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Calculus III Midterm Test 2, November 2001 - MA 227, Exams of Advanced Calculus

Symbiosis InternationalAdvanced Calculus

The calculus iii midterm test 2 held on november 20, 2001, for ma 227 students. The test covers various topics such as finding partial derivatives, directional derivatives, double integrals, and calculating volumes. Students are required to solve problems related to finding derivatives, comparing values, computing partial derivatives, finding directions, finding closest points, maximizing and minimizing values, and calculating double integrals.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

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MA 227: Calculus III
Midterm Test #2, November 20, 2001
Time limit: 105 min.
Your name:
Your student ID:
1. Find ∂z/∂ x and ∂z/∂ y if
xy2z3+x3y2z=x+y+z.
10 points
2. If z=x2−xy + 3y2and (x, y) changes from (3,−1) to 2.96,−0.95, compare the values of ∆z
and dz.
10 points
3. u=xy +yz +zx,x=st,y=est ,z=t2. Compute ∂ u/∂s and ∂u/∂ t.
10 points
1
pf3
pf4

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MA 227: Calculus III Midterm Test #2, November 20, 2001

Time limit: 105 min. Your name: Your student ID:

  1. Find ∂z/∂x and ∂z/∂y if

xy^2 z^3 + x^3 y^2 z = x + y + z. 10 points

  1. If z = x^2 − xy + 3y^2 and (x, y) changes from (3, −1) to 2. 96 , − 0 .95, compare the values of ∆z and dz. 10 points
  2. u = xy + yz + zx, x = st, y = est, z = t^2. Compute ∂u/∂s and ∂u/∂t. 10 points
  1. Find all directions u (||u|| = 1) in which the directional derivative of the function f (x, y) = x^3 + xy + y^3 at the point (1, 0) is equal to 3. 10 points
  2. Find the points on the surface x^2 y^2 z = 1 that are closest to the origin. 10 points
  3. Find the maximum and minimum values of the function f (x, y, z) = 2x − z subject to the condition x^2 + 10y^2 + z^2 = 5. 10 points
  1. Calculate the double integral ∫ ∫

R

(x^2 + y^2 )^2 dA,

where R = {(x, y)| 1 ≤ x^2 + y^2 ≤ 4 }. 10 points