Math 21a, Fall 2008
Exams page of Math 21a, Multivariable Calculus
Course head: Oliver Knill
Office: SciCtr 434

Final exam

Final exam distribution. The average score was 78.
Our final exam took place January 13'th at 9:15 AM - 12:15 AM in lecture halls B and C. Here is the exam schedule of the registrar. We had final global comprehensive reviews on Thursday, January 8, 2009 in Hall C at 3:30-5 (with material covering the first half of the course) and on Friday, January 9, 2009 again in Hall C at 2-3:30 (with material featuring the second half of the course). Here are the review slides.

This is a list of practice exams. The idea is to do one or two and to look for practice in topics, which you need to do more. It is unrealistic look at all the 100 problems. Practice exam 1 had essentially been the exam of last year. Without a handfull of exceptions, all other exam problems have appeared in previous exams.

Practice exam I Practice exam II Practice exam III Practice exam IV Practice exam V Practice exam VI Practice exam VII Final exam
Solution exam I Solution exam II Solution exam III Solution exam IV Solution exam V Solution exam VI Solution exam VII Solution of Finals
Errata for practice exams: the above files were corrected at the time indicated
  • 5. January, 3 PM: Practice I Problem 8, clarified UFO situation with a second picture.
  • 5. January, 4 PM: Practice 7,Problem 2, curve picture 4 had appeared twice.
  • 7. January, 3 PM: Practice 2,Problem 14c, explanation not clear. Thanks to Leslie Beh
  • 7. January, 11 PM: Practice 5, Problem 5, replace inclusive by inconclusive. Practice 6,Problem 6c, at the end of the solution (0,1), not (0,0). Thanks to Jenny Xia
  • 8. January, 0:30 AM: Practice 6, TF 3, replace cylinder by paraboloid, answer stays. Thanks to Kenric Tam.
  • 9. January, 4 PM: Practice 6, Problem 8, gradient <4x,4y>, not <<8x,8y>, answer stays. Thanks to Dominik Nieszporowski
  • 9. January, 6 PM: Practice 7, Problem 2, order III,I,II,IV, Thanks to Connie Lee and Samir Patel and Ben Li
  • 10. January 6 PM: Practice 4, Problem 8 b) sign error in the result. Thanks to Jenny Xia.
  • 10. January 6 PM, Practice 7, Problem 9 b). In the integration computation, the part F(r(t)).r'(t) had been evaluated already. The result is the same. Thanks to Jason Varnell.
  • 11. January 1 AM, Practice 5, Problem 6, two typos in gradient and discriminant. Result stays. Thanks to Corinne Tu.
  • 11. January 1 AM, Practice 1, Problem TF15, divergence is 3, not 1, result stays. Thanks to James Danz.
  • 11. January 1 AM, Practice 2, Problem 14a, the parametrization must be r(phi,theta), not r(theta,phi) to have the orientation outwards. The result stays. Thanks go Hanna Retallack.
  • 11. January 1:30 AM, Practice 4, Problem 6, typo with one 6 too much. Thanks to Jenny Xia
  • 11. January 5 PM, Practice 3, Problem 4, discriminant is -1-2y. Thanks to Jenny Xia
  • 11. January 11 PM, Practice 7, Problem 3, vector field mixup (and error introduced early January when rerunning some graphics in mathematica 7), Thanks to Justin Koh.
  • 11. January 11 PM, Practice 5, Problem 8, typo pi3/ instead of pi/3. Thanks to Daniel Norris
  • 11. January 11:30 PM, Practice 2, TF4, correction of L(x,y,z) at (1,1,1). Result does not change. Thanks to Jim Danz.
  • 12. January 7 PM, Practice 6, TF20, the region under considering is an octant intersected with a solid sphere. Result stays. Thanks to Sanjay Misra.
  • 12. January 4 PM, Practice 1, 14), instead of fxx, it should rather be faa. Thanks to Taylor Shimberg.
  • 12. January 8 PM, Practice 3, TF 16, div (F) = 2x + 2y + 2z, Problem 14, differentials in wrong order. Thanks to James Danz.

The final exam in exam group 1 is a three-hour final exam administered by the Exams Office for all enrolled students. As such, the exam is subject to the following criteria:
  • All students enrolled in the course are required without exception to attend the exam administered by the FAS Exams Office. Students not present at this exam will receive a grade of ABS for the course.
  • Instructors may not approve or administer a makeup exam to any student absent from the exam. Makeup exams may only be approved by the Administrative Board for undergraduates or the Deputy Registrar for graduate students, Special Students, or cross-registered students and will be administered by the FAS Exams Office during the scheduled makeup exam period.

Second hourly

The second hourly took place on Tuesday November 18 at 7 PM in Hall B. It covers week 5-8. There was a global review on Sunday 11/16 from 7-8:30pm in Science center lecture hall C. Review slides. Here is the exam and practice exams The average score of the class for the second midterm was 81 points, the standard deviation was 10.

Errata for practice exams: since many problems were rewritten or added newly, there quite a few typos and errors: thanks for everybody for hunting them down. (The name indicates the first person noticing the error, the date and time indicates, when a corrected version has been posted on the website).
  • 11/13/08, 11:59 PM, Denise Xu, I 3), typo in final boxed result, in b), typo in 3y2, does not affect answer
  • 11/13/08, 11:59 PM, Edward Li, III 8) , typo 2s-t changes answer.
  • 11/14/08, 12:05 AM, Adam Lee Purvis, I 2), solution had been missing
  • 11/14/08, 12:15 AM, Anirudh Penumaka, III 2a), a(y) and b(y) for type II integral
  • 11/15/08, 7:15 AM, Adam Lee Purvis, I TF4) direction <0,-1> in explanation. Result remains, I6 (2,1) -> (1,2) Result remains, I 7) x=y=-1, result remains
  • 11/15/08: 7:25 AM, Denise Xu, I4), exponent -2/3, not 2/3, result remains.
  • 11/15/08: 7:30 AM, Denise Xu and Dominik Nieszporowski, I 5bc) were false, the result changed.
  • 11/14/08: 7:30 AM, Edward Li, I 10), incorrect upper bound. The result changed.
  • 11/14/08: 6:25 PM, Nan Du, I 4), Typo (0,-90) should be (0,-3). Result unchanged.
  • 11/14/08: 6:40 PM, Dominik Nieszporowski, II 10a), the answer changed
  • 11/14/08: 6:45 PM, Christopher Madl, I 10a), still a bug in the final result
  • 11/15/08: 1:20 PM, Jacob Weatherly, II 4a) factor of r missing in computation. Result unchanged.
  • 11/15/08: 6:30 PM, Samantha Keyser, II 2,10) missing or incomplete solutions.
  • 11/16/08: 2:00 AM, Denise Xu, III 9) integral goes to pi^2, the final result changes
  • 11/17/08: 3:20 PM, Hanzi Zhang, III 6) typos in computation, final result remains
  • 11/17/08: 8:00 PM, William Hawley, I 3), clean out typo 6y result remains
  • 11/17/08: 11:00 PM, Lakshmi Parthasarthy, Sean Singh, Jeremy Cushman III 4), typos after changing order of integration, result unchanged
  • 11/17/08: 11:30 PM, Sean Singh and Kevin McNamara IV TF1, wrong answer. The condition implies that it is a local minimum

First hourly

The first hourly took place Tuesday October 21 at 7 PM in Hall B. It covered the material from week 1-4. There was a global review on Sunday October 19'th from 7-8:30 PM in lecture hall B. slides [PDF]. Here are the practice exams: The average score of the class for the first hourly was 75 points, the standard deviation was 13.5.
Errata for practice exams: (the name indicates the first person noticing the error, the date and time indicates, when a corrected version has been posted on the website).
  • 10/16/08, 7 AM: Sean Singh: Practice 1, TF12 "in" missing, TF11/TF16 had been identical.
  • 10/18/08, 9 AM, Edward Li: Practice 4, Problem 9), sign error for z= ...
  • 10/18/08, 9 AM, Edward Li, Practice 4, TF 19, is true, not false
  • 10/18/08 12:30 PM, Jacob Weatherly, Practice 4, TF 4, is false, not true
  • 10/18/08 23 PM, Jeremy Feng, Practice 3, TF 1, the answer is false
  • 10/19/08 7:30 AM, Denise Xu, Practice 3, 7b), typo in z-part of acceleration
  • 10/19/08/ 11:30 AM, Tom Yu , Practice 2, 9b), parametrization of ellipsoid.
  • 10/20/08 20:00 PM, Ezgi Bereketli, Practice 4, 5), <5,0,2> instead of <5,2,0>, result stays

Official policy at Harvard is that "out of sequence exams" are only offered for students observing religious holidays and students who need out of sequence exam as ordered from the accessible education office. We sometimes can accommodate students who happen to have an important sports event at the same time (it must be an actual game or tournament) or a concert or theater event (regular practice or rehearsals of course do not qualify to miss the exam) or a political election event (for which you are a major candidate). In case of a qualifying conflict, contact the course head to see whether it is possible to arrange the exam earlier at the same day. We will need a letter from your coach or director. The request has to come at least a week before the exam.
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