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Unformatted text preview: Caldamm MAKEUP FINAL EXAM, MATH 234: CALCULUS —— FUNCTIONS OF SEVERAL VARIABLES
MAY 14, 2008 No calculators, books or papers may be used. This examination consists of seven questions. In total7 thereare 110 points on this exam, but the
maximum score achievable is 100. If your total score exceeds 100, your ﬁnal exam grade will count
as 100 towards your ﬁnal grade. If your total score is less than 100, the actual score will be used
towards your ﬁnal grade. (So you may consider one problem as extra credit.) Partial credits will
be given only when a substantial part of a problem has been worked out. Merely displaying some
formulas is not sufﬁcient ground for receiving partial credits. PLEASE BOX YOUR ANSWERS. Hint: Use of Stokes’ and divergence theorems may be needed for solving
some of the questions. 0 YOUR NAME, PRINTED: 0 YOUR LECTURE SECTION (CIRCLE ONE): ADDINGTON AMORIM TU
ZHU n
— Name: 1. (20 points) Compute where S is the hemisphere z = / f5 curl (13er do, —» F = y?— scii— Z213, 1 — $2 — y2, and 733 is the upward unit normal. Name: 2 // ﬁ‘ﬁdO‘
6V on the portion of the solid cylinder :52 + y2 g 4 lying in the ﬁrst octant and below the plane
z = 9:, Where 2. (20 points) Evaluate 15" = (9577+ 35+ 212),
and 85 denotes the boundary of S . Name: 3 //G(xy+z)da Where G is the part of the plane 23: — 2y + z = 7 above the triangle in the xy—plane With
vertices (0,0), (1,0), (1,1). 3. (15 points) Evaluate Name:
4. (20 points) Find the global maximum and global minimum of
f(x,y) = 3 + 2x2 + 3/2 on the closed set
S: {(x,y) : 9:2 +43;2 3 4}. Name: 5. (15 points) Find the directional derivative of ﬁlmy) = (x + y)2  06y in the direction of 4
E ﬁ‘éz— _. at the point 15’: (1, 2). Name:
6. (20 points) Find the surface area of the portion of the sphere
x2 + y2 + 22 = 9 which lies above the plane z = 2. ...
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 Spring '08
 DICKEY
 Multivariable Calculus

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