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

Max grade | 12 | 14 | 12 | 10 | 14 | 12 | 14 | 12 | 93 |

Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 13 |

Mean grade | 8.35 | 7.92 | 4.86 | 5.66 | 5.58 | 7.99 | 7.57 | 3.86 | 51.79 |

Median grade | 9 | 7 | 4 | 7 | 4 | 10 | 8 | 1 | 53 |

Numerical grades will be retained for use in computing
the final letter grade in the course. Students with grades of D or
F on this exam should be very concerned about their likely success in
this course. Indeed, students with low grades should evaluate the
amount of time and effort they can devote to this course as
part of their entire plan of work for the balance of the
semester. Such students may wish to drop this course or drop
another course in order to increase their chances of success in what
remains. I strongly recommend considering such actions.

**Class attendance and watching the instructors do
problems is not adequate practice. Individual time and effort must be
given.**

We will continue to use improper integrals, estimation techniques,
integration by parts, and familiarity with functions and with
algebraic manipulation a great deal in this course, and much of what
we will do is important to your later technical education and
practice. Please don't waste your time and effort with false
expectations. I want students to be successful and will work
diligently to help this occur, but I don't want to deceive them.

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] |

Minor errors (missing factor in a final answer, sign error, etc.) will
be penalized minimally. Students whose errors materially simplify the
problem will *not* be eligible for most of the problem's credit.

**Problem 1** (12 points)

2 points for setting up the pieces in the partial fraction
expansion. 2 points for getting an equation relating the top to the
constants in the pieces. 3 points for correct solution of this
equation. 3 points for antidifferentiation. 2 points for the answer (1
of these for using ln properties to verify the displayed result).

**Problem 2** (14 points)

a) (10 points) 3 points for the formula: Pi_{0}^{2}(sin(x))^{5}dx. 5 points for
the antidifferentiation and 2 points for the answer.

b) (4 points) 2 points for the correct integrand and 2 points for the
correct bounds on the definite integral.

**Problem 3** (12 points)

This is a force·distance problem. Modeling the force (the
weight of the bucket) is worth 5 points. This should be a linear
function: `A+B`(*Variable*). Also
this function should be at least 1 in the variable's interval (the
student is advised that the weight of bucket+water is at least 1
pound). The limits of integration should be the length that the bucket
is raised: 2 points. The distance part is d*Variable* and earns 2
points if the limits of integration are correct.

3 points remain for the integration and answer (which is worth 1 of
the 3 points). A student's answer is *eligible* for these three
points if the function being integrated is linear, not constant, and
positive everywhere in the interval of integration whose length must
be the length the bucket is lifted, otherwise the student is not doing
the requested problem. An integration of any random function (a number
of people had quadratic integrands) on any random interval will not
earn credit.

**Problem 4** (10 points)

Stating one *useful* substitution gets 4 points (2 points if the
dx and x are not consistent!). Correctly
carrying out the substitution to get an antiderivative (including
substituting back to x's!) is 6 points.

Beginning a good integration by parts strategy also will earn 4
points, and then carrying it out can earn the remaining 6 points.

**Problem 5** (14 points)

a) (3 points) The sketch should have a __straight line segment__
and a __curve__ which
__intersect at (0,0) and (1,Pi/2)__ for arcsin
and which
__intersect at (0,0) and (1,Pi/4)__ for arctan. The curve
should be __increasing and concave up__ for arcsin and
__increasing and concave down__ for arctan. 1 point will be
deducted for each indicated item which is not correct up to a possible
3 points.

b) (9 points) A correct setup (definite integral and integrand) earns
2 points. 5 points for correct antidifferentiation, and 2 points for
evaluation.

**Problem 6** (12 points)

6 points for correct integration by parts and getting the
antiderivative (4 points for the first u,v step and then 2 points for
the final antiderivative). 3 points for setting up an integral from 1
to a large number and substituting (1 of these points was earned for
writing the improper integral as a limit of a proper integral with a
parameter). 1 point for finding the limit correctly, and 2 points for
showing where L'H is used and using it correctly. Just "plugging in"
infinity in the antiderivative is *not* an acceptable
strategy!

**Note** *No points* were earned if the student ignored the
integral and applied L'H to the integrand and presented some related
result as the "solution". This is a totally different and much easier
problem. It is not the problem students were asked to solve.

**Problem 7** (14 points)

a) (6 points) There should be seven terms, values of square root at
the appropriate x's, "weighted" appropriately (1/4/2/4/2/4/1), and
these should be multiplied by (6-0)/(6·3). The multiplier is
worth 1 point, the weights are worth 2 points, the appropriate x's are
worth 2 points, and square root appearing correctly is worth 1
point. No function evaluations need to be done. If the function in the
sum is only identified as "f" then the 1 point for square root is not
earned.

b) (8 points) Use of the second derivative graph to get some estimate
of the size of |f´´(x)| on [1,3] is worth 2
points. *Use* of the Trapezoidal Rule error term is worth 2
points. The correct N earns 1 point, and getting some correct
*numerical* error estimate is 1 point. A correct (interval!)
answer earns the last 2 points and this can only be earned with a
correct error estimate.

**Comment** Some answers confuse the estimate of the integral and
the error estimate. These are *not* the same.

**Problem 8** (12 points)

2 points for trying the correct substitution (some multiple of
tangent); 1 point for getting the multiple correct; 1 point for
getting secant out of the quadratic term; 1 point for the correct dx;
1 point for getting to the integral of secant; 1 point for applying
integral of secant correctly; 3 points for getting back to x
correctly; 2 points for the correct answer (evaluating and using log
properties).

The student also could earn all points by evaluating the integral
correctly in the substituted variable (no student did this).

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

Max grade | 16 | 12 | 12 | 10 | 12 | 12 | 12 | 14 | 95 |

Min grade | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |

Mean grade | 11.44 | 8.66 | 6.73 | 3.03 | 2.29 | 4.63 | 4.95 | 8.44 | 50.14 |

Median grade | 13 | 10 | 7 | 2 | 1 | 3 | 8 | 9 | 51 |

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] |

Minor errors (missing factor in a final answer, sign error, etc.) will
be penalized minimally. Students whose errors materially simplify the
problem will *not* be eligible for most of the problem's credit.

**Problem 1** (12 points)

a) (6 points) 2 points for horizontal; 4 points for the others (1
point off for each element which is tilted up where it should be down
or vice versa, and 1 point off if the elements do not vary correctly,
more or less tilt, as scanned horizontally: max of 2 off for "tilt"
errors) and max of 2 off for sign errors.

b) (2 point) one equilibrium solution and specification of the
constant with the right equation (so x=`CONSTANT` loses a point because it does
*not* describe a solution to the differential equation).

c) (8 points) 1 point for correct separation; 3 points for integration
(1 of these is for a correct "+C"); 2 points for use of the initial
condition; 2 points for solving for y as a function of x.

**Problem 2** (12 points)

Use of the arc length formula: 1 point; correct computation of
f´(x) earns 2 points; 2 points for the correct integrand;
2 points for the change to sec(x); 2 points for integrating secant;
2 points for evaluating sec and tan at the endpoints; 1 point for the
final answer recognizing ln(1).

**Note** This is essentially problem 10 in section 8.1 of the
textbook.

**Problem 3** (12 points)

a) (5 points) 2 points for the first and second derivatives; 2 points
for correct evaluation of the function value, and the first and second
derivatives; 1 point for final assembly of T_{2}(x). ?????

b) (1 point) Earned for the answer. If the student gives a reasonable
polynomial as answer for a) this is earned by the value of the
student's polynomial at 5.

c) (6 points) The correct third derivative earns 2 points; estimation
of K in the correct way earns 2 points (selecting the correct endpoint
earns 1 of the 2 points but just selecting the incorrect endpoint
earns 0; the other point is earned for some reason why the correct
endpoint was selected); 2 points for putting everything together
(factorials, powers, etc.).

**Problem 4** (10 points)

2 points for starting with a valid and relevant polynomial related to
e^{x}; 2 points for correct substitution of -x^{2}; 2
points for indicating multiplication of this polynomial by the
appropriate linear polynomial; 2 points for carrying out the
multiplication; 1 point for further algebraic expansion into standard
form (sum of constants times powers of x); 1 point for the answer. The
answer point is *not* earned if the polynomial does not have the
correct degree.

**Problem 5** (12 points)

a) (6 points) Transforming the sequence to a form where
L'Hôpital's Rule could be used is worth 2 points; 1 point for
remarking on eligibility; 2 points for using L'Hôpital's Rule
(differentiation of the top and bottom); 1 point for the answer.

b) (6 points) 2 points for separating into two geometric series; 2
points each for the correct use of geometric series and the answer.

**Problem 6** (12 points)

3 points for realizing or using the "key observation" that dropping
the square root increases the size of the fraction (using the other
part of the formula doesn't earn credit since it cannot be estimated
usefully); 3 points for showing that the infinite tail must be
estimated (here either a geometric series or a relevant [improper]
definite integral can be given); 3 points for estimating the infinite
tail by finding the sum of the relevant geometric series or with a
*correct* antiderivative; 3 points for using the sum and the
tabular information (or a *correct* integral!) to get a correct
answer. Merely asserting an answer is not sufficient to get credit
here (lots of data was supplied). Asserting that an infinite tail is
small because one or a few terms are small also earns no credit.

No credit will be given for a purely arithmetic approach to this
problem because this is a calculus course.

**Problem 7** (12 points)

4 points for some evidence connecting the sum to the integral. One
acceptable item would be a picture similar to that displayed on the
answer sheet. An explicit inequality connecting the N^{th}
partial sum with a definite integral would also be acceptable. Also
useful would be mention of a relevant function *decreasing*. But
some evidence should be given.

4 points for evaluating a relevant definite integral. 4 points for a
correct answer with evidence showing that specific N is valid.
Students who use N in place of N+1 in an otherwise correct solution
will be penalized 2 points.

**Problem 8** (14 points)

a) (8 points) 4 points for computing the ratio and correctly obtaining
a simple fraction (1 point is lost if the absolute value is missing);
3 points for obtaining the limit of the ratio; 1 point for the answer.

b) (6 points) 3 points for considering the case x=`RIGHT END POINT`. 1 of these 3 points is for
the answer, and 2 points for correct support of the answer.

3 points for considering the case x=`LEFT END
POINT`. 1 of these 3 points is for the answer, and 2 points
for correct supporting of the answer.

**
Maintained by
greenfie@math.rutgers.edu and last modified 4/20/2008.
**