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 (1/2n)3 as the nth 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.