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Questions on Multivariable and Vector Calculus for Quiz 7 | MATH 275, Quizzes of Calculus

Material Type: Quiz; Class: Multivariable and Vector Calculus; Subject: Mathematics; University: Boise State University; Term: Fall 2002;

Typology: Quizzes

Pre 2010

Uploaded on 08/19/2009

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MATH 275 Section 001 Quiz 7
You may work with other class members on this quiz, but you may not receive
assistance from people not in MATH 275 (Section 001). You must show all of
your work to receive full credit. Do all your work on other sheets of paper and
be sure to staple all the pieces of paper together or YOU WILL GET A ‘ZERO’
ON THE QUIZ. Do not use decimal approximations unless asked to do so. Your
work on this quiz must be handed in by Monday, 21 October 2002 at 9:40 a.m.
GOOD LUCK!
1) Consider a triangle whose angles are A,B, and C. Use Lagrange multipiers
to prove that the expression
sin Asin Bsin C(1)
is maximized when the triangle is equilateral. Provide an argument that explains
why your result maximizes (and not minimizes) (1).
2) Use Lagrange multipliers to find the point closest to the origin on the inter-
section of the planes y+ 2z= 12 and x+z= 6. Provide an argument to explain
why the point you found represents the closest point and not some other type
of extreme point.
3) Let R={(x, y) : 0 x4,0y6}. Compute the integral
Z
R
Z(2x+ 3) dA
in three different ways:
a) as an iterated integral, integrating first with respect to x, then with
respect to y;
b) as an iterated integral, integrating first with respect to y, then with
respect to x;
c) using geometrical arguments (i.e., do not compute any antiderivatives).
4) Find the volume of the solid bounded by:
the x-yplane
the y-zplane
the x-zplane
the plane x= 2
the place y= 3
the plane 3x+ 2yz+ 6 = 0.

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MATH 275 – Section 001 – Quiz 7

You may work with other class members on this quiz, but you may not receive assistance from people not in MATH 275 (Section 001). You must show all of your work to receive full credit. Do all your work on other sheets of paper and be sure to staple all the pieces of paper together or YOU WILL GET A ‘ZERO’ ON THE QUIZ. Do not use decimal approximations unless asked to do so. Your work on this quiz must be handed in by Monday, 21 October 2002 at 9:40 a.m. GOOD LUCK!

  1. Consider a triangle whose angles are A, B, and C. Use Lagrange multipiers to prove that the expression

sin A sin B sin C (1)

is maximized when the triangle is equilateral. Provide an argument that explains why your result maximizes (and not minimizes) (1).

  1. Use Lagrange multipliers to find the point closest to the origin on the inter- section of the planes y + 2z = 12 and x + z = 6. Provide an argument to explain why the point you found represents the closest point and not some other type of extreme point.

  2. Let R = {(x, y) : 0 ≤ x ≤ 4 , 0 ≤ y ≤ 6 }. Compute the integral ∫

R

(2x + 3) dA

in three different ways:

a) as an iterated integral, integrating first with respect to x, then with respect to y;

b) as an iterated integral, integrating first with respect to y, then with respect to x;

c) using geometrical arguments (i.e., do not compute any antiderivatives).

  1. Find the volume of the solid bounded by:
  • the x-y plane
  • the y-z plane
  • the x-z plane
  • the plane x = 2
  • the place y = 3
  • the plane 3x + 2y − z + 6 = 0.