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Math 241 Exam 3 Instructions and Problems - Prof. Thomas E. Carty, Exams of Advanced Calculus

University of Illinois - Urbana-ChampaignAdvanced Calculus
Prof. Thomas E. Carty

The instructions and problems for exam 3 of math 241. The exam covers topics such as vector calculus, parametric curves, and integrals. Students are prohibited from using hats, dark sunglasses, book bags, cell phones, music systems, or calculators during the exam. The exam is worth 50 points and lasts for fifty minutes.

Typology: Exams

2010/2011

Uploaded on 06/28/2011

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Math 241 – Exam 3 – 12PM V1 Name:
November 12, 2010 Section Registered In:
50 points possible
1. No hats or dark sunglasses. All hats are to be removed.
2. All book bags are to be closed and placed in a way that makes them inaccessible. Do
not reach into your bag for anything during the exam. If you need extra pencils, pull
them out now.
3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during
the exam, it will be construed as cheating and you will be asked to leave. This includes
using it as a time-piece.
4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell
phones.
5. If you have a question, raise your hand and a proctor will come to you. Once you stand
up, you are done with the exam. If you have to use the facilities, do so now. You will
not be permitted to leave the room and return during the exam.
6. Every exam is worth a total of 50 points. Check to see that you have all of the pages.
Including the cover sheet, each exam has 7 pages.
7. Be sure to print your proper name clearly and then write down the title of the dis-
cussion section (i.e. BD7) for which you are registered.
8. If you finish early, quietly and respectfully get up and hand in your exam. You need
to show your student ID when you hand in the exam. (Drivers license, passport, etc.
will work also.) No exam will be accepted without ID.
9. When time is up, you will be instructed to put down your writing utensil, close the
exam and remain seated. Anyone seen continuing to write after this announcement
will have their exam marked and lose all points on the page they are writing on. We
will come and collect the exams from you. Have your ID ready.
10. Good luck. You have fifty minutes to complete the exam.
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Download Math 241 Exam 3 Instructions and Problems - Prof. Thomas E. Carty and more Exams Advanced Calculus in PDF only on Docsity!

Math 241 – Exam 3 – 12PM V1 Name:

November 12, 2010 Section Registered In:

50 points possible

  1. No hats or dark sunglasses. All hats are to be removed.
  2. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now.
  3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as a time-piece.
  4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell phones.
  5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave the room and return during the exam.
  6. Every exam is worth a total of 50 points. Check to see that you have all of the pages. Including the cover sheet, each exam has 7 pages.
  7. Be sure to print your proper name clearly and then write down the title of the dis- cussion section (i.e. BD7) for which you are registered.
  8. If you finish early, quietly and respectfully get up and hand in your exam. You need to show your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID.
  9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready.
  10. Good luck. You have fifty minutes to complete the exam.
  1. Let F(x, y, z) := (sin (xy), cos (yz), sin (xz)). (a) (5pts) Compute the divergence and curl of F.

(b) (2pts) Is the vector field F irrotational or incompressible? Briefly justify your answer.

  1. Let a wire be parametrized by r(t) := (sin^2 (t), sin(t)), −π/ 2 ≤ t ≤ π/2. (a) (3pts) Sketch the wire.

(b) (7pts) The mass density of the wire is given by ρ(x, y) := |y|. Compute the total mass of the wire.

  1. (8pts) Compute the integral

D

√^2 y x^4 + 1 dA^ where^ D^ is the region in^ xy-space bound by x = 1, x = 2, y = x^3 /^2 and y = 0.

  1. (8pts) Compute the integral

D

( (^) y − x y + 2x + 1

dA where D is the region bounded by the parallelogram formed by y = x + 1, y = x + 2, y = − 2 x and y = − 2 x + 4.

Math 241 – Exam 3 – 12PM V2 Name:

November 12, 2010 Section Registered In:

50 points possible

  1. No hats or dark sunglasses. All hats are to be removed.
  2. All book bags are to be closed and placed in a way that makes them inaccessible. Do not reach into your bag for anything during the exam. If you need extra pencils, pull them out now.
  3. No cell phones. Turn them off now. If you are seen with a cell phone in hand during the exam, it will be construed as cheating and you will be asked to leave. This includes using it as a time-piece.
  4. No music systems – IPODs, MP3 players, etc. – or calculators; same rules as with cell phones.
  5. If you have a question, raise your hand and a proctor will come to you. Once you stand up, you are done with the exam. If you have to use the facilities, do so now. You will not be permitted to leave the room and return during the exam.
  6. Every exam is worth a total of 50 points. Check to see that you have all of the pages. Including the cover sheet, each exam has 7 pages.
  7. Be sure to print your proper name clearly and then write down the title of the dis- cussion section (i.e. BD7) for which you are registered.
  8. If you finish early, quietly and respectfully get up and hand in your exam. You need to show your student ID when you hand in the exam. (Drivers license, passport, etc. will work also.) No exam will be accepted without ID.
  9. When time is up, you will be instructed to put down your writing utensil, close the exam and remain seated. Anyone seen continuing to write after this announcement will have their exam marked and lose all points on the page they are writing on. We will come and collect the exams from you. Have your ID ready.
  10. Good luck. You have fifty minutes to complete the exam.

(a) (3pts) Compute ∇(x^2 y^2 z^2 ).

(b) (5pts) Compute

r^ F·ds^ where^ F(x, y, z) :=^ ∇(x

(^2) +y (^2) +z (^2) ) and r(t) := (t/π, sin(t), cos(t)),

0 ≤ t ≤ π.

  1. Let a wire be parametrized by r(t) := (sin(t), sin^2 (t)), −π/ 2 ≤ t ≤ π/2. (a) (3pts) Sketch the wire.

(b) (7pts) The mass density of the wire is given by ρ(x, y) := |x|. Compute the total mass of the wire.

  1. (9pts) Find the volume of the solid bounded by the paraboloids z = x^2 + y^2 and z = 32 − x^2 − y^2.
  1. (8pts) Compute the integral

D

( (^) y + x y − 3 x + 1

dA where D is the region bounded by the parallelogram formed by y = −x + 1, y = −x + 2, y = 3x and y = 3x + 2.