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

Max grade | 15 | 12 | 10 | 12 | 8 | 8 | 17 | 9 | 8 | 86 |

Min grade | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 14 |

Mean grade | 12.51 | 4.16 | 6.03 | 4.48 | 5.76 | 1.73 | 6.85 | 3.07 | 3.76 | 48.35 |

Median grade | 14 | 2 | 6 | 4 | 7 | 1 | 6 | 2 | 4 | 50 |

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] | [75,79] | [65,74] | [60,64] | [50,59] | [45,49] | [0,44] |

Statistical measures of the performance of the 46 students who took version A and the 47 students who took version B were close. The overall low grades were not anticipated by the instructor.

**Problem 1 (15 points)**

a) 6 points. 1 point for the derivative, 2 points for the critical
numbers, and 3 points for the intervals.

b) 4 points. 1 point for the (second) derivative, 1 point for the
inflection point, and 2 points for the intervals.

c) 5 points. 2 points for sketching a continuous function which
wiggles up and down in a reasonably correct manner; 3 points for the
requested labels. Since students are assumed to have a graphing
calculator and the axes given were very suggestively scaled axes, 1
point will be deducted if the graph egregiously does not fill the
indicated window correctly. Identifying the ends of the graph as
relative extrema is incorrect and will earn a 1 point penalty.

**Problem 2 (12 points)**

4 points for each asymptote. There are two horizontal asymptotes
(**+** infinity and **-**infinity) and one vertical
asymptote. Of the 4 points, 2 are earned for the correct answer, and 2
for some justification. "Justification" will be interpreted very
generously. 2 points are lost once in the problem for giving an
asymptote only as a constant, since the equations of the asymptotes
are requested. 1 point is lost once in the problem for giving a
numerical approximation since the question states that "approximations
are *not* acceptable."

**Problem 3 (10 points)**

2 points for a correct equation relating D and L and W.

1 point for finding D at "the certain time."

3 points for differentiating the equation connecting D and L and W.

3 points for substituting correctly in the differentiated equation and
finding "how fast is the length of the rectangle's diagonal" is
changing.

1 point for the answer, "decreasing."

If the problem has been simplified too much by an incorrect and much
simpler equation relating D and L and W, only 3 points can be awarded
of the 6 available for differentiation and substitution.

**Problem 4 (12 points)**

a) 2 points. Evidence should be given, or else the correct answer
(both points needed!) earns only 1 point.

b) 10 points. 5 points for computing the derivative. 4 points for
further computation (2 for symbolic work and 2 for numerical work) and
1 point for the answer.

**Problem 5 (8 points)**

a) 1 point.

b) 2 points for the derivative and 1 point for evaluation.

c) 2 points for the formula and 2 points for the evaluation.

**Problem 6 (8 points)**

**Comment** This is a composition and is *not*
multiplication. Students treating this as multiplication will not
receive any credit.

1 point for correct evaluation of the function value.

2 points for a correct formula for the first derivative and 1 point
for the value of the first derivative.

3 points for a correct formula for the second derivative and 1 point
for the value of the second derivative.

**Problem 7 (18 points)**

a) 2 points. 1 point for the answer, and 1 point for a supporting
reason.

b) 3 points. 2 points for the answers (1 for each limit), and 1 point
for supporting reasoning.

c) 9 points. 2 points for computing the derivative, and 3 points for
manipulating the derivative so that information can be obtained. 2
points for the answers (1 point for each extremum) and 2 points for
supporting reasoning.

d) 4 points. 2 points for the answer, and 2 points for supporting
reasoning, which should refer both to results of b) and c). It is not
sufficient to use the evidence from c), because a function with a
relative max and relative min may well have a larger domain. One very
simple example is given by the function in problem 1: look at its
graph and compare the graph of the function in this problem!

**Problem 8 (9 points)**

3 points for the setup: constraint and objective function. 3 points
for finding critical numbers. 3 points for checking values at
endpoints and the interior critical number.

**Comment** I thought the phrasing "the product of the square of
one multiplied by the other" was unequivocal, but some students seemed
to misunderstand. I regret this. Perhaps I should have written, "the
product of one of these numbers multiplied by the square of the other
number." Improvement is good.

**Problem 9 (8 points)**

a) 4 points. Compute the derivative.

b) 4 points. Declare that f´(x)>0 appropriately, so that f(x)
is increasing.