career tech ap calculus

8/8/2019 Career Tech AP Calculus

1/15

AP CALCULUS

The Calculus objectives that follow are meant as a framework forCareerTech Centers and educators. They should aide in alignment ofcurriculum, supply additional resources, and assist in the assignment ofhigh school calculus academic credit.A committee of Mathematicsinstructors compiled the list of necessary objectives that all CareerTechstudents should master upon completion of the course. CareerTech

upper division math courses demonstrate the Content and ProcessStandards of the National Council of Teachers of Mathematics (NCTM)and the Process Standards of Oklahoma Priority Academic Student Skills(PASS). Students are expected to problem solve, do reasoning andproofs, demonstrate mathematical communication, connect and linkmathematical ideas to real-world and other disciplines, and usemathematical representations for modeling, interpreting, andcommunicating. Technology will be integrated into the course in order toprepare students for real-world situations.

Course Description:

This is a college preparatory course that offers extreme rigor in aspecialized field of study. It will enable the student to be successful onthe Advanced Placement AB Calculus exam and/or in college calculus.Topics covered will be: limits, differentiation, and basic integrationtechniques. An emphasis will be placed on real world applications asthey relate to the various engineering fields as well as development ofproblem-solving skills. Prerequisites for the course are: Algebra I,Algebra II, Geometry, Trigonometry/Pre-Calculus

Requirements for College Admission Status (Title 70 O.S. 11-103.6)These courses are to be taught by a highly qualified teacher with anOklahoma Advanced Mathematics teaching certification. The studentsshould be in the eleventh or twelfth grade or if a sophomore, they shouldbe in a Focused Field of Career Study program. The course will have at aminimum, but may exceed, a duration of 120 hours within a school year.


2/15

CalculusSyllabus

Objective NCTM Standard OklahomaPASS

ProcessStandard

I. Preparation forCalculus

A. Sketch the graph ofan equation

AlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 5.1,5.2

B. Find the intercepts

of a graph

Algebra

Measurement

1.1, 1.2, 2.1, 2.2, 5.1,

5.2C. Test a graph forsymmetry with respectto an axis and theorigin

Number & OperationsAlgebraMeasurementGeometryData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 3.3,3.4, 5.1, 5.2

D. Find the points ofintersection of twographs

Algebra 1.1, 1.2, 2.1, 2.2, 5.1,5.2

E. Interpret

mathematical modelsfor real-life data

Number & Operations

AlgebraMeasurementData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,

3.1, 3.3, 4.1, 4.3, 5.1,5.2, 5.3

F. Find the slope of aline passing throughtwo points

AlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 5.1,5.2

G. Write the equationsof a line with a givenpoint and slope

Number & OperationsAlgebra

1.1, 1.2, 2.1, 2.2, 5.1,5.2

H. Interpret slope as aratio or as a rate in areal-life application

Number & OperationsAlgebraMeasurement

Data Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

I. Sketch the graph ofa linear equation inslope-intercept form

Algebra 1.1, 1.2, 2.1, 2.2, 5.1,5.2


3/15

J. Write equations oflines that are parallelor perpendicular to agiven line


1.1, 1.2, 2.1, 2.2, 5.1,5.2

K. Use function

notation to representand evaluate afunction

Number & Operations

AlgebraMeasurementData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 5.1,

5.2

L. Find the domainand range of afunction



1.1, 1.2, 2.1, 2.2, 5.1,5.2

M. Sketch the graphof a function

AlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 5.1,5.2

N. Identify differenttypes oftransformations offunctions

Number & OperationsAlgebraData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 5.1,5.2

O. Classify functionsand recognizecombinations offunctions


1.1, 1.2, 2.1, 2.2, 5.1,5.2

P. Fit a linear modelto a real-life data set

Number & OperationsAlgebraMeasurement DataAnalysis & Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

Q. Fit a quadraticmodel to a real-lifedata set

Number & OperationsAlgebraMeasurementData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

R. Fit a trigonometricmodel to a real-life

data set


MeasurementData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

II. Limits and TheirProperties

A. Understand whatcalculus is and how itcompares to pre-

Data Analysis &ProbabilityAlgebra

1.1, 1.2, 2.1, 2.2, 2.3,4.4, 5.1, 5.2


4/15

calculus

B. Understand thatthe tangent lineproblem is basic tocalculus


1.1, 1.2, 2.1, 2.2, 2.3,4.4, 5.1, 5.2

C. Understand thatthe area problem isalso basic to calculus

Number & OperationsData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,4.4, 5.1, 5.2

D. Estimate a limitusing a numerical orgraphical approach


1.1, 1.2, 2.1, 2.2, 5.1,5.2

E. Learn differentways that a limit canfail to exist

Number & OperationsData Analysis &ProbabilityMeasurement

1.1, 1.2, 2.1, 2.2, 5.1,5.2

F. Study and use aformal definition of alimit


1.1, 1.2, 2.1, 2.2, 5.1,5.2

G. Evaluate a limitusing properties oflimits


1.1, 1.2, 2.1, 2.2, 5.1,5.2

H. Develop and use astrategy


1.1, 1.2, 2.1, 2.2, 5.1,5.2

I. Evaluate a limitusing dividing out andrationalizingtechniques


1.1, 1.2, 2.1, 2.2, 5.1,5.2

J. Evaluate a limitusing the SqueezeTheorem


1.1, 1.2, 2.1, 2.2, 5.1,5.2

K. Determinecontinuity at a pointand continuity on anopen interval


1.1, 1.2, 2.1, 2.2, 5.1,5.2

L. Determine one-sided limits andcontinuity on a closedinterval


1.1, 1.2, 2.1, 2.2, 5.1,5.2


5/15

M. Use properties ofcontinuity


1.1, 1.2, 2.1, 2.2, 5.1,5.2

N. Understand and

use the IntermediateValue Theorem

Number & Operations


1.1, 1.2, 2.1, 2.2, 5.1,

5.2

O. Determine infinitelimits from the left andfrom the right


1.1, 1.2, 2.1, 2.2, 5.1,5.2

P. Find and sketch thevertical asymptotes ofthe graph of a function


1.1, 1.2, 2.1, 2.2, 5.1,5.2

III. Differentiation

A. Find the slope ofthe tangent line to acurve at a point


1.1, 1.2, 2.1, 2.2, 5.1,5.2

B. Use the limitdefinition to find thederivative of a function


1.1, 1.2, 2.1, 2.2, 5.1,5.2

C. Understand therelationship betweendifferentiability andcontinuity


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

D. Find the derivativeof a function using theConstant Rule

Number & OperationsAlgebraMeasure Data Analysis& Probability ment

1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

E. Find the derivativeof a function using thePower Rule


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

F. Find the derivativeof a function using theConstant Multiple Rule


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

G. Find the derivativeof a function using theSum and Difference


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2


6/15

Rules Data Analysis &Probability

H. Find the derivativeof the sine functionand of the cosine

function



1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

I. Use derivatives tofind rates of change


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

J. Find the derivativeof a function using theProduct Rule

Number & OperationsAlgebraMeasurementData Analysis &

Probability

1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

K. Find the derivativeof a function using theQuotient Rule


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

L. Find the derivativeof a trigonometricfunction


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

M. Find a higher-order derivative of afunction


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

N. Find the derivativeof a compositefunction using theChain Rule


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

O. Find the derivativeof a function using theGeneral Power Rule


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

P. Simplify thederivative of a functionusing algebra


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

Q. Find the derivativeof a trigonometricfunction using theChain Rule


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

R. Distinguish Number & Operations 1.1, 1.2, 2.1, 2.2, 2.3,


7/15

between functionswritten in implicit andexplicit forms

AlgebraMeasurement

3.3, 4.2, 5.1, 5.2

S. Use implicitdifferentiation to find

the derivative of afunction


Measurement

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

T. Find a related rate Number & OperationsAlgebraMeasurementData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

U. Use related rates tosolve real-life problems

Number & OperationsAlgebraMeasurementData Analysis &

Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

IV. Applications ofDifferentiation

A. Understand thedefinition of extrema ofa function on aninterval


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

B. Understand thedefinition of relativeextrema of a functionon an open interval


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

C. Find extrema on aclosed interval


1.1, 1.2, 2.1, 2.2, 2.3,5.1, 5.2

D. Understand anduse Rolles Theorem


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

E. Understand anduse the Mean ValueTheorem


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

F. Determine intervals

on which a function isincreasing ordecreasing

Data Analysis &

ProbabilityAlgebraNumbers & Operations

1.1, 1.2, 2.1, 2.2, 2.3,

3.3, 4.2, 5.1, 5.2

G. Apply the FirstDerivative Test to findrelative extrema of afunction


1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

H. Determine intervals Data Analysis & 1.1, 1.2, 2.1, 2.2, 2.3,


8/15

on which a function isconcave upward ordownward


3.1, 3.3, 4.1, 4.3, 5.1,5.2, 5.3

I. Find any points ofinflection of the graph

of a function


Algebra

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.31.1, 1.2, 2.1,2.2, 2.3, 3.1, 3.3, 4.1,

4.3, 5.1, 5.2, 5.3

J. Apply the SecondDerivative Test to findrelative extrema of afunction

Data Analysis &ProbabilityAlgebraNumbers & Operations

1.1, 1.2, 2.1, 2.2, 2.3,3.1, 3.3, 4.1, 4.3, 5.1,

5.2, 5.3

K. Determine (finite)limits at infinity


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

L. Determine the

horizontal asymptotesof the graph of afunction

Data Analysis &


1.1, 1.2, 2.1, 2.2, 2.3,

3.3, 4.2, 5.1, 5.2

M. Determine infinitelimits at infinity


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

N. Analyze and sketchthe graph of a function


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

O. Solve appliedminimum andmaximum problems


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

P. Approximate a zeroof a function usingNewtons Method


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

Q. Understand theconcept of a tangentline approximation


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

R. Compare the value

of the differential, dy,with the actual changein y, delta y

Data Analysis &


1.1, 1.2, 2.1, 2.2, 2.3,

3.3, 4.2, 5.1, 5.2

S. Estimate apropagated error usinga differential


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

T. Find the differential Data Analysis & 1.1, 1.2, 2.1, 2.2, 2.3,


9/15

of a function usingdifferentiationformulas


3.3, 4.2, 5.1, 5.2

V. Integration

A. Write the general

solution of adifferential equation

Numbers & Operations

MeasurementAlgebra

1.1, 1.2, 2.1, 2.2, 5.1,

5.2

B. Use indefiniteintegral notation forantiderivatives

Numbers & OperationsMeasurementAlgebra

1.1, 1.2, 2.1, 2.2, 5.1,5.2

C. Use basicintegration rules tofind antiderivatives


1.1, 1.2, 2.1, 2.2, 5.1,5.2

D. Find a particularsolution of adifferential equation


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

E. Use sigma notationto write and evaluate asum


1.1, 1.2, 2.1, 2.2, 5.1,5.2

F. Understand theconcept of area


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

G. Approximate theareas of a plane region


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

H. Find the area of aplane region usinglimits


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

I. Understand thedefinition of aRiemann sum


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2

J. Evaluate a definiteintegral using limits

Numbers & OperationsMeasurementAlgebraData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

K. Evaluate a definite

integral usingproperties of definiteintegrals


MeasurementAlgebraData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,

3.3, 4.2, 5.1, 5.2, 5.3

L. Evaluate a definiteintegral using theFundamental Theorem


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3


10/15

of Calculus Data Analysis &Probability

M. Understand anduse the Mean Value

Theorem for Integrals

Numbers & OperationsMeasurement

AlgebraData Analysis &Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

N. Find the averagevalue of a functionover a closed interval


1.1, 1.2, 2.1, 2.2, 5.1,5.2

O. Use patternrecognition to evaluatean indefinite integral



1.1, 1.2, 2.1, 2.2, 5.1,5.2

P. Use a change ofvariables to evaluatean indefinite integral


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

Q. Use the GeneralPower Rule forIntegration to evaluatean indefinite integral

Numbers & OperationsMeasurementAlgebraData Analysis &

Probability

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

R. Use a change ofvariables to evaluate adefinite integral


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

S. Evaluate a definiteintegral involving aneven or odd function


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

T. Approximate adefinite integral usingthe Trapezoidal Rule


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

U. Approximate adefinite integral using

Numbers & OperationsMeasurement

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3


11/15


12/15

functions Measurement

K. Integrate naturalexponential functions

Numbers & OperationsAlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

L. Define exponential

functions that havebases other than e


AlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 5.1,

5.2

M. Differentiate andintegrate exponentialfunctions that havebases other than e


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

N. Use exponentialfunctions to modelcompound interest andexponential growth


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

O. Use separation of

variables to solve asimple differentialequation


AlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 2.3,

3.3, 4.2, 5.1, 5.2, 5.3

P. Use exponentialfunctions to modelgrowth and decay inapplied problems


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

Q. Use initialconditions to findparticular solutions ofdifferential equations


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

R. Recognize andsolve differentialequations that can besolved by separation ofvariables


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

S. Recognize and solvehomogeneousdifferential equations


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

T. Use a differentialequation to model and

solve an appliedproblem

Numbers & OperationsAlgebra

Measurement

1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

U. Develop propertiesof the six inversetrigonometricfunctions


1.1, 1.2, 2.1, 2.2, 5.1,5.2

V. Differentiate aninverse trigonometric


1.1, 1.2, 2.1, 2.2, 5.1,5.2


13/15

function Measurement

W. Review the basicdifferentiationformulas forelementary functions


1.1, 1.2, 2.1, 2.2, 5.1,5.2

X. Integrate functionswhose antiderivativesinvolve inversetrigonometricfunctions


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

Y. Use the method ofcompleting the squareto integrate a function


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

Z. Review the basicintegration formulasinvolving elementary

functions


1.1, 1.2, 2.1, 2.2, 5.1,5.2

AA. Develop propertiesof hyperbolic functions


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

BB. Differentiate andintegrate hyperbolicfunctions


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

CC. Developproperties of inversehyperbolic functions


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

DD. Differentiate andintegrate functionsinvolving inversehyperbolic functions


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

VII. Applications ofIntegration

A. Find the area of aregion between twocurves usingintegration


1.1, 1.2, 2.1, 2.2, 5.1,5.2

B. Find the area of a

region betweenintersecting curvesusing integration


AlgebraMeasurement

1.1, 1.2, 2.1, 2.2, 5.1,

5.2

C. Describeintegration as aaccumulation process


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3

D. Find the volume ofa solid of revolution


1.1, 1.2, 2.1, 2.2, 5.1,5.2


14/15

using the disk method Measurement

E. Find the volume ofa solid of revolutionusing the washermethod


1.1, 1.2, 2.1, 2.2, 5.1,5.2

F. Find the volume ofa solid with knowncross sections


1.1, 1.2, 2.1, 2.2, 5.1,5.2

G. Find the volume ofa solid of revolutionusing the shell method


1.1, 1.2, 2.1, 2.2, 5.1,5.2

H. Compare the usesof the disk method andthe shell method


1.1, 1.2, 2.1, 2.2, 2.3,3.3, 4.2, 5.1, 5.2, 5.3


15/15

References and Resources

Referenced Standards

Principles and Standards for School Mathematics(4th ed.). (2005).

National Council of Teachers of Mathematics, Reston, VA

Oklahoma Priority Academic Student Skills(2003). Oklahoma StateDepartment of Education-PASS-{www.sde.state.ok.us}

Suggested Text and Supplemental Materials

Larson, Ron, Hostetler, Bob, Edwards, Bruce. (2006). Calculus withAnalytic Geometry (8thed)/Calculus of a Single Variable (8thed). Boston:Houghton Mifflin.

Larson, Ron, Hostetler, Bob, Edwards, Bruce. (2006). Calculus withAnalytic Geometry (8thed)/Calculus of a Single Variable (8thed)-Instructors Resource Guide. Boston: Houghton Mifflin.

Varberg, Dale, Edwin J. Purcell, and Steven E. Rigdon. (2007). Calculus(9 ed.) th . Boston: Pearson Prentice Hall.

Thomas, George B., Jr., and Ross L. Finney (1996). Calculus withAnalytic Geometry (9 ed.)th . Boston: Addison-Wesley.

Finney, Ross L., Franklin D. Demana, Bert K. Waits, and DanielKennedy. (2007). Calculus: Graphical, Numerical, Algebraic: AP Edition (3ed.)

rd

. Boston: Pearson Prentice Hall.

career tech ap calculus

Documents