6760, Math 21a, Fall 2009
Exams page of Math 21a, Multivariable Calculus
Exams
Course head: Oliver Knill
Office: SciCtr 434

Final exam

The final exam of December 19 with solution draft.

The average of the exam was 76/100 points with a standard deviation of 10 (everything scaled to 100).


The final exam took place Saturday, December 19'th at 2 PM in Halls C,D and E. See the registrars page for the rooms. We had global reviews on
review slides


Some statistics about the exam:

Practice exams:

As usual, we have much more practice exams than you can possibly do, but the idea is that you find enough practice problems for a topic, you still need to practice.


Errata: (corrected)
Tips for the reading period collected by peer advisors

Second Hourly:



The second hourly took place on November 3rd at 7 PM in Hall C. The practice exams below should give you a pretty good idea what type of questions are asked. Sections in the book: 11.3 - 12.5 are relevant. Additionally, the PDE handout is relevant (that handout tells what you have to know). You do not have to worry about terms introduced in 12.5. There was a global review from 7-8:30 PM on Sunday November 1, in Hall C, the same room as the exam room. The average score was 87 with a standard deviation of 9. The high test score can be explained by the coach program, the problem sessions and the bureau of study council and certainly also by excellent preparation by the class.
Practice exams Errata: (corrected)
  • October 29, 2009 4 PM, Exam 5: there are some problems which involve parametrization. While you should still remember what a parametrized surface is for later parts of the course, this is not subject of this midterm. Thanks to Christine Shrock for pointing this out.
  • October 29, 2009, 9 PM, Exam 1: Problem TF 14, minimum should have been replaced by maximum in the explanation. Thanks to Christina Guo for pointing this out!
  • October 31, 2009, 9 AM, Exam 1: Problem 10, The formulas do not quite fit with the story. Without changing the math, the story has been changed. Thanks to Phillip Yao for noticing this.
  • October 31, 2009, 5 PM, Exam 1: problem 5: root(2), not root(3). problem 6)a) the components of nabla(x,y,z) should be divided by 2. Thanks to Carl Daher for these corrections. Nevin Raj for noticed the first mistake too. Nolan Pollock corrected of the correction.
  • October 31, 2009, 11 PM, Exam 3: TF 13, typo on solution. Should be "smaller" rather than larger. Thanks to Christian Yoo.
  • November 1, 2009, 9 AM, Exam 4: Problem 6: Typo in Lagrange equations: second equation 4x, not 4y. Thanks to Insup Lee for pointing this out.
  • November 1, 2009, 2 PM, Exam 2, TF 4: explanation not good. While it is true that the directional derivative in the direction nabla f/|nabla f| is the length of the gradient, this is irrelevant here. Thanks to Phillip Yao.
  • November 1, 2009, 6 PM, Exam 2, Typo in TF 4 (the -> then) Thanks to Allen Shih.
  • November 1, 2009, 9 PM, Exam 3, TF 20, x+y is a counter example, not xy. Thanks Phillip Gao. for this correction.
  • November 2, 2009, 8 AM, Exam 5, Problem 8, gradient at (1,0) not (1,1) and <3,0> instead of <2,0>. End result not affected. Thanks to Sarvagna Patel.
  • November 2, 2009, 5 PM, Exam 1, TF 15, f(x,y) should be f(x,y,z), thanks to Sam Himel.
  • November 3, 2009, 3 PM, Exam 2, TF 4, answer key: directional derivative 1 not -1. Thanks to Eric Liao.
  • November 3, 2009, 5 PM, Exam 3, r(0)=(0,0) not r(0,0)=0. Thanks to Christian Yoo.


First Hourly:



The first hourly took place on Tuesday, Oct 6. 2009, Hall C, 7-8:30 PM. The average of the exam was 79, the standard deviation 11. The maximal score was 96. We had a global review on Sunday October 4th at 7 PM in Hall B.
The exam: Practice exams Errata:
  • October 1, 2009, 6 PM, Exam 1: Problem 10, the hyperboloid is x2+y2=z2+1. The explanations for TF14 and TF18 had some typos or unclear statement. Thanks to Christine Shrock to point these out!
  • October 3, 2009 4 PM, Exam 1: TF 1, typo in completion of square. Thanks to Mishal Rahman for correcting this.
  • October 3, 2009, 5 PM, Exam 1: 10c) the z coordinate is 2 cos(phi), not 2 cos(theta). Thanks to Carl Daher for correcting this.
  • October 4, 2009, 9 AM, Exam 5: problem 4). In the symmetric equation for the line, the second term had a sign error. Thanks to Daniel Kim. In the same exam, TF 1, the explanation was improved thanks to Carolyn Stein.
  • October 4, 2009, 4 PM, Exam 5, Problem 5, one vector was (5,2,0) instead of (5,0,2). Thanks to William Sun for spotting that typo.
  • October 4, 2009, 5 PM, Exam 3, Problem 9c). The question should ask for the curvature at (2,5,3). The answer is unchanged. In 9b) some coefficients were doubled. T Thanks to Seungsoo Kim for pointing this out.
  • October 5, 2009 10 PM, Exam 3 TF 4. The function is not continuous everywhere, because on the sphere rho = pi or any sphere rho=k pi the function blows up. Note that the function is continuous at the origin rho=0, because Hopital shows you can assign a value 1 there which makes it continuous. Thanks for Wenting Cao for asking about this.


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.
Questions and comments to knill@math.harvard.edu
Math21b | Math 21a | Fall 2009 | Department of Mathematics | Faculty of Art and Sciences | Harvard University