6. Let $f$ be a differentiable function, defined for all real numbers $\mathbf { x }$, with the following properties.
(i) $f ^ { \prime } ( x ) = a x ^ { 2 } + b x$
(ii) $f ^ { \prime } ( 1 ) - 6$ and $f ^ { \prime \prime } ( 1 ) = 18$
(iii) $\int _ { 1 } ^ { 2 } f ( x ) d x = 18$
Find $f ( x )$. Show your work.
Solution Distribution of Points
| \(\begin{aligned} | a + b = 6 |
| f ^ { \prime \prime } ( x ) = 2 a x + b |
| 2 a + b = 18 |
| \therefore a = 12 , b = - 6 |
| \text { and } f ^ { \prime } ( x ) = 12 x ^ { 2 } - 6 x \end{aligned}\) | 4: |
| \(\begin{aligned} f ( x ) | - \int \left( 12 x ^ { 2 } - 6 x \right) d x |
| = 4 x ^ { 3 } - 3 x ^ { 2 } + C \end{aligned}\) | 5: 1: for antiderivative of $f$ |
Grades for the Advanced Placement Examinations are reported on a five-point scale, ranging from 1 to 5. Before these grades are determined, a number of intermediate scoring steps take place. First, the answer sheet for the multiple-choice section is machine-scored. This score is the number of correct answers minus a fraction of the number of incorrect answers; negative scores are set equal to zero. Second, scores are assigned to individual problem sets in the free-response section by Readers at the AP Reading. These scores are based on detailed scoring standards established by the Chief Reader and the Readers. Third, the scores on the free-response questions and the multiple-choice section are weighted according to formulas determined in advance by the AP Mathematics Development Committee to yield a single composite score for each candidate. In Calculus AB, the six problems that constitute the free-response section are weighted equally to form the total free-response score. The total free-response score and the total multiple-choice score are then weighted equally in forming the composite score, for which the maximum value was 108 in 1988. Finally, the conversion from the composite scores to the reported grades is determined by setting four cut-points on the composite score scale which are used to determine the ranges of composite scores that make up the five possible grades. The setting of these cut-points for each examination is based on the judgment of the Chief Reader in consultation with ETS professional staff.
A variety of information is available to assist the Chief Reader in judging which papers have scored high enough to receive each of the grades. Computer printouts with complete distributions of scores on each portion of the multiplechoice and free-response sections of the examination are provided along with totals for each section and the composite score total. With these figures and special statistical tables presenting score distributions from previous years, the Chief Reader can calibrate the examination against the results of other years and can evaluate the section-by-section performance on the current examination. Assessments are also made of the examination itself as well as the reliability of the grading. Finally, computer rosters containing the complete breakdown of scores for thousands of candidates enable the Chief Reader to analyze patterns of performance. On the basis of professional judgment regarding the quality of performance represented by the achieved scores, the Chief Reader determines the candidates' final AP grades.
The grade distributions for the 1988 AP Mathematics Examination: Calculus AB are shown below, with the percentage indicated at each grade level, together with the mean and standard deviation.
| Examination Grade | Number of Students | Percent at Grade |
| Extremely well qualified | 5 | 9,666 | 17.8 |
| Well qualified | 4 | 12,295 | 22.7 |
| Qualified | 3 | 15,003 | 27.7 |
| Possibly qualified | 2 | 8,547 | 15.8 |
| No recommendation | 1 | 8,724 | 16.1 |
| Total Number of Students* | \multicolumn{3}{|c|}{54,235} |
| Mean Grade | \multicolumn{3}{|c|}{3.10} |
| Standard Deviation | \multicolumn{3}{|c|}{1.31} |
*This total differs from the total number of candidates given on pages 60 and 61 because the statistical summaries were produced at different times and hence different candidate data were available.
The classification reliability of AP grades can be examined by using a recently developed statistical technique that makes it possible to estimate the consistency and accuracy of decisions based on those grades. The consistency of the decisions is the extent to which they would agree with the decisions that would have been made if the candidate had taken a different form of the $A P$ Calculus $A B$ Exam, equal in difficulty and covering the same content as the form the candidate actually took. The accuracy of the decisions is the extent to which they would agree with the decisions that would be made if each candidate could somehow be tested with all possible forms of the exam.
The table below shows the decision consistency and accuracy of the 1988 AP Calculus AB Examination. Each number in the table indicates the estimated percentage of candidates who would be consistently classified as above or below the 2 to 3 and the 3 to 4 grade boundaries.
\begin{table}[h]
Estimated Consistency and Accuracy of Decisions Based on AP Grades for the 1988 AP Calculus AB Examination | \multicolumn{4}{|c|}{Estimated Percentage of Candidates Who Would Be Reclassified the Same Way on the Basis of:} |
| \multicolumn{2}{|c|}{Another Form} | \multicolumn{2}{|c|}{Average of All Forms} |
| 2-3 boundary | 3-4 boundary | 2-3 boundary | 3-4 boundary |
| 93\% | 92\% | 95\% | 94\% |
\end{table}
The percentages in the table are estimates-candidates never actually took more than one form of the exam--and are based on data from a representative sample of the total group of candidates who took the 1988 AP Calculus AB Exam. Research results indicate that these estimates are biased in an upward direction and, therefore, overestimate the actual consistency and accuracy of decisions based on AP grades.
SCORING WORKSHEET 1988 AP CALCULUS AB EXAMINATION
SECTION I: MULTIPLE-CHOICE (TOTAL):
- $( 1 / 4 \times \underbrace { } _ { \begin{array} { l } \text { Number } \\ \text { correct } \end{array} } = \frac { } { \begin{array} { l } \text { Multiple-choice score } \\ \text { (Round to nearest whole number. } \\ \text { If less than zero, enter zero.) } \end{array} }$
SECTION II: FREE-RESPONSE:
Scores for individual questions Question 1 $\_\_\_\_$ (Out of 9)
2 $\_\_\_\_$ (Out of 9)
3 $\_\_\_\_$ (Out of 9)
4 $\_\_\_\_$ (Out of 9)
5 $\_\_\_\_$ (Out of 9)
6 $\_\_\_\_$ (Out of 9) Sum $=$ $\_\_\_\_$ COMPOSITE SCORE:
1.200 x $\_\_\_\_$ $=$ $\_\_\_\_$ score choice score
$$\begin{gathered}
\overline { \text { Weighted multiple- } } + \frac { } { \text { Free-response } } = \frac { } { \text { score } } \\
\text { choice score }
\end{gathered}$$
AP GRADE:
| Composite Score Range* | AP Grade |
| $83 - 108$ | 5 |
| $68 - 82$ | 4 |
| $48 - 67$ | 3 |
| $32 - 47$ | 2 |
| $0 - 31$ | 1 |
\footnotetext{*This composite score range is for the 1988 examination only. }
DISTRIBUTION OF SCORES SECTION II 1988 AP CALCULUS AB EXAMINATION
| \multirow[b]{2}{*}{Score} | \multicolumn{6}{|c|}{Free-Response Questions} |
| 1 (9*) | 2 (9*) | 3 (9*) | 4 (9*) | 5 (9*) | 6 (9*) |
| \multirow{10}{*}{\begin{tabular}{l} 9 |
| 8 |
| 7 |
| 6 |
| 5 |
| 4 |
| 3 |
| 2 |
| 1 |
| 0 |
} & 3,364 & 8,926 & 6,783 & 7,048 & 11,050 & 21,138 \hline & 9,108 & 9,725 & 3,421 & 3,878 & 5,721 & 5,596 \hline & 11,382 & 7,406 & 4,643 & 6,502 & 4,101 & 2,333 \hline & 10,626 & 5,900 & 5,894 & 5,320 & 2,704 & 4,156 \hline & 7,938 & 4,447 & 7,904 & 5,713 & 7,411 & 6,997 \hline & 5,421 & 3,946 & 5,146 & 5,437 & 3,611 & 3,407 \hline & 3,323 & 3,351 & 4,189 & 4,756 & 5,539 & 2,456 \hline & 1,384 & 3,508 & 3,707 & 4,679 & 4,565 & 2,387 \hline & 1,016 & 2,514 & 3,567 & 3,677 & 4,219 & 2,235 \hline & 928 & 4,767 & 9,236 & 7,480 & 5,569 & 3,785 \hline Total Number of Candidates** & 54,490 & 54,490 & 54,490 & 54,490 & 54,490 & 54,490 \hline Mean & 5.90 & 5.56 & 4.40 & 4.58 & 4.98 & 6.23 \hline Standard Deviation & 2.00 & 2.90 & 2.99 & 2.97 & 3.09 & 2.98 \hline Mean as \% of Max. Possible Score & 65.56 & 61.78 & 48.89 & 50.89 & 55.33 & 69.22 \hline \end{tabular}
\footnotetext{*Maximum possible score **This total differs from the total number of candidates given on pages 57 and 61 because the statistical summaries were produced at different times and hence different candidate data were available. }
| \multirow{2}{*}{MultipleChoice Score} | \multicolumn{5}{|c|}{AP Grade} | \multicolumn{2}{|c|}{\multirow[b]{2}{*}{Total}} |
| 1 | 2 | 3 | 4 | 5 | | |
| \multirow[t]{2}{*}{32 to 45} | 0 | 0 | 56 | 1,975 | 8,411 | 10,442 | |
| 0.0\% | 0.0\% | 0.5\% | 18.9\% | 80.5\% | 100.0\% | (19.1\%) |
| \multirow[t]{2}{*}{26 to 31} | 0 | 28 | 2,380 | 8,030 | 1,294 | 11,732 | |
| 0.0\% | 0.2\% | 20.3\% | 68.4\% | 11.0\% | 100.0\% | (21.5\%) |
| \multirow[t]{2}{*}{18 to 25} | 52 | 2,278 | 10,926 | 2,344 | 4 | 15,604 | |
| 0.3\% | 14.6\% | 70.0\% | 15.0\% | 0.0\% | 100.0\% | (28.6\%) |
| \multirow[t]{2}{*}{12 to 17} | 1,514 | 5,162 | 1,685 | 3 | 0 | 8,364 | |
| 18.1\% | 61.7\% | 20.1\% | 0.0\% | 0.0\% | 100.0\% | (15.3\%) |
| \multirow[t]{2}{*}{0 to 11} | 7,252 | 1,132 | 31 | 0 | 0 | 8,415 | |
| 86.2\% | 13.5\% | 0.4\% | 0.0\% | 0.0\% | 100.0\% | (15.4\%) |
| \multirow[t]{2}{*}{Total} | 8,818 | 8,600 | 15,078 | 12,352 | 9,709 | 54,557* | |
| 16.2\% | 15.8\% | 27.6\% | 22.6\% | 17.8\% | 100.0\% | (100.0\%) |
This table shows the statistical relationship between candidates' $A P$ grades on the 1988 AP Calculus AB Examination and their scores on the multiple-choice portion of the examination. The multiple-choice scores have been divided into five categories, corresponding to the five AP grade levels. Each multiple-choice score category contains approximately the same percentage of the scores as the corresponding AP grade level. The table shows the number and the percentage of the students in each multiple-choice score category who received each $A P$ grade. For example, there were 10,442 students with multiple-choice scores of 32 to 45 , and 8,411 of these students, or 80.5 percent, received an AP grade of 5 . The percentages shown in parentheses at the far right of the table indicate the percentage of all the candidates who had multiple-choice scores in that category. For example, 19.1 percent of all the candidates had multiple-choice scores of 32 to 45.
Of the candidates with multiple-choice scores of 32 or higher (corrected for guessing), all but 0.5 percent earned AP grades of 4 or 5, and the majority earned a 5. Of those with multiple-choice scores of 26 to 31 , the majority earned a 4 , while most others earned a 3 or a 5 . Of those with multiple-choice scores of 18 to 25 , the majority earned a 3 , while most others earned a 2 or a 4 . Of those with multiple-choice scores of 12 to 17 , most earned a 2 , while most others earned a 1 or a 3. Of the candidates with multiple-choice scores of 11 or less, the majority earned a 1 , while most others earned a 2 .
\footnotetext{*This total includes only those candidates who requested that their AP grades be reported. It also differs from the total number of candidates given on pages 57 and 60 because the statistical summaries were produced at different times and hence different candidate data were available. }