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Test InformationGuide:College-LevelExaminationProgram®

2011-12

Calculus

© 2011 The College Board. All rights reserved. College Board, College-Level ExaminationProgram, CLEP, and the acorn logo are registered trademarks of the College Board.

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CLEP TEST INFORMATIONGUIDE FOR CALCULUS

History of CLEP

Since 1967, the College-Level Examination Program(CLEP®) has provided over six million people withthe opportunity to reach their educational goals.CLEP participants have received college credit forknowledge and expertise they have gained throughprior course work, independent study or work andlife experience.

Over the years, the CLEP examinations have evolvedto keep pace with changing curricula and pedagogy.Typically, the examinations represent material taughtin introductory college-level courses from all areasof the college curriculum. Students may choose from33 different subject areas in which to demonstratetheir mastery of college-level material.

Today, more than 2,900 colleges and universitiesrecognize and grant credit for CLEP.

Philosophy of CLEP

Promoting access to higher education is CLEP’sfoundation. CLEP offers students an opportunity todemonstrate and receive validation of theircollege-level skills and knowledge. Students whoachieve an appropriate score on a CLEP exam canenrich their college experience with higher-levelcourses in their major field of study, expand theirhorizons by taking a wider array of electives andavoid repetition of material that they already know.

CLEP Participants

CLEP’s test-taking population includes people of allages and walks of life. Traditional 18- to 22-year-oldstudents, adults just entering or returning to school,homeschoolers and international students who needto quantify their knowledge have all been assisted byCLEP in earning their college degrees. Currently,58 percent of CLEP’s test-takers are women and52 percent are 23 years of age or older.

For over 30 years, the College Board has worked toprovide government-funded credit-by-examopportunities to the military through CLEP. Militaryservice members are fully funded for their CLEP examfees. Exams are administered at military installations

worldwide through computer-based testing programsand also — in forward-deployed areas — throughpaper-based testing. Approximately one-third of allCLEP candidates are military service members.

2010-11 National CLEP Candidates by Age*

These data are based on 100% of CLEP test-takers who responded to this survey question during their examinations.

*

Under 189%

18-22 years39%

23-29 years22%

30 years and older30%

2010-11 National CLEP Candidates by Gender

41%

58%

Computer-Based CLEP Testing

The computer-based format of CLEP exams allowsfor a number of key features. These include:

• a variety of question formats that ensure effectiveassessment

• real-time score reporting that gives students andcolleges the ability to make immediate credit-granting decisions (except College Composition,which requires faculty scoring of essays twice amonth)

• a uniform recommended credit-granting score of50 for all exams

• “rights-only” scoring, which awards one point percorrect answer

• pretest questions that are not scored but providecurrent candidate population data and allow forrapid expansion of question pools

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CLEP Exam Development

Content development for each of the CLEP examsis directed by a test development committee. Eachcommittee is composed of faculty from a widevariety of institutions who are currently teachingthe relevant college undergraduate courses. Thecommittee members establish the test specificationsbased on feedback from a national curriculumsurvey; recommend credit-granting scores andstandards; develop and select test questions; reviewstatistical data and prepare descriptive material foruse by faculty (Test Information Guides) and studentsplanning to take the tests (CLEP Official Study Guide).

College faculty also participate in CLEP in otherways: they convene periodically as part ofstandard-setting panels to determine therecommended level of student competency for thegranting of college credit; they are called upon towrite exam questions and to review forms and theyhelp to ensure the continuing relevance of the CLEPexaminations through the curriculum surveys.

The Curriculum Survey

The first step in the construction of a CLEP exam isa curriculum survey. Its main purpose is to obtaininformation needed to develop test-contentspecifications that reflect the current collegecurriculum and to recognize anticipated changes inthe field. The surveys of college faculty areconducted in each subject every three to five yearsdepending on the discipline. Specifically, the surveygathers information on:

• the major content and skill areas covered in theequivalent course and the proportion of the coursedevoted to each area

• specific topics taught and the emphasis given toeach topic

• specific skills students are expected to acquire andthe relative emphasis given to them

• recent and anticipated changes in course content,skills and topics

• the primary textbooks and supplementary learningresources used

• titles and lengths of college courses thatcorrespond to the CLEP exam

The Committee

The College Board appoints standing committees ofcollege faculty for each test title in the CLEP battery.Committee members usually serve a term of up tofour years. Each committee works with contentspecialists at Educational Testing Service to establishtest specifications and develop the tests. Listedbelow are the current committee members and theirinstitutional affiliations.

Daniel Frohardt,Chair

Wayne State University

Sharon Sledge San Jacinto College

Richard West Francis Marion University

The primary objective of the committee is to producetests with good content validity. CLEP tests must berigorous and relevant to the discipline and theappropriate courses. While the consensus of thecommittee members is that this test has high contentvalidity for a typical introductory Calculus course orcurriculum, the validity of the content for a specificcourse or curriculum is best determined locallythrough careful review and comparison of testcontent, with instructional content covered in aparticular course or curriculum.

The Committee Meeting

The exam is developed from a pool of questionswritten by committee members and outside questionwriters. All questions that will be scored on a CLEPexam have been pretested; those that pass a rigorousstatistical analysis for content relevance, difficulty,fairness and correlation with assessment criteria areadded to the pool. These questions are compiled bytest development specialists according to the testspecifications, and are presented to all the committeemembers for a final review. Before convening at atwo- or three-day committee meeting, the membershave a chance to review the test specifications andthe pool of questions available for possible inclusionin the exam.

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At the meeting, the committee determines whetherthe questions are appropriate for the test and, if not,whether they need to be reworked and pretestedagain to ensure that they are accurate andunambiguous. Finally, draft forms of the exam arereviewed to ensure comparable levels of difficulty andcontent specifications on the various test forms. Thecommittee is also responsible for writing anddeveloping pretest questions. These questions areadministered to candidates who take the examinationand provide valuable statistical feedback on studentperformance under operational conditions.

Once the questions are developed and pretested,tests are assembled in one of two ways. In somecases, test forms are assembled in their entirety.These forms are of comparable difficulty and aretherefore interchangeable. More commonly,questions are assembled into smaller,content-specific units called testlets, which can thenbe combined in different ways to create multiple testforms. This method allows many different forms tobe assembled from a pool of questions.

Test Specifications

Test content specifications are determined primarilythrough the curriculum survey, the expertise of thecommittee and test development specialists, therecommendations of appropriate councils andconferences, textbook reviews and other appropriatesources of information. Content specifications takeinto account:

• the purpose of the test

• the intended test-taker population

• the titles and descriptions of courses the test isdesigned to reflect

• the specific subject matter and abilities to be tested

• the length of the test, types of questions andinstructions to be used

Recommendation of the AmericanCouncil on Education (ACE)

The American Council on Education’s CollegeCredit Recommendation Service (ACE CREDIT)has evaluated CLEP processes and procedures for

developing, administering and scoring the exams.Effective July 2001, ACE recommended a uniformcredit-granting score of 50 across all subjects, withthe exception of four-semester language exams,which represents the performance of students whoearn a grade of C in the corresponding collegecourse.

The American Council on Education, the majorcoordinating body for all the nation’s higher educationinstitutions, seeks to provide leadership and a unifyingvoice on key higher education issues and to influencepublic policy through advocacy, research and programinitiatives. For more information, visit the ACECREDIT website at www.acenet.edu/acecredit.

CLEP Credit Granting

CLEP uses a common recommended credit-grantingscore of 50 for all CLEP exams.

This common credit-granting score does not mean,however, that the standards for all CLEP exams arethe same. When a new or revised version of a test isintroduced, the program conducts a standard settingto determine the recommended credit-granting score(“cut score”).

A standard-setting panel, consisting of 15–20 facultymembers from colleges and universities across thecountry who are currently teaching the course, isappointed to give its expert judgment on the level ofstudent performance that would be necessary toreceive college credit in the course. The panelreviews the test and test specifications and definesthe capabilities of the typical A student, as well asthose of the typical B, C and D students.* Expectedindividual student performance is rated by eachpanelist on each question. The combined average ofthe ratings is used to determine a recommendednumber of examination questions that must beanswered correctly to mirror classroom performanceof typical B and C students in the related course. Thepanel’s findings are given to members of the testdevelopment committee who, with the help ofEducational Testing Service and College Boardpsychometric specialists, make a final determinationon which raw scores are equivalent to B and C levelsof performance.

*Student performance for the language exams (French, German and Spanish)is defined only at the B and C levels.

5

Calculus

Description of the ExaminationThe Calculus examination covers skills and concepts that are usually taught in a one-semester college course in calculus. The content of each examination is approximately 60% limits and differential calculus and 40% integral calculus. Algebraic, trigonometric, exponential, logarithmic and general functions are included. The exam is primarily concerned with an intuitive understanding of calculus and experience with its methods and applications. Knowledge of preparatory mathematics, including algebra, geometry, trigonometry and analytic geometry is assumed.

The examination contains 44 questions, in two sections, to be answered in approximately 90 minutes. Any time candidates spend on tutorials and providing personal information is in addition to the actual testing time.

• Section 1: 27 questions, approximately 50 minutes. No calculator is allowed for this section.

• Section 2: 17 questions, approximately 40 minutes. The use of an online graphing calculator (non-CAS) is allowed for this section. Only some of the questions will require the use of the calculator.

Graphing CalculatorA graphing calculator is integrated into the exam software, and it is available to students during Section 2 of the exam. Since only some of the questions in Section 2 actually require the calculator, students are expected to know how and when to make appropriate use of it. The graphing calculator, together with a brief tutorial, is available to students as a free download for a 30-day trial period. Students are expected to download the calculator and become familiar with its functionality prior to taking the exam.

For more information about downloading the practice version of the graphing calculator, please visit the Calculus exam description on the CLEP website, www.collegeboard.org/clep.

In order to answer some of the questions in Section 2 of the exam, students may be required to use the online graphing calculator in the following ways:

• Perform calculations (e.g., exponents, roots, trigonometric values, logarithms).

• Graph functions and analyze the graphs.• Find zeros of functions.• Find points of intersection of graphs of

functions.• Find minima/maxima of functions.• Find numerical solutions to equations.• Generate a table of values for a function.

Knowledge and Skills RequiredQuestions on the exam require candidates to demonstrate the following abilities:

• Solving routine problems involving the techniques of calculus (approximately 50% of the exam)

• Solving nonroutine problems involving an understanding of the concepts and applications of calculus (approximately 50% of the exam)

The subject matter of the Calculus exam is drawn from the following topics. The percentages next to the main topics indicate the approximate percentage of exam questions on that topic.

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C A L C U L U S

10% Limits• Statement of properties, e.g., limit of a

constant, sum, product or quotient• Limit calculations, including limits

involving infi nity, e.g., lim sin ,x

xx→

=0

1

limx x→0

1 is nonexistent, and lim sin

x

xx→∞

= 0

• Continuity

50% Differential CalculusThe Derivative• Defi nitions of the derivative

e.g., ′( ) = ( ) − ( )−→

f a f x f ax ax a

lim

and ′( ) = +( ) − ( )→

f x f x h f xhh

lim0

• Derivatives of elementary functions• Derivatives of sums, products, and

quotients (including tan x and cot x )• Derivative of a composite function (chain

rule), e.g., sin ,ax b+( ) aekx , ln k x( )

• Implicit differentiation• Derivative of the inverse of a function

(including arcsin x and arctan x )• Higher order derivatives• Corresponding characteristics of graphs of

f, ′f , and ′′f• Statement of the Mean Value Theorem;

applications and graphical illustrations• Relation between differentiability and

continuity• Use of L’Hospital’s Rule (quotient and

indeterminate forms)

Applications of the Derivative• Slope of a curve at a point• Tangent lines and linear approximation• Curve sketching: increasing and

decreasing functions; relative and absolute maximum and minimum points; concavity; points of infl ection

• Extreme value problems• Velocity and acceleration of a particle

moving along a line• Average and instantaneous rates of change• Related rates of change

40% Integral CalculusAntiderivatives and Techniques of Integration• Concept of antiderivatives• Basic integration formulas• Integration by substitution (use of

identities, change of variable)

Applications of Antiderivatives• Distance and velocity from acceleration

with initial conditions• Solutions of ′ =y k y and applications to

growth and decay

The Defi nite Integral• Defi nition of the defi nite integral as the

limit of a sequence of Riemann sums and approximations of the defi nite integral using areas of rectangles

• Properties of the defi nite integral• The Fundamental Theorem: ddx f t dt f x

a

x( ) = ( )∫ and

′( ) = ( ) − ( )∫ F x dx F b F a

a

b

Applications of the Defi nite Integral• Average value of a function on an interval• Area, including area between curves• Other (e.g., accumulated change from a

rate of change)

7

C A L C U L U S

Notes and Reference Information

(1) Figures that accompany questions are intended to provide information useful in answering the questions. All fi gures lie in a plane unless otherwise indicated. The fi gures are drawn as accurately as possible EXCEPT when it is stated in a specifi c question that the fi gure is not drawn to scale. Straight lines and smooth curves may appear slightly jagged.

(2) Unless otherwise specifi ed, all angles are measured in radians, and all numbers used are real numbers.

(3) Unless otherwise specifi ed, the domain of any function f is assumed to be the set of all real numbers x for which f x( ) is a real number. The range of f is assumed to be the set of all real numbers f x( ) , where x is in the domain of f.(4) In this test, ln x denotes the natural logarithm of x (that is, the logarithm to the base e).

(5) The inverse of a trigonometric function f may be indicated using the inverse function notation f −1 or with the prefi x “arc” (e.g., sin arcsin− =1x x ).

Sample Test QuestionsThe following sample questions do not appear on an actual CLEP Examination. They are intended to give potential test-takers an indication of the format and diffi culty level of the examination, and to provide content for practice and review. Knowing the correct answers to all of the sample questions is not a guarantee of satisfactory performance on the exam.

Section IDirections: A calculator will not be available for questions in this section. Some questions will require you to select from among fi ve choices. For these questions, select the BEST of the choices given. Some questions will require you to enter a numerical answer in the box provided.

1. If f x x( ) = − −2 3, then ′( ) =f x

(A) 6 2x (B) 6 2x− (C) 6 4x− (D) − −6 2x (E) − −6 4x

2. At which of the ve points on the graph in the

gure above are dydx

and d ydx

2

2 both negative?

(A) A (B) B (C) C (D) D (E) E

3. Which of the following is an equation of the line tangent to the graph of f x x x( ) = −3 at the point where x = 2 ?

(A) y x− = −( )6 4 2(B) y x− = −( )6 5 2(C) y x− = −( )6 6 2(D) y x− = −( )6 11 2(E) y x− = −( )6 12 2

4. e e dxx +( ) =∫(A) e Cx +(B) e e Cx + +(C) e ex Cx + +

(D) ex ex Cx+

+ + +1

1

(E) ex

e Cx+

+ + +1 2

1 2

8

C A L C U L U S

5. What is lim ?x

xx x→∞−

+ −

2

24

2 4(A) −2

(B) − 14

(C) 12

(D) 1 (E) The limit does not exist.

6. The graph of ′′f , the second derivative of the function f , is shown in the gure above. On what intervals is the graph of f concave up?

(A) −( )∞ ∞,(B) − −( )∞, 1 and 3,∞( )(C) −( )∞, 1(D) −( )1 3,(E) 1, ∞( )

7. x x dx−( ) =∫ 1

(A) 25

23

52

32x x C− +

(B) 12 22

23x x x C+ − +

(C) 12

2x x C− +

(D) 23 2

32

12x x C+ +

(E) 32

12

12x x C− +

−

8. Let f and g be the functions de ned by f x x( ) = sin and g x x( ) = cos . For which of

the following values of a is the line tangent to the graph of f at x a= parallel to the line tangent to the graph of g at x a= ?

(A) 0 (B) π4 (C) π2 (D) 34π (E) π

9. The acceleration, at time t , of a particle moving along the x-axis is given by a t t( ) = +20 63 . At time t = 0, the velocity of the particle is 0 and the position of the particle is 7. What is the position of the particle at time t ?

(A) 120 7t +(B) 60 72t t+(C) 5 6 74t t+ +(D) t t5 23 7+ +(E) t t t5 23 7+ +

9

C A L C U L U S

10. If f x xx( ) = sin ,2 then ′( ) =f x

(A) cos x2

(B) x x xx

cos sin−2 2

(C) x x xx

cos sin−4 2

(D) sin cosx x xx

−2 2

(E) sin cosx x xx

−4 2

x f x( ) ′( )f x g x( ) ′( )g x

10 35 15 6 4

20 8 5 12 10

30 24 25 20 10

11. Selected values of the functions f and g and their derivatives, ′f and ′g , are given in the table above. If h x f g x( ) = ( )( ), what is the value of ′( )h 30 ?

(A) 5 (B) 15 (C) 35 (D) 50 (E) 250

12. What is limcos cos

?h

h

h→

+( ) −0

2 2π π

(A) −∞ (B) −1 (C) 0 (D) 1 (E) ∞

13. If x y x y2 3 3 2+ = , then dydx =

(A) 2 3 32

2 2 2

3x y x y

x y+ −

(B) 2 3 23

3 2 2

2x y x y x

y+ −

(C) 3 23 2

2 2

2 3x y xy x y

−−

(D) 3 23 2

2 3

2 2y x yx y x

−−

(E) 6 23

2

2x y xy−

14. For which of the following functions does d ydx

dydx

3

3 = ?

I. y ex=II. y e x= −

III. y x= sin

(A) I only (B) II only (C) III only (D) I and II (E) II and III

15. The vertical height, in feet, of a ball thrown upward from a cliff is given by s t t t( ) = − + +16 64 2002 , where t is measured in seconds. What is the height of the ball, in feet, when its velocity is zero?

10

C A L C U L U S

16. Which of the following statements about the curve y x x= −4 32 is true?

(A) The curve has no relative extremum.(B) The curve has one point of in ection and

two relative extrema.(C) The curve has two points of in ection and

one relative extremum.(D) The curve has two points of in ection and

two relative extrema.(E) The curve has two points of in ection and

three relative extrema.

17. ddx xsin cos( )( ) =

(A) cos cos x( )(B) sin sin−( )x(C) sin sin cos−( )( )x x(D) − ( )( )cos cos sinx x(E) − ( )( )sin cos sinx x

18. Let f be the function de ned by

f xxx x

x( ) =

−− ≠

=

⎧⎨⎪

⎩⎪

for

for

2 255 5

0 5.

Which of the following statements about f are true?

I. limx

f x→

( )5

exists.

II. f 5( ) exists.III. f x( ) is continuous at x = 5.

(A) None (B) I only (C) II only (D) I and II only (E) I, II, and III

19. What is the average rate of change of the function f de ned by f x x( ) = ⋅100 2 on the interval 0 4, ?[ ]

(A) 100(B) 375(C) 400(D) 1,500(E) 1,600

20. If the functions f and g are de ned for all real numbers and f is an antiderivative of g , which of the following statements is NOT necessarily true?

(A) If g x( ) > 0 for all x , then f is increasing.

(B) If g a( ) = 0, then the graph of f has a horizontal tangent at x a= .

(C) If f x( ) = 0 for all x , then g x( ) = 0 for all x .

(D) If g x( ) = 0 for all x , then f x( ) = 0 for all x .

(E) f is continuous for all x .

21. If f x x( ) = ( )arctan ,π then ′( ) =f 0

(A) −π (B) −1 (C) 0 (D) 1 (E) π

11

C A L C U L U S

22. A rectangle with one side on the x-axis and one side on the line x = 2 has its upper left vertex on the graph of y x= 2 , as indicated in the gure above. For what value of x does the area

of the rectangle attain its maximum value?

(A) 2 (B) 43

(C) 1 (D) 34

(E) 23

23. Let f x x x( ) = +3 . If h is the inverse function of f , then ′( ) =h 2

(A) 113

(B) 14 (C) 1 (D) 4 (E) 13

24. Let F be the number of trees in a forest at time t , in years. If F is decreasing at a rate

given by the equation dFdt F= −2 and if

F 0 5000( ) = , then F t( ) =

(A) 5000 2t− (B) 5000 2e t− (C) 5000 2− t (D) 5000 2+ −t (E) 5000 2+ −e t

25. The function f is given by f x x( ) = ( )sin .12 Which of the following is the local linear approximation for f at x = 0 ?

(A) y x= 12(B) y x= −12(C) y x= +1 12(D) y x= −1 12(E) y x= − +1 12

26. What is the area of the region in the rst quadrant that is bounded by the line y x= 6 and the parabola y x= 3 2 ?

27. Let f be a differentiable function de ned on the closed interval a b,[ ] and let c be a point in the open interval a b,( ) such that

• ′( ) =f c 0,• ′( ) >f x 0 when a x c≤ < , and• ′( ) <f x 0 when c x b< ≤ .

Which of the following statements must be true?

(A) f c( ) = 0(B) ′′( ) =f c 0(C) f c( ) is an absolute maximum value of f

on a b,[ ] .(D) f c( ) is an absolute minimum value of f

on a b,[ ] .(E) The graph of f has a point of in ection at

x c= .

12

C A L C U L U S

28. The function f is continuous on the open

interval −( )π π, . If f x xx x( ) = −cos

sin1 for

x ≠ 0, what is the value of f 0( ) ?

(A) −1 (B) − 12 (C) 0 (D) 12 (E) 1

g 20 0( ) =′( ) >g t 0 for all values of t

29. The function g is differentiable and satis es the conditions above. Let F be the function given

by F x g t dtx

( ) = ( )∫0. Which of the following

must be true?

(A) F has a local minimum at x = 20.(B) F has a local maximum at x = 20.(C) The graph of F has a point of in ection at

x = 20.(D) F has no local minima or local maxima on

the interval 0 ≤ < ∞x .(E) ′( )F 20 does not exist.

30. The Riemann sum ii 50

150

2

1

50 ( )=∑ on the closed

interval 0 1,[ ] is an approximation for which of the following de nite integrals?

(A) x dx20

1∫

(B) x dx20

50∫

(C) x dx502

0

1( )∫

(D) x dx502

0

50( )∫

(E) x dx2

30

1

50∫

31. x dx+ =−∫ 2

3

3

(A) 0(B) 9(C) 12(D) 13(E) 14

32. A particle moves along the x-axis, and its velocity at time t is given by v t t t t( ) = - + +3 23 12 8. What is the maximum acceleration of the particle on the interval 0 3£ £t ?

(A) 9(B) 12(C) 14(D) 21(E) 44

13

C A L C U L U S

Section IIDirections: A graphing calculator will be available for the questions in this section. Some questions will require you to select from among fi ve choices. For these questions, select the BEST of the choices given. If the exact numerical value of your answer is not one of the choices, select the choice that best approximates this value. Some questions will require you to enter a numerical answer in the box provided.

33. ln 10

5

10 xx dx( ) =⌠

⌡(A) 1.282(B) 2.952(C) 5.904(D) 6.797(E) 37.500

34. The graph of the function f is shown in the gure above. What is lim ?

xf x

→( )

2

(A) −1 (B) 0 (C) 1 (D) 2 (E) The limit does not exist.

35. If the function f is continuous for all real

numbers and lim ,h

f a h f ah→

+( ) − ( ) =0

7 then

which of the following statements must be true?

(A) f a( ) = 7(B) f is differentiable at x a= .(C) f is differentiable for all real numbers.(D) f is increasing for x > 0.(E) f is increasing for all real numbers.

36. Let f be a function with second derivative given by ′′( ) = ( ) − ( )f x x xsin cos .2 4 How many points of in ection does the graph of f have on the interval 0 10, ?[ ]

(A) Six (B) Seven (C) Eight (D) Ten (E) Thirteen

37. The area of the region in the rst quadrant between the graph of y x x= −4 2 and the x-axis is

(A) 23 2

(B) 83

(C) 2 2 (D) 2 3

(E) 163

14

C A L C U L U S

38. The function f is given by f x x( ) = +3 12 . What is the average value of f over the closed interval 1 3, ?[ ]

39. Starting at t = 0, a particle moves along the x-axis so that its position at time t is given by x t t t t( ) = − +4 25 2 . What are all values of t for which the particle is moving to the left?

(A) 0 0 913< <t .(B) 0 203 1 470. .< <t(C) 0 414 0 913. .< <t(D) 0 414 2 000. .< <t(E) There are no values of t for which the

particle is moving to the left.

40. The function f has a relative maximum value of 3 at x = 1, as shown in the gure above. If h x x f x( ) = ( )2 , then ′( ) =h 1

(A) −6 (B) −3 (C) 0 (D) 3 (E) 6

41. cos sin2 x x dx∫ =

(A) − +cos3

3x C

(B) − +cos sin3 2

6x x C

(C) sin2

2x C+

(D) cos3

3x C+

(E) cos sin3 2

6x x C+

′( ) = − +( ) −f x x x e x1 2

42. The rst derivative of the function f is given above. At what value of x does the function f attain its minimum value on the closed interval −[ ]5 5, ?

(A) −5 000. (B) −1 235. (C) −0 618. (D) 0.160(E) 1.618

43. The function f is differentiable on a b,[ ] and a c b< < . Which of the following is NOT necessarily true?

(A) f x dx f x dx f x dxa

b

a

c

c

b( ) = ( ) + ( )∫ ∫ ∫

(B) There exists a point d in the open interval

a b,( ) such that ′( ) = ( ) − ( )−f d f b f a

b a .

(C) f x dxa

b( ) ≥∫ 0

(D) limx c

f x f c→

( ) = ( )(E) If k is a real number,

then k f x dx k f x dxa

b

a

b( ) = ( )∫ ∫ .

15

C A L C U L U S

′( ) =+

⎛⎝⎜

⎞⎠⎟

g xx

tan 21 2

44. Let g be the function with rst derivative given above and g 1 5( ) = . If f is the function de ned by f x g x( ) = ( )( )ln , what is the value of ′( )f 1 ?

(A) 0.311(B) 0.443(C) 0.642 (D) 0.968(E) 3.210

45. Let r t( ) be a differentiable function that is positive and increasing. The rate of increase of r3 is equal to 12 times the rate of increase of r when r t( ) =

(A) 43 (B) 2 (C) 123 (D) 2 3 (E) 6

46. The function f is shown in the gure above. At which of the following points could the derivative of f be equal to the average rate of change of f over the closed interval −[ ]2 4, ?

(A) A (B) B (C) C (D) D (E) E

47. ddx t dt

x 21∫ =

(A) 2x (B) x2 1−

(C) x2

(D) x3

313−

(E) x C3

3 +

48. A college is planning to construct a new parking lot. The parking lot must be rectangular and enclose 6,000 square meters of land. A fence will surround the parking lot, and another fence parallel to one of the sides will divide the parking lot into two sections. What are the dimensions, in meters, of the rectangular lot that will use the least amount of fencing?

(A) 1,000 by 1,500(B) 20 5 by 60 5(C) 20 10 by 30 10(D) 20 15 by 20 15(E) 20 15 by 40 15

x 1 2 3 4 5

f x( ) 15 10 9 6 5

49. The function f is continuous on the closed interval 1 5,[ ] and has values that are given in the table above. If two subintervals of equal length are used, what is the midpoint Riemann

sum approximation of f x dx( )∫15

?

(A) −3 (B) 9 (C) 14 (D) 32 (E) 35

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C A L C U L U S

50. If f is continuous for all x , which of the following integrals necessarily have the same value?

I. f x dxa

b( )∫

II. f x a dxb a

+( )−

∫0

III. f x c dxa c

b c

+

+∫ +( )

(A) I and II only(B) I and III only(C) II and III only(D) I, II, and III(E) No two necessarily have the same value.

51. A spherical balloon is being in ated at a constant rate of 25 cm3/sec. At what rate, in cm/sec, is the radius of the balloon changing when the radius is 2 cm? (The volume of a

sphere with radius r is V r= 43

3π .)

(A) 2516π

(B) 258π

(C) 7516π

(D) 3225π

(E) 323π

52. R is the region below the curve y x= and above the x-axis from x = 0 to x b= , where b is a positive constant. S is the region below the curve y x= cos and above the x-axis from x = 0 to x b= . For what value of b is the area of R equal to the area of S ?

(A) 0.739(B) 0.877(C) 0.986(D) 1.404(E) 4.712

53. Let f be the function de ned by f x e x( ) = 3 , and let g be the function de ned by g x x( ) = 3. At what value of x do the graphs of f and g have parallel tangent lines?

(A) -0 657.(B) -0 526.(C) -0 484.(D) -0 344.(E) -0 261.

54. ddx xsin− ( )( ) =1 5

(A) cos− ( )1 5x

(B) 5 51cos− ( )x

(C) 11 5 2− x

(D) 51 25 2− x

(E) 51 25 2+ x

17

C A L C U L U S

55. The population P of bacteria in an experiment

grows according to the equation dPdt kP= ,

where k is a constant and t is measured in hours. If the population of bacteria doubles every 24 hours, what is the value of k ?

(A) 0.029(B) 0.279(C) 0.693(D) 2.485(E) 3.178

18

C A L C U L U S

Study ResourcesTo prepare for the Calculus exam, you should study the contents of at least one introductory college-level calculus textbook, which you can fi nd in most college bookstores. You would do well to consult several textbooks, because the approaches to certain topics may vary. When selecting a textbook, check the table of contents against the knowledge and skills required for this exam.

Visit www.collegeboard.org/clepprep for additional calculus resources. You can also fi nd suggestionsfor exam preparation in Chapter IV of the Offi cialStudy Guide. In addition, many college faculty post their course materials on their schools’ websites.

Answer Key

Section 1 Section 21. C 33. B2. B 34. E3. D 35. B4. C 36. B5. B 37. B6. D 38. 147. A 39. B8. D 40. E9. D 41. A

10. B 42. B11. D 43. C12. B 44. A13. C 45. B14. D 46. B15. 264 47. C16. C 48. C17. D 49. D18. D 50. A19. B 51. A20. D 52. D21. E 53. C22. B 54. D23. B 55. A24. B25. A26. 427. C28. B29. A30. A31. D32. D

19

C A L C U L U S

Test Measurement Overview

Format

There are multiple forms of the computer-based test,each containing a predetermined set of scoredquestions. The examinations are not adaptive. Theremay be some overlap between different forms of atest: any of the forms may have a few questions,many questions, or no questions in common. Someoverlap may be necessary for statistical reasons.

In the computer-based test, not all questionscontribute to the candidate’s score. Some of thequestions presented to the candidate are beingpretested for use in future editions of the tests andwill not count toward his or her score.

Scoring Information

CLEP examinations are scored without a penalty forincorrect guessing. The candidate’s raw score issimply the number of questions answered correctly.However, this raw score is not reported; the rawscores are translated into a scaled score by a processthat adjusts for differences in the difficulty of thequestions on the various forms of the test.

Scaled Scores

The scaled scores are reported on a scale of 20–80.Because the different forms of the tests are notalways exactly equal in difficulty, raw-to-scaleconversions may in some cases differ from form toform. The easier a form is judged to be, the higherthe raw score required to attain a given scaled score.Table 1 indicates the relationship between numbercorrect (raw score) and scaled score across all forms.

The Recommended Credit-GrantingScore

Table 1 also indicates the recommendedcredit-granting score, which represents theperformance of students earning a grade of C in thecorresponding course. The recommended B-levelscore represents B-level performance in equivalentcourse work. These scores were established as theresult of a Standard Setting Study, the most recenthaving been conducted in 2008. The recommendedcredit-granting scores are based upon the judgmentsof a panel of experts currently teaching equivalentcourses at various colleges and universities. These

experts evaluate each question in order to determinethe raw scores that would correspond to B and Clevels of performance. Their judgments are thenreviewed by a test development committee, which, inconsultation with test content and psychometricspecialists, makes a final determination. Thestandard-setting study is described more fully in theearlier section entitled “CLEP Credit Granting” onpage 4.

Panel members participating in the most recent studywere:

Judy Broadwin Baruch College of CUNYDon Campbell Middle Tennessee State

UniversityBen Cornelius Oregon Institute of TechnologyJohn Emert Ball State UniversityDaria Filippova Bowling Green State UniversityLaura Geary South Dakota School of Mines

and TechnologyJohn Gimbel University of Alaska —

FairbanksStephen Greenfield Rutgers, The State University of

New JerseyMurli Gupta George Washington UniversityErick Hofacker University of Wisconsin —

River FallsJohn Jensen Rio Salado CollegeBen Klein Davidson CollegeStephen Kokoska Bloomsburg UniversityKeith Leatham Brigham Young UniversityGlenn Miller Borough of Manhattan

Community CollegeSteven Olson Northeastern UniversityDavid Platt Front Range Community

CollegeLola Swint North Central Missouri CollegeMary Wagner-Krankel St. Mary’s UniversityRichard West Francis Marion University

To establish the exact correspondences between rawand scaled scores, a scaled score of 50 is assigned tothe raw score that corresponds to the recommendedcredit-granting score for C-level performance. Thena high (but in some cases, possibly less than perfect)raw score will be selected and assigned a scaledscore of 80. These two points — 50 and 80 —determine a linear raw-to-scale conversion forthe test.

20

Table 1: Calculus Interpretive Score DataAmerican Council on Education (ACE) Recommended Number of Semester Hours of Credit: 3

Course Grade Scaled Score Number Correct80 39-4079 3878 3877 3776 3675 35-3674 3573 3472 33-3471 3370 3269 3168 30-3167 3066 2965 28-29

B 64 27-2863 2762 26-2761 25-2660 24-2559 2458 23-2457 22-2356 2255 21-2254 20-2153 19-2052 19-2051 18-19

C 50* 17-1849 16-1748 16-1747 15-1646 14-1545 13-1544 13-1443 12-1342 11-1341 11-1240 10-1139 9-1038 8-1037 8-936 7-835 6-834 5-733 5-632 4-631 3-530 2-429 2-328 1-327 0-226 0-125 0-124 -23 -22 -21 -20 -

*Credit-granting score recommended by ACE.Note: The number-correct scores for each scaled score on different forms may vary depending on form dif culty.

21

C A L C U L U S

Validity

Validity is a characteristic of a particular use of thetest scores of a group of examinees. If the scores areused to make inferences about the examinees’knowledge of a particular subject, the validity of thescores for that purpose is the extent to which thoseinferences can be trusted to be accurate.

One type of evidence for the validity of test scoresis called content-related evidence of validity. It isusually based upon the judgments of a set of expertswho evaluate the extent to which the content of thetest is appropriate for the inferences to be madeabout the examinees’ knowledge. The committeethat developed the CLEP Calculus examinationselected the content of the test to reflect the contentof the general Calculus curriculum and courses atmost colleges, as determined by a curriculum survey.Since colleges differ somewhat in the content of thecourses they offer, faculty members should, and areurged to, review the content outline and the samplequestions to ensure that the test covers core contentappropriate to the courses at their college.

Another type of evidence for test-score validity iscalled criterion-related evidence of validity. Itconsists of statistical evidence that examinees whoscore high on the test also do well on other measuresof the knowledge or skills the test is being used tomeasure. Criterion-related evidence for the validityof CLEP scores can be obtained by studiescomparing students’ CLEP scores with the gradesthey received in corresponding classes, or othermeasures of achievement or ability. At a college’srequest, CLEP and the College Board conduct thesestudies, called Admitted Class Evaluation Service, orACES, for individual colleges that meet certaincriteria. Please contact CLEP for more information.

Reliability

The reliability of the test scores of a group ofexaminees is commonly described by two statistics:the reliability coefficient and the standard error ofmeasurement (SEM). The reliability coefficient isthe correlation between the scores those examineesget (or would get) on two independent replicationsof the measurement process. The reliabilitycoefficient is intended to indicate thestability/consistency of the candidates’ test scores,and is often expressed as a number ranging from.00 to 1.00. A value of .00 indicates total lack ofstability, while a value of 1.00 indicates perfectstability. The reliability coefficient can be interpretedas the correlation between the scores examineeswould earn on two forms of the test that had noquestions in common.

Statisticians use an internal-consistency measure tocalculate the reliability coefficients for the CLEPexam. This involves looking at the statisticalrelationships among responses to individualmultiple-choice questions to estimate the reliabilityof the total test score. The formula used is known asKuder-Richardson 20, or KR-20, which is equivalentto a more general formula called coefficient alpha.The SEM is an index of the extent to which students’obtained scores tend to vary from their true scores.1

It is expressed in score units of the test. Intervalsextending one standard error above and below thetrue score (see below) for a test-taker will include68 percent of that test-taker’s obtained scores.Similarly, intervals extending two standard errorsabove and below the true score will include95 percent of the test-taker’s obtained scores. Thestandard error of measurement is inversely related tothe reliability coefficient. If the reliability of the testwere 1.00 (if it perfectly measured the candidate’sknowledge), the standard error of measurementwould be zero.

Scores on the CLEP examination in Calculusare estimated to have a reliability coefficient of0.91. The standard error of measurement is 4.39scaled-score points.1 True score is a hypothetical concept indicating what an individual’s score on a

test would be if there were no errors introduced by the measuring process. It isthought of as the hypothetical average of an infinite number of obtained scoresfor a test-taker with the effect of practice removed.

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