Chapter III: Answers tothe 2008 AP Calculus AB andCalculus BC Exams

Section I: Multiple Choice• Section I Answer Key and Percent Answering

Correctly• Analyzing Your Students' Performance on the

Multiple-Choice Section• Diagnostic Guide for the 2008 AP Calculus AB and

Calculus BC Exams

Section II: Free Response• Comments from the Chief Reader» Commentary, Scoring Guidelines, and Sample

Student Responses• AB/BC Question 1• AB/BC Question 2

• AB Question 3• AB/BC Question 4• AB Question 5• AB Question 6• BC Question 3• BC Question 5• BC Question 6

Section I: Multiple ChoiceOn the following pages are the correct answers to themultiple-choice questions for Form Q, the percent ofAP students who answered each question correctly byAP grade, and the total percent answering correctly.

Section I Answer Key and Percent Answering CorrectlyCalculus AB

ItemNo.

1

2345678910111213141516171819*202122232425

CorrectAnswer

BDDBAABEDCBDAECDCA-DABEBB

Percent Correct by Grade5 4 3 2 1

889672957474978695909889725093748647-734880508055

778448865955897482799279513082466223-502664265426

687135744946766465698269362572304315-351758173316

585726604039555243586957242361203112-251155121814

363214372430222916354433141841112611-197507913

TotalPercentCorrect

656739704949676160667765402970375022-412262234026

ItemNo.

2627287677787980818283848586878889909192

CorrectAnswer

ACABCBBEEBBCBCDCEACB

Percent Correct by Grade5 4 3 2 1

5370389889978170909690917686849667679258

2952139373916652718876805371558440378145

174168356805438557763643858356523206843

12313

6541614426426148412843214516115241

92033024282814212727141818112110102736

TotalPercentCorrect

2543147357715540566961584355426232306445

'Although 45 multiple-choice items were administered in Section I,item #19 was not used in scoring for statistical reasons. Manystudents who scored well on the exam chose option A or option C,instead of the correct answer E. It is important for teachers toemphasize that when determining the horizontal asymptotes for thegraph of y = f (x), students must examine the limit of / (x) as xapproaches infinity and the limit of / (x) as x approaches negativeinfinity.

91

Section I Answer Key and Percent Answering CorrectlyCalculus BC

ItemNo.

1

234567891011121314151617181920212223

CorrectAnswer

BACDCADBDBADBEADBAACAEE

Percent Correct by Grade5 4 3 2 1

9796889594819089969075647563856279596149583180

9592778685617371858048355539633860334123351157

9387677476525856747234254032452745233016237

42

8980556166434339585927212927312137162312166

28

81673640403523263339211623221716281318910515

TotalPercentCorrect

9389738080636968797652425545614259384330381856

Analyzing Your Students' Performance onthe Multiple-Choice SectionIf you give your students the 2008 Calculus AB orCalculus BC Exam for practice, you may want to analyzethe results to find overall strengths and weaknesses in theirunderstanding of calculus. The following diagnostic work-sheets will help you do this. You are permitted to photocopyand distribute them to your students for completion.

1. In each category, students should insert a check mark foreach correct answer.

2. Add together the total number of correct answers foreach category.

ItemNo.

24252627287677787980818283848586878889909192

CorrectAnswer

ABDADDDCDEBEBACCDAEDDB

Percent Correct by Grade5 4 3 2 1

63575683269399947070849393958973918579918590

372735665

8498814556687882867442686055796673

251925523

7294663644566571736228484338624754

1913194125687492733415159544919312924442933

131213282

3669262121243739283513181614221518

TotalPercentCorrect

41353864137793744954657577787147655954716167

3. To compare the student's number of correct answers fora given category with the average number correct forthat category, copy the number of correct answers to the"Number Correct" table at the end of the DiagnosticGuide.

In addition, under each item, the percent of AP studentswho answered correctly is shown, so students can analyzetheir performance on individual items. This informationwill be helpful in deciding how students should plan theirstudy time. Please note that one item may appear in severaldifferent categories, as questions can cover different topics.

92

Diagnostic Guide for the 2008 AP Calculus AB Exam

Differential Calculus (Average number correct = 13.0)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

1

65

24

40

3

39

25

26

5

49

26

25

6

49

27

43

8

61

28

14

11

77

76

73

12

65

77

57

13

40

78

71

14

29

80

40

16

37

82

69

18

22

84

58

20

4J

88

62

21

22

89

32

22

62

90

30

Integral Calculus (Average number correct = 8.9)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

2

67

4

70

7

67

9

60

10

66

15

70

17

50

23

23

79

55

81

56

83

61

85

43

86

55

87

42

91

64

92

45

Part A—No Calculator (Average number correct = 12.8)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

1

65

15

70

2

67

16

37

3

39

17

50

4

70

18

22

5

49

20

41

6

49

21

22

7

67

22

62

8

61

23

23

9

60

24

40

10

66

25

26

11

77

26

25

12

65

27

43

13

40

28

14

14

29

Part B—With Graphing Calculator (Average number correct = 9.1)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

76

73

77

57

78

71

79

55

80

40

81

56

82

69

83

61

84

58

85

43

86

55

87

42

88

62

89

32

90

30

91

64

92

45

Calculator-Active Items (Average number correct = 3.5)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

78

71

80

40

82

69

83

61

87

42

91

64

93

Diagnostic Guide for the 2008 AP Calculus AB Exam (continued)

Number Correct

Number ofQuestions

AverageNumberCorrect

My NumberCorrect

DifferentialCalculus

28

13.0(46.4%)

IntegralCalculus

16

8.9(55.6%)

Part A — NoCalculator

27

12.8(47.4%)

Part B— withGraphingCalculator

17

9.1(53.5%)

Calculator-Active Items

6

3.5(58.3%)

Diagnostic Guide for the 2008 AP Calculus BC Exam

Calculus AB Topics (Average number correct = 17.0)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

2

89

77

93

6

63

78

74

8

68

80

54

9

79

81

65

10

76

83

77

13

55

85

71

14

45

86

47

15

61

87

65

17

59

88

59

18

38

89

54

21

38

90

71

25

35

91

61

27

64

92

67

76

77

Calculus BC Only Topics (Average number correct = 9.7)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

1

93

3

73

4

80

5

80

1

69

11

52

12

42

16

42

19

43

20

30

22

7S

23

56

24

41

26

38

28

13

79

49

82

75

84

78

Differential Calculus (Average number correct = 12.6)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

1

93

3

73

6

63

7

69

10

76

14

45

15

61

17

59

18

38

21

38

25

35

27

64

28

J3

76

77

78

74

80

54

85

71

89

54

90

77

91

61

92

67

94

Diagnostic Guide for the 2008 AP Calculus BC Exam (continued)

Integral Calculus (Average number correct = 9.7)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

2

89

5

80

8

68

9

79

11

52

13

55

19

43

22

18

24

41

26

38

77

93

81

65

83

77

86

47

87

65

88

59

Part A—No Calculator (Average number correct = 15.4)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

1

93

15

61

2

89

16

42

3

73

17

59

4

80

18

38

5

80

19

43

6

63

20

30

7

69

21

38

8

68

22

18

9

79

23

56

10

76

24

41

11

52

25

35

12

42

26

38

13

55

27

64

14

45

28

13

Part B—With Graphing Calculator (Average number correct = 11.4)

Question #

Correct/Incorrect

Percent of StudentsAnswering Correctly

76

77

77

93

78

74

79

49

80

54

81

65

82

75

83

77

84

78

85

71

86

47

87

65

88

59

89

54

90

71

91

61

92

67

Series Items (Average number correct = 4.5)

Question f

Correct/Incorrect

Percent of StudentsAnswering Correctly

4

80

12

42

16

42

20

30

23

56

79

49

82

75

84

78

Calculator-Active Items (Average number correct = 4.2)

Question f

Correct/Incorrect

Percent of StudentsAnswering Correctly

77

93

80

54

83

77

85

71

87

65

91

61

95

Diagnostic Guide for the 2008 AP Calculus BC Exam (continued)

Number Correct

Number ofQuestions

AverageNumberCorrect

My NumberCorrect

CalculusAB

Topics

27

17.0(63.0%)

CalculusBC OnlyTopics

18

9.7(53.9%)

DifferentialCalculus

21

12.6(60.0%)

IntegralCalculus

16

9.7(60.6%)

Part A—No Calculator

28

15.4(55.0%)

Part B—With

GraphingCalculator

17

11.4(67.1%)

SeriesItems

8

4.5(56.3%)

Calculator-Active Items

6

4.2(70.0%)

96