentation of 2.5.11.B,sums using left, right, and knowledge of -Model the effects on the problem solving 2.5.11.C,midpoint evaluation points integral calculus to interrelationships between -applications to real 2.5.11.D -Definite integral as a limit of model problems position, velocity, and life problemsRiemann sums over equal involving rates of acceleration -AP Calculus examsubdivisions change in a variety -Use integral calculus to practice questions -Definite integral of the rate model net change in data -technologyof change of a quantity over -Review applications of applicationsan interval integral calculus to the area -Basic properties of definite of physicsintegrals -Review applications of -Use of the Fundamental integral calculus to the areaTheorem of Calculus to of statisticsevaluate definite integrals -Use of the FundamentalTheorem of Calculus torepresent a particularantiderivative, and theanalytical and graphicalanalysis of functions sodefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line of applications. -classroom discussion -Solving separable differential equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Understand and apply the -written 2.11.11.E -Computation of Riemann able to use limits to concept of improper documentation ofsums using left, right, and evaluate improper integrals with infinite problem solvingmidpoint evaluation points integrals and use the integration limits -applications to real -Definite integral as a limit of direct comparison -Understand and apply the life problemsRiemann sums over equal test and the limit concept of improper -AP Calculus examsubdivisions comparison test to integrals with infinite practice questions -Definite integral of the rate determine the discontinuities -technologyof change of a quantity over convergence or -Test for convergence or applicationsan interval divergence of divergence of an integral -Basic properties of definite improper integrals. using the direct comparisonintegrals test -Use of the Fundamental -Test for convergence orTheorem of Calculus to divergence of an integralevaluate definite integrals using the limit comparison -Use of the Fundamental testTheorem of Calculus to -Apply the concepts ofrepresent a particularantiderivative, and theanalytical and graphicalanalysis of functions sodefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line improper integrals to finding -classroom discussion -Solving separable differential the volume of an infinite equations and using them in solidmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Utilize and apply the three -written 2.11.11.E -Computation of Riemann able to approximate basic rectangular documentation ofsums using left, right, and the area under the approximation methods problem solvingmidpoint evaluation points graph of a function (LRAM, RRAM, MRAM) -applications to real -Definite integral as a limit of by using rectangle -Apply rectangular life problemsRiemann sums over equal approximation approximation methods to -AP Calculus examsubdivisions methods, interpret three dimensional volume practice questions -Definite integral of the rate the area under a estimations -technologyof change of a quantity over graph, and express -Express the definite applicationsan interval the area under a integral as a limit of -Basic properties of definite curve as a definite Riemann sumsintegrals integral and as a -Understand and utilize the -Use of the Fundamental limit of Riemann terminology and notation ofTheorem of Calculus to sums. integrationevaluate definite integrals -Apply the concept of a -Use of the Fundamental definite integral to computeTheorem of Calculus to the area under a curverepresent a particular -Evaluate the integral of aantiderivative, and the constantanalytical and graphical -Use technology to evaluateanalysis of functions so an integraldefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line -classroom discussion -Solving separable differential equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and



AP Calculus BC


Integrals The students will be -Understand and apply rules -written 2.11.11.E -Computation of Riemann able to apply rules for definite integrals documentation ofsums using left, right, and for definite integrals, -Apply the Mean Value problem solvingmidpoint evaluation points find the average Theorem for definite -applications to real -Definite integral as a limit of value of a function, integrals life problemsRiemann sums over equal and apply the -Examine the connections -AP Calculus examsubdivisions Fundamental between differential and practice questions -Definite integral of the rate integral calculus -technologyof change of a quantity over -Understand the proof of applicationsan interval the Fundamental Theorem -Basic properties of definite of Calculusintegrals -Apply the Fundamental -Use of the Fundamental Theorem of CalculusTheorem of Calculus to -Use technology to graph anevaluate definite integrals integral function -Use of the Fundamental -Review applications of theTheorem of Calculus to Fundamental Theorem ofrepresent a particular Calculusantiderivative, and theanalytical and graphicalanalysis of functions sodefined -Antiderivatives followingdirectly from derivatives ofbasic functions -Antiderivatives bysubstitution of variables, parts, and simple partial fractions -Improper integrals -Finding specificantiderivatives using initialconditions, includingapplications to motion along a



AP Calculus BC


line Theorem of -classroom discussion -Solving separable differential Calculus. equations and using them inmodeling. In particular,studying the equation y'=kyand exponential growth -Solving logistic differentialequations and using them inmodeling -Use of Riemann andtrapezoidal sums toapproximate definite integralsof functions representedalgebraically, graphically, and


Instructional Unit Polynomial Approximations and Series

AP Calculus BC


Polynomial Approximations and The students will be -Express a series In both -written 2.11.11.D Series able to apply the expanded form and using documentation of -The concept of a series properties of summation notation problem solving -Motivating examples, geometric series and -Identify a geometric series -applications to realincluding decimal expansion be able to and determine the sum of life problems -Geometric series with differentiate, the geometric series if it -AP Calculus examapplications integrate, or converges practice questions -Alternating series with error substitute into a -Write an expression for a -technologybound known power series power series centered at applications -Terms of series as areas of in order to find zero and at some arbitraryrectangles and their relationship additional power value a to improper integrals, including series -Perform a term by term the integral test and its use in representations. differentiation of a powertesting the convergence of seriesp-series -Perform a term by term -The ratio test for integration of a power seriesconvergence and divergence -Comparing series to test forconvergence or divergence -Taylor polynomialapproximation with graphicaldemonstration of convergence

-Maclaurin series and thegeneral Taylor series centeredat x=a -Maclaurin series for thefunctions e to the x power, sinx, cos x, and 1/(1-x) -Formal manipulation ofTaylor series and shortcuts tocomputing Taylor series,including substitution,



AP Calculus BC


differentiation, -classroom discussionantidifferentiation, and theformation of new series fromknow series -Functions defined by powerseries -Radius and interval ofconvergence of power series -Lagrange error bound forTaylor polynomials



AP Calculus BC


Polynomial Approximations and The students will be -Construct a series that can -written Series able to use be used as a polynomial documentation of -The concept of a series derivatives to find approximation of a function problem solving -Motivating examples, the Maclaurin series -applications to realincluding decimal expansion or Taylor series -Construct a power series life problems -Geometric series with generated by a approximation for the sine -AP Calculus examapplications differentiable and cosine functions practice questions -Alternating series with error function and -State the definition of both -technologybound approximate a a Taylor and Maclaurin applications -Terms of series as areas of function with a seriesrectangles and their relationship -Generate an nth term to improper integrals, including Taylor polynomial for a the integral test and its use in function centered at sometesting the convergence of arbitrary value ap-series -Construct and apply basic -The ratio test for Maclaurin seriesconvergence and divergence representations of functions -Comparing series to test for convergence or divergence -Utilize Taylor's Theorem to -Taylor polynomial determine the remainder forapproximation with graphical the approximationdemonstration of convergence -Estimate the value of the

remainder for a Taylor -Maclaurin series and thegeneral Taylor series centeredat x=a -Maclaurin series for thefunctions e to the x power, sinx, cos x, and 1/(1-x) -Formal manipulation ofTaylor series and shortcuts tocomputing Taylor series,including substitution,



AP Calculus BC


differentiation, Taylor polynomial. polynomial approximation of -classroom discussionantidifferentiation, and the a functionformation of new series fromknow series -Functions defined by powerseries -Radius and interval ofconvergence of power series -Lagrange error bound forTaylor polynomials



AP Calculus BC


Polynomial Approximations and The students will be -Understand the difference -written Series able to use the between convergence and documentation of -The concept of a series nth-term test, direct divergence in relationship to problem solving -Motivating examples, comparison test, a series -applications to realincluding decimal expansion ratio test, integral -Apply the nth term test for life problems -Geometric series with test, and alternating divergence -AP Calculus examapplications series test to -Utilize the direct practice questions -Alternating series with error determine the comparison test to -technologybound convergence or determine the convergence applications -Terms of series as areas of divergence of a or divergence of a seriesrectangles and their relationship series, in addition -Utilize the ratio test to to improper integrals, including the students will be determine the convergence the integral test and its use in able to determine or divergence of a seriestesting the convergence of the interval and -Utilize the ratio test top-series radius of determine the convergence -The ratio test for or divergence of theconvergence and divergence endpoints of a series -Comparing series to test for -Utilize the integral test toconvergence or divergence determine the convergence -Taylor polynomial or divergence of a seriesapproximation with graphical -Identify a p-series anddemonstration of convergence determine its convergence

or divergence -Maclaurin series and the -Utilize the limit comparisongeneral Taylor series centered test to determine theat x=a convergence or divergence -Maclaurin series for the of a seriesfunctions e to the x power, sin -Utilize the alternating seriesx, cos x, and 1/(1-x) test to determine the -Formal manipulation of convergence of a seriesTaylor series and shortcuts to -Utilize the ratio test tocomputing Taylor series, determine the interval ofincluding substitution, convergence and radius of



AP Calculus BC


differentiation, convergence. convergence of a power -classroom discussionantidifferentiation, and the seriesformation of new series from -Develop a strategy forknow series determining the -Functions defined by power convergence or divergenceseries of a series -Radius and interval ofconvergence of power series -Lagrange error bound forTaylor polynomials


ap calculus bc - hershey.k12.pa.us€¦ · instructional unit derivatives ap calculus bc unit...

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