Problem #1 |
Problem #2 |
Problem #3 |
Problem #4 |
Problem #5 |
Problem #6 |
Problem #7 |
Extra credit (BLOCK) |
Total | |
---|---|---|---|---|---|---|---|---|---|

Max grade | 12 | 22 | 16 | 12 | 18 | 12 | 8 | 5 | 93 |

Min grade | 2 | 4 | 0 | 2 | 5 | 0 | 0 | 0 | 34 |

Mean grade | 7.82 | 17.76 | 4.15 | 6.97 | 12.62 | 9.92 | 4.68 | 2.21 | 66.12 |

Median grade | 7 | 20 | 2.5 | 7 | 13 | 11 | 4 | 0 | 68.5 |

34 students took this exam. 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] | [60,64] | [55,59] | [0,54] |

**Please**

- Do my problems,
*not*those you invent. - I cannot read your mind or imagine your answers: what you write is what I will read (and grade).
- If you do extensive computation, you are probably not doing my problem (see 1) or you are doing my problem inefficiently.

**Problem 1** (12 points)

6 points for each part. I will *read* what was *written*. I
can *not* guess what you meant to write. I required that what was
written be responsive to what was asked. In part a), an important "If
... then ..." needed to be inferred from what you wrote (for example,
"If [a certain linear combination is 0] then [*all* the
coefficients are 0]". Or "The only way a certain linear combination is
0 is if all the coefficients are 0". I also accepted statements about
none of the vectors being linear combinations of the others. But I
read what you wrote, and tried to read it carefully.

I took 2 points off in part b) if the definition used AX=X and did not specify that X must be
non-zero. This specification is important. If a definition using the
equation det(A-I_{n})=0
was given, there was no such risk. Again, I read what students wrote,
and did not guess at what they did not write.

**Problem 2** (22 points)

a) (2 points) Take the determinant of A-I. You may compute this determinant in any way you like.

b) (2 points) You may have already simplified the characteristic
polynomial in part a), or you can do it here. The roots should be
obvious.

c) (4 points) You need to solve three (3) homogeneous systems of
linear equations. But they all are rather simple.

d) (2 points) You are merely asked to write P and D, which you
certainly should be able to do after parts b) and c). Verification of
your statements occurs in the next few parts of the problem. If
incorrect results from b) and c) are used *correctly* here, I
gave full credit.

e) (3 points) You may find P^{-1} in any way you like. The
answer is easy enough to check, so I gave 1 point out of 3 for an
incorrect answer. The exam I handed out, by the way, had two parts labeled
d. That should be fixed in the posted version.

f) (2 points) Compute the product requested.

g) (2 points) Compute the product requested. If you do not get a
correct diagonal matrix, I gave no points.

h) (5 points) 2 points for setting up the requested relationship
A^{6}=PD^{6}P^{-1}. 1 point for computing
D^{6} and 2 points for computing A^{6} (information
was given allowing you to check your answer).

**Problem 3** (16 points)

a) (8 points) A restatement of the definition in the language of the
problem (that is, writing an *arbitrary* linear combination of
the functions of the problem, setting this equal to 0, etc.) earns 2
points. The balance is earned when the answer contains verification
that the coefficients of the linear combination must be 0.

b) (8 points) The correct answer (**No**) gets 2 points. The cover
page states, "An answer alone may not receive full credit." Correct
supporting evidence is needed for the other 6 points.

**Problem 4** (12 points)

In this problem, I expected students to evaluate the determinant (8
points). A few students tried other strategies (using RREF), and one
or two were successful. Students who "plugged in" values for a and b
and c were simplifying the problem too much and can earn at most
3 of these 8 points.

I hoped the determinant evaluation would be combined with the
knowledge that a matrix is singular (*not* invertible) exactly
when the determinant is 0 (2 points). This then could be matched up
with the perpendicularity condition to get the desired conclusion (2
points).

**Problem 5** (18 points)

It is possible to make mistakes in a) and have serious effects on work
in b) and c). If the result of a) were as complex as the correct
answer, points were only taken off in a). A similar approach was
followed for errors in b).

a) (5 points) One application of integration by parts. Keep track
of the n's and the signs.

b) (4 points) Evaluate the antiderivative. Notice that sin(nPi) and
sin(n 0) are 0 and that cos(nPi)=(-1)^{n} and
cos(n 0) is 1. I took off 2 points for errors which really fouled
up the answers in c) (examples: omitting the sign "flip" or omitting
the Pi).

c) (2 points) The points were earned if the result of b) was used
correctly.

d) (7 points)

The left-hand graph (4 points)

In the graph of the partial
sum, I looked for the following qualitative behavior:

- Continuity on [0,Pi] with matching values at 0 and Pi
- The value 0 at both 0 and Pi.
- Closeness (with "wiggling) to the line segment inside the interval.
- Gibbs phenomena (overshoot) at both ends

The right-hand graph (3 points)

This is supposed to be the graph of the sum of the whole Fourier sine series. Here the behavior required was much simpler:

- Identical to x+1 except at the ends (1 point)
- 0 at both ends (1 point each)

**Problem 6** (12 points)

With orthogonality, there is almost no computation in this problem
(yes, other than small integers). I'll take off 1 point if the
normalization constant is misquoted (I did this in my own solution of
the problem, so I would have scored 99 at most). If the integrals are
computed as the sum of squares of the coefficients with no constant
(or, better, with the constant=1) I will take 2 points off.

Some students antidifferentiated instead of differentiating, and this
will lose 2 points.

I took off 3 points for an error I didn't anticipate, an answer which
essentially declares that the integral of (a negative
number)^{2} is negative. I can't read people's minds, and I
don't know on what level this error was made: through fatigue and
nervousness under exam conditions, or because of serious
misunderstanding.

**Problem 7** (8 points)

I am not satisfied with the statement of this problem. The statement
might have made the problem more difficult for students. I wanted to
ask: what is the polynomial formula when x<0 for the odd
(respectively, even) extension of the polynomial x+x^{4}? I
think *now* I should just have asked exactly that. The statement
of the question(s) seemed to invite misinterpretation.

4 points for each part, with 2 points for the specification of x<0,
1 point for x>0, and 1 point for identifying which Fourier
coefficients must be 0.

I'll give 1 point on each part to people who correctly write the
Fourier sine (respectively, cosine) series for F(x) (respectively,
G(x)).

I'll put a "cleaner" version of the question in the posted exam, but
I'll also show the wording I actually used, since I believe that
affected student work on the problem.