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MATH2263 Summer 2009 Exam 2: Multivariable Calculus Problems, Exams of Calculus

Dr. Bhim Rao Ambedkar UniversityCalculus

The summer 2009 exam 2 for math2263, a multivariable calculus course. The exam covers various topics such as finding local maxima, minima, and saddle points, evaluating iterated integrals, sketching regions of integration, finding double integrals, finding gradients and directional derivatives, and using lagrange multipliers. The exam consists of 7 problems, each worth a certain number of points.

Typology: Exams

2012/2013

Uploaded on 02/11/2013

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MATH 2263 Name (Print):
Summer 2009 Student ID:
Exam 2
July 8, 2009 Signature:
Time limit: 55 minutes
This exam contains 9 pages (including this cover page and a scratch page) and 7 problems.
Check to make sure you have all 9 pages. Enter all requested information at the top of this
page, and put your initials on the top of every page, in case the pages become separated.
No calculators or note-sheets are allowed.
The following rules apply:
•Show your work, in a reasonably neat and coherent way, in the space provided. All
answers must be justified by valid mathematical reasoning. To receive full
credit on a problem, you must show enough work so that your solution can be followed
by someone without a calculator.
•Mysterious or unsupported answers will not receive full credit. Your work should be
mathematically correct and carefully and legibly written.
•A correct answer, unsupported by calculations, explanation, or algebraic work will
receive no credit; an incorrect answer supported by substantially correct calculations
and explanations might still receive partial credit.
•In the event that you cannot fit your entire answer in the space provided, clearly
indicate where the answer continues.
1 15 pts
2 15 pts
3 15 pts
4 10 pts
5 10 pts
6 15 pts
7 20 pts
Total 100 pts
pf3
pf4
pf5
pf8
pf9

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Download MATH2263 Summer 2009 Exam 2: Multivariable Calculus Problems and more Exams Calculus in PDF only on Docsity!

MATH 2263 Name (Print): Summer 2009 Student ID: Exam 2 July 8, 2009 Signature: Time limit: 55 minutes

This exam contains 9 pages (including this cover page and a scratch page) and 7 problems. Check to make sure you have all 9 pages. Enter all requested information at the top of this page, and put your initials on the top of every page, in case the pages become separated. No calculators or note-sheets are allowed.

The following rules apply:

  • Show your work, in a reasonably neat and coherent way, in the space provided. All answers must be justified by valid mathematical reasoning. To receive full credit on a problem, you must show enough work so that your solution can be followed by someone without a calculator.
  • Mysterious or unsupported answers will not receive full credit. Your work should be mathematically correct and carefully and legibly written.
  • A correct answer, unsupported by calculations, explanation, or algebraic work will receive no credit; an incorrect answer supported by substantially correct calculations and explanations might still receive partial credit.
  • In the event that you cannot fit your entire answer in the space provided, clearly indicate where the answer continues.

1 15 pts 2 15 pts 3 15 pts 4 10 pts 5 10 pts 6 15 pts 7 20 pts Total 100 pts

  1. (15 pts) Find and classify all the local maxima, local minima, and saddle points of

f (x, y) = 4xy − x^4 − y^4.

  1. (15 pts) Sketch the region of integration, reverse the order of integration, and evaluate the integral (^) ∫ 8

0

√ (^3) x

y^4 + 1

dy dx.

  1. (10 pts) Using a double integral, find the area of the region bounded by the curves

y = 1 − x and y = 1 − x^2.

(Be sure to sketch the region.)

  1. (15 pts) Let F (x, y, z) = z − xey^ cos(z) + 1.

(a) Find the gradient ∇F (x, y, z).

(b) Find the directional derivative of F at the point (1, 0 , 0) in the direction of v = i− 2 j+4k.

(c) Find the equation of the tangent plane to the surface

z + 1 = xey^ cos(z)

at the point (1, 0 , 0).

  1. (20 pts) Use the method of Lagrange multipliers to find the maximum and minimum values of the function f (x, y) = 2x^2 − 8 y + 5, subject to the constraint x^2 + 4y^2 = 4. (You may assume that both a maximum and minimum value exist.)