Problem #1 |
Problem #2 |
Problem #3 |
Problem #4 |
Problem #5 |
Problem #6 |
Problem #7 |
Total | |
---|---|---|---|---|---|---|---|---|

Max grade | 12 | 10 | 20 | 10 | 20 | 18 | 10 | 92 |

Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 11 |

Mean grade | 7.21 | 7.07 | 8.49 | 3.13 | 14.85 | 11.10 | 1.43 | 53.16 |

Median grade | 9 | 8 | 9 | 3 | 17 | 12 | 1 | 57 |

Numerical grades will be retained for use in computing the final letter grade in the course. Here are approximate letter grade assignments for this exam:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [80,100] | [70,79] | [60,69] | [55,59] | [50,54] | [45,49] | [0,44] |

**Problem #1**

The definition was worth 2 points. Part b) was worth 10
points. Writing *only* a correct formula for f'(x) there earned
nothing.

**Problem #2**

a) and c) were straightforward computations. In
part b), the graph of the cubic earned points for showing
the correct asymptotic information, the correct roots, and the correct
approximate locations of "bumps". The line should have been drawn
*tangent* to the curve.

**Problem #3**

Each part was worth 5 points. The correct answer earned a point. The
other 4 points for each part were given for some correct
justification. Numerical
substitution alone was not considered sufficient justification.

**Problem #4**

This was the most difficult problem to grade (indeed, grading it took
more time than grading **4** other problems!). A strong effort
was made to grade fairly, with partial credit given for specific facts
such as the computation of f(0) in part a). Statements about
x^{2} getting "very big" in part b) did not earn credit. The
problem asks for very specific information about f(x) for x>=2, and
so precise information (including estimates!) are needed. Please note
that the input to sine will always be in radians in this course unless
otherwise specified. Therefore the sign of sin(80)
or sin(140) are not obvious without more discussion. (One is
positive and one is negative, actually!) Successful arguments in both
parts used estimates of sine's values: all that is needed is the range
of sine, the interval [-1,1]. The use of the Intermediate
Value Theorem in part a) needed citation of the necessary hypotheses,
including continuity of the function and the difference in signs of
the function's values at the endpoints. There are simple examples of
discontinuous functions with no roots whose values are both positive
and negative. The Intermediate Value Theorem is not relevant to part b).

**Problem #5**

Again, each part was worth 5 points. The lecturer missed a minus sign
in one of these answers, so he would have gotten 99 on this exam.

**Problem #6**

Specific points were assigned to each feature of
the graph of f'(x). Certainly there is room for honest disagreement
since only qualitative information is given. For example, students
received full credit if their graph of f'(x) appproached a finite
limit (with the correct sign!) as x-->B^{+}. The lecturer's
answer indicates an asymptote on the graph of y=f'(x), but that
does not seem to be forced by the graph of f(x). The vertical and
horizontal asymptotes should have been written as
equations: 1 point was deducted if they were not. Erroneously giving
more or fewer points of discontinuity and non-differentiability was
penalized.

**Problem #7**

Few students were successful in this problem, either because it was
difficult or because it was the last problem on the exam. A few points
of partial credit were given for a useful sketch or for writing the
derivative of 1/x.

**Overall results**

Overall class performance was definitely weaker than the instructional
staff anticipated and wanted. The lecturer wanted students to do
"well" on problems 4 and 7. Most of the other problems are algorithmic
and could be done by a computer program. The
lecturer believes that problems 4 and 7 distinguish those students who
can use the ideas of the course.

Problem #1 |
Problem #2 |
Problem #3 |
Problem #4 |
Problem #5 |
Problem #6 |
Problem #7 |
Problem #8 |
Problem #9 |
Subtotal | Bonus questions | Total | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Max grade | 8 | 8 | 10 | 10 | 14 | 16 | 10 | 15 | 7 | 84 | 29 | 109 |

Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 2 | 8 |

Mean grade | 3.43 | 2.90 | 4.11 | 7.38 | 3.53 | 7.49 | 6.35 | 7.16 | 1.08 | 43.44 | 20.01 | 61.95 |

Median grade | 4 | 2 | 4 | 9 | 2 | 7 | 8 | 7 | 1 | 44 | 22 | 66 |

Numerical grades (the **Total**) will be retained for use in
computing the final letter grade in the course. Here are approximate
letter grade assignments for this exam:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [85,100] | [80,84] | [70,79] | [65,69] | [55,64] | [50,54] | [0,49] |

**Problem #1**

a) was worth 4 points. No formula for linear
approximation/linearization/differential received 0 points. f'(x) computed
correctly was worth 1 point, and the correct value of f'(2) received another point.
2 points were given for applying the formula correctly.

b) Asserting that the function was concave down earned 1 point. 1
point was given for computing f''(x) and 1 point for computing f"(2)
or f"(1.97). Giving a reason why the graph was concave down earned
the fourth point.

**Problem #2**

a) 2 points for the drawing and 2 points for the limit of the sequence.
In version B, we felt that the limit resulting from a_{0}
could be argued about: the exam writer wanted the limit to be
+infinity, but some students drew convincing tangent lines and got the
limit to be R_{2}. If there was a coherent picture and
discussion, these students received full credit.

b) 2 points for the drawing and 2 points for the
limit.

In each part, 1 point was given for having only the first point correctly drawn.

**Problem #3**

4 points if y' is correct. 1 point if the result is set equal to 1. 5
more points for finding the specific points on the ellipse which are
requested. Partial credit was given for partial progress. 8 points
were given if only one correct point was presented as the answer.

**Problem #4 **

4 points for a correct first antiderivative (that is, f'(x), with +C)
and 1 point for evaluating that constant using the initial
condition. Another 4 points were given for a correct second
antiderivative (f(x), with +C) and 1 last point for evaluating the
second constant. Note that the first constant appears as part of the
second antiderivative.

**Problem #5**

Correct statement of the constraint (an equation connecting V and r
and h) earned 2 points. Correct statement of the objective
function (the sum of the area of the bottom plus the area of the side)
earned 3 points. Use of the constraint to obtain a function of
one variable which must be minimized earned 3
points. Differentiation of this equation earned <3
points. Solving for the unique critical point earned 1
point and solving for the value of the other variable, 1
point. I gave 2 points for explaining why a minimum was
found.

I do *not* apologize for any "abstraction" in this problem (that
is, using "V" rather than 38, say). The problem is quoted directly
from the text. Students in this course should be able to handle a
some abstraction.

**Problem #6**

I found this problem difficult to grade, and tried to give partial
credit carefully. I believe that the key to this problem is doing as
little (!) work as possible by carefully using the information
given. Since f'(x) is given in factored form, parts a) and b) should
be easy. Please do note that x^{2}-3 has two roots, not just
one!

a) 4 points, one for each interval.

b) 3 points, one for each extreme value.

c) 1 point for correct computation of f''(x) and then 3 points, one
for each interval of concavity.

d) 5 points. I tried to see if the graph either was correct or was
consistent with the student's previous information. I looked for
correctly labeled extreme points and inflection points.

**Problem #7**

10 points: 2 points for a general "area of a triangle
formula", and 1 point for using the formula to compute the base of the
triangle at the time given; 3 points for differentiating
correctly; 2 points< for *using* the given rates of change
correctly; finally, **2 points** for obtaining the correct
answer. Students can also do this problem by "solving" for the base as
a function of area and height, and then using the quotient rule, and
full credit was given for solutions of that type.

Students who wrote a blatantly incorrect version of, say, the product
rule, were not given high grades on this problem.

**Problem #8**

Each part was worth 5 points. 1 point was reserved for the correct
numerical answer, and the other points were earned by supporting
evidence. As mentioned in class, calculator evidence alone is not
valid. L'Hopital's rule is *not* relevant to part c) at all, and
attempts to use it (usually characterized by writing some variant of
the derivative) were given no credit.

**Problem #9**

8 points. From the first exam. I looked for some evidence
that more was known. Similar computations have been needed in a number
of applications since the first exam, so I hoped that additional
"practice" would have resulted in more success here. This does not
seem to be the case.

**Overall results on the first 9 problems**

This exam was probably too long. I don't believe any single problem
was inappropriate or too difficult, but a better exam would have had
only 8 of these problems rather than 9.

**Grading the Bonus Problems**

Problem 1 was graded mostly by glancing at the answer. If it was
wrong, further investigation was needed.

In problem 2, various routes to the correct answer are possible, and
a number of algebraic versions of the correct answer are
possible. Therefore the process and answer were both checked. Some
students gave only the first derviative which is not what was
requested.

Few people were totally successful doing problem 3. 1 point was given
for a correct algebraic statement of the constraint, and another point for
correctly using the constraint to reduce the objective function from
two variables to one. If the one variable objective function was
correctly differentiated, a third point was earned.

Many people were successful doing the fourth problem. A few students,
however, seemed to invent and try to answer their own problem(s),
which usually seemed more difficult.

Problem 5 needed some explanation to score full credit. 1 point was
given for the correct answer, and another point for differentiating a
polynomial (or two). Finding the roots of these polynomials earned
another point. But full credit was only awarded for making correct
logical deductions from the evidence to support the correct
answer.

Most students got problem 6 correct. Partial credit was given for
small mistakes, but more points were taken off for serious misuse of
the chain and/or product rules.

73 students gave answers to these bonus problems. The correlation
coefficient for the bonus problem scores and the subtotals was .689,
quite high (not surprisingly).

Course grades should be submitted to the Rutgers computer system some time today, Thursday, December 18. Within a day or two these grades should be accessible online by students. Course grades were assigned in consultation with Ms. Calinescu. The entire student record (two exams during the semester, the final exam, textbook and workshop homework, and QotD scores) was assessed. Although both the lecturer and the workshop instructor were involved in this consideration, the course grades are the responsibility of the lecturer and questions about these grades should be addressed to him.

**The final exam itself**

Current course policy is to keep the questions on the final exam
confidential. Also, university regulations say that final exams must
remain in the custody of the department. Students may examine their
exams by making an appointment with the lecturer.

The grading of the final exam was straightforward. My personal opinion
is that the exam was slightly easier than other recent Math 151
finals. I will write no more about the grading, except to declare that
Ms. Calinescu and I have taken enough *virtual airplane flights* and
searched diligently for ways to make *functions continuous*.

The median grade for the approximately 600 students who took the final exam was 135. The median grade for students taking the exam in sections 4, 5, and 6 was 148, and the mean for these students was 126.

Numerical grades (the **Total**) were retained for use in
computing the final letter grade in the course. Here are
letter grade assignments for this exam:

Letter equivalent | A | B+ | B | C+ | C | D | F |
---|---|---|---|---|---|---|---|

Range | [170,200] | [160,169] | [150,159] | [135,149] | [120,134] | [105,119] | [0,104] |

**
Maintained by
greenfie@math.rutgers.edu and last modified 1/9/2003.
**