**Some general comments**

If an indefinite integral is asked for and a "+C" is omitted, 1 point
will be deducted, but only 1 time in the exam.

If a student makes a mistake early in a problem, I will try to "read
with" the student and give them an appropriate amount of credit. But
if a student's errors materially simplify a later part of a problem,
then full credit cannot be earned for that part of the problem.

1. (8 points)

2 points for correctly separating the differential equation.

2 points for correctly integrating the two sides.

2 points for having a constant of integration and using the initial
condition correctly.

2 points for correctly writing y as a function of x.

2. (12 points)

a) 2 points for correctly listing the k's.

b) 6 points for a correct sketch. I looked for an increasing graph
entirely contained within the strip 0 < y < 1. I wanted the
correct asymptotics and some correct concavity.

Each correct limit was itself worth 2 points.

3. (12 points)

a) 6 points. 2 points deducted for doing the problem with each cube
seeming to have one edge. A setup as an infinite geometric series got
2 points, with an additional point if it was a correct geometric
series.

b) 6 points, as above. 2 points for a correct infinite series with
(1/2^{n})^{3} as the n^{th} term.

4. (12 points)

2 points for the algebraic "preparation" for the Root or Ratio
Tests.

2 points for taking the limit correctly.

1 point for correctly diagnosing the restriction on x from the
limit.

1 point each for correctly analyzing the endpoints.

1 point for correctly reporting the radius of convergence and 2 points
for correctly reporting the interval of convergence.

5. (10 points)

1 point for a valid comparison with a simpler series which helps in
the solution of the problem.

2 points for comparing with a valid integral. 2 points for a correct
antiderivative.

3 points for getting an appropriate restriction on how large the
partial sum should be.

2 points for writing or indicating an appropriate partial sum.

6. (10 points)

a) (4 points)

1 point for a correct answer.

3 points for supporting algebraic manipulation and limit facts.

b) (6 points)

1 point for a correct answer.

1 point for taking logs correctly.

1 point for correct algebraic manipulation setting up l'Hospital's
Rule.

2 points for correct differentiation in l'Hospital's Rule.

1 point for a correct limit of the result of the preceding step.

7. (10 points)

4 points for clearly indicating a specific valid polynomial.

3 points for a "setup" of an error estimate.

3 points for valid completion of the error estimate.

8. (12 points)

a) (6 points)

1 point for citation of the Taylor series for the exponential
function.

1 point for substitution of -x^2, and then 1 point for algebraically
treating it in the sum.

1 point for setting up the integral and 1 point for correct
integration.

1 point for valid use of the upper and lower limits of
integration.

b) (6 points)

1 point for recognizing an alternating series.

3 points for treating the error correctly (deduct 1 point for not
putting in an absolute value sign correctly).

2 points for reporting or indicating a correct answer.

9. (10 points)

3 points for correctly identifying the polynomial as an appropriate
Taylor polynomial of cosine.

3 points for setting up a Taylor's Theorem error estimate.

4 points for correctly carrying out the error estimate.

Describing the full Taylor series of cosine as an alternating series
is actually incorrect. The successive terms need not decrease in
absolute value (example: x=10, and n=2 and 3, say). It is true that
when x is in the interval [-2,2], the omitted terms after T_7 do form
an alternating series satisfying all hypotheses of the alternating
series test. So it is possible to do this problem correct with these
stipulations, but only part credit will be earned by just citing
"alternating series" without additional discussion.

Comment: a better problem for the purposes of this exam probably would
have resulted from writing "Use the error estimate in Taylor's Theorem
to estimate ..." instead of letting the student select the method of
error estimation.

10. (6 points)

a) 3 points. Some valid computation will earn a point, but full credit
comes with the correct answer and a supporting statement.

b) 3 points, earned by a correct explanation.