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

Max grade | 10 | 12 | 14 | 20 | 8 | 8 | 18 | 10 | 96 |

Min grade | 0 | 2 | 0 | 5 | 0 | 0 | 0 | 0 | 20 |

Mean grade | 6.35 | 9.40 | 10.55 | 14.43 | 2.56 | 1.94 | 11.10 | 5.94 | 62.27 |

Median grade | 8 | 10 | 11.5 | 15 | 2 | 1 | 11.5 | 6 | 61 |

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

**Problem 1 (10 points)**

2 points for correctly substituting *this* f(x) in the
definition, including f(x+h). 2 points for correctly combining the
fractions. 2 points for factoring out and canceling the h's. 2 points
for taking the limit, and 2 points for recognizing the derivative.

0 points for a correct answer which is not supported by algebra. 0
points for quoting the definition of derivative, since the definition
is given on the formula sheet. 2 points off for reporting that the
derivative is a formula with two variables. This seems to have
occurred because a limit was not taken.

The answer can be checked using the quotient rule.

**Problem 2 (12 points)**

a) 8 points. 2 points for getting a point (two coordinates) the line
must go through. 2 points for computing the derivative. 2 points for
getting a correct value of the derivative. 2 points for assembling and
reporting a correct equation for the tangent line.

b) 4 points. 2 points for a rudimentarily acceptable curve (up and
down, and through the origin). 2 points for a *tangent* line at
the correct point.

**Calculators and consistency**

Please note that the general information about this course states:

*A graphing calculator is required for this course.*

Therefore you should have such a device with you, and the curve sketch
requested in this problem can be graded strictly. But this also means
that you should have been able to compare the graph of the curve and
the answer obtained in a), and the consistency of the two (picture and
the tangent line equation) can be checked. For example, even a rough
check should suggest a positive slope for the tangent line.

**Problem 3 (14 points)**

a) 8 points: 1 point each for the correct values of A and B, and 6
points for discussing how/why: some reasoning must be given. In
particular, there must be explicit reference to either left/right
limits at 0 and 1 and/or explicit consideration of the "other"
functions, 3-x^{2} and 2^{x}. Since we have
studied continuity, the correct words and techniques (involving
limits) are available.

b) 6 points: the graph should be continuous (!) otherwise -2 points. 2
points for the correctly drawn parabolic curve segment, 2 points for
the correctly drawn line segment, and 2 points for the correctly drawn
exponential curve segment. Each curve segment should connect the
correct two end points (1 point) and should bend in the correct way (1
point): the parabolic curve should bend down, the straight line should
not bend, and the exponential curve should bend up. Again,
availability of a graphing calculator means that this aspect of the
problem can be graded rather strictly.

2 points will be deduced for drawing something which is not the graph
of a function (that is, several curves over the same x).

**Problem 4 (20 points)**

This problem was graded quite *generously*, perhaps to make up
for the strict grading of the exam's graphical problems. Acceptable
"evidence" was sometimes rather minimal.

Each part is worth 5 points: the answer alone is worth 1 point, and
other work (how/why/explanation) is worth 4 points. Graphical
information was acceptable verification on part b); alternatively,
some comment on the behavior of the function must be given. In part
d), you should know the value of cos([even integer]Pi) (1 point).

**Problem 5 (8 points)**

Parts a) and b) together are worth 5 points. Observing that
2^{5} is positive and (-2)^{5} is negative is worth 2
of the 5 points. To earn the remainder, some idea should be given
about the size (+/-) of the trig part of the function compared to the
monomial.

c) 3 points: a citation of continuity and/or the Implicit Function
Theorem should be given (1 point) along with a specific interval as
answer.

Please note that verifying the conclusions of the problem using a
specific value of K does *not* solve the problem written on the
exam.

**The instructor's error** Version B changes the cosine to sine,
which makes part c) rather simple: for *any* value of K, f(0)
will be 0. Three students who took version B earned points with this
observation.

**Problem 6 (8 points)**

The restrictions in domain obtained by considering the top are worth 5
points. The restriction from the bottom are worth the other 3 points
when correct conclusions are obtained.

Since the instructions specified "Explain your answer algebraically" a
graph cannot support the explanation, but a graphing calculator
probably can be used to get hints.

**Problem 7 (18 points)**

2 points for each correct answer. Parts e) and g) are therefore worth
4 points each.

If an answer is an interval, I'll give 1 point for each correct
endpoint. I'll be a bit sloppy and not worry about whether the
endpoints themselves are in the intervals. When considering points of
continuity and differentiability, I will deduct 1 point if any
endpoints of the domain are included. The endpoints are not
part of the domain and therefore are not eligible for continuity or
differentiability of f(x).

Consistent with this, if the answer to a part of the problem is one or
two values of x, I'll deduct a point for each extra (incorrect!)
number you supply. The deduction will be limited by a score of 0 for
the part!

**Problem 8 (10 points)**

a) and b) are worth 3 points each. Small errors (+/- signs, for
example) will be penalized 1 point, while errors in the product or
quotient rule will receive 2 point deductions. "Bald" answers with no
supporting work will receive full credit here.

c) 4 points. 1 point for the answer, and 3 points for the process,
roughly 1 point for each possible invocation of the product rule (so
the first derivative can earn 1 point, and the second derivative can
earn 2 points). I must see explicit evidence (with g(x), g´(x)
and g´´(x)) that the product rule has been correctly
applied.

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.

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

Range | [175,200] | [165,174] | [150,164] | [135,149] | [110,134] | [100,109] | [0,99] |

I looked at the grading of each exam and then checked the addition on each exam. I corrected grading and addition errors (there were a few, but really not very many, and none of significance).

University rules state that I must retain possession of the exams but that you may certainly have access to them. If you wish to look at your final exam, please get in touch with me to find a mutually agreeable time. E-mail (to greenfie@math.rutgers.edu) is probably best.

**
Maintained by
greenfie@math.rutgers.edu and last modified 5/13/2005.
**