6760, Math 21a, Fall 2009
Exams page of Math 21a, Multivariable Calculus
Exams
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
• Wednesday 12/9 from 3-4:30 in Hall B (part I until double integrals)
• Thursday 12/10 from 3-4:30 In Hall B (part II until the end)
review slides

 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)
• December 12, 2009: Exam I, problem 13, a bound 3 had been wrong in the integral. The final result was right. Thanks to Audrey Young.
• December 15, 2009: Sign errors in exam I, Problem 10. Final answer is ok. Thanks to Adrienne Smallwood.
• December 15, 2009: Exam IV, problem 2, the 2D vector field should be F(x,y) = (cos(x), sin(y) ) . Thanks to Christine Shrock.
• December 15, 2009: Exam IV, problem 11, integration bounds are from -1 to 1. Thanks to Christine Shrock.
• December 15, 2009, Exam VIII, Problem 7: Value of f at (1/2,-1/2,0) is 1/4 not 3/4. Thanks to Allen Shih.
• December 17, 2009, Exam I, Problem 11, 42 = -20-22 not 42 = -21-21 even so it we have Math 21a. Some times there is more than one way to write the answer to the ultimate question, but not all answers are the right explanation. Thanks to Phillip Yao.
• December 17, 2009, Exam V, Problem 5, Typo 20 should be 5, Thanks to Tommy MacWilliam and Shirley Zhou
• December 17, 2009, Exam II, Question 7), Sign typo in 6x=4-6y. End result correct. Thanks to Scott Crouch.
• December 18, 2009, Exam VII, TF question 11, The curl is constant 2, not 1. Thanks to Yacoub Kureh.

## 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. Here is the second midterm Exam, and the solutions .
 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. Here is the first midterm Exam, and the solutions .
 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.

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