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Unformatted text preview: MATH'234, Lec. 2, FINAL EXAM
YOUR NAME
T.A.'s NAME DISC. SEC. (time and day) Show. all your work. No calculators or references. W—m—ﬁ
‘ 1.(2o pts) I .(20 pts) 5.(2o pts) . 9.(2o pts) : 1o.(2o pts) 1. (a) Find the equation of the line in parametric form passing
through the points (1,2,3) and (2,3,1 ). (b) Find the equation of
the plane containing the three points (1,2,3), (2,3,1) and
(1,1,1). 2. 1119 position of a particleA as a function of the time is given by
r(t) = (t2 + 4) i + (2t — 3) j. Find the velocity, tangential and
normal components of acceleration. 3. The potential V(x,y,z) is given by V = 5x2  3xy + xyz.
E) find the rate of change of the potential in the direction
u =‘i 4‘]  k at the point (1,1,1). (b) In what direction does
V increase most rapidly at (1,1,1)? (0) What is the maximum
rate of increase at (1,1,1)? 4. Find the minimum of x2 + 4y2 + 1622 under the constraint
xyz = 1. 5. Find the area of the parabolic surface 2 = x2 + y2 which lies
under the plane 2 = 9. §‘\ 6. Evaluate jff y dV where R lies under the plane 2 = x + y
R and above the region in the xy plane bounded by y = x,
y = O, x = 1. ,3 . .A a;
7. Evaiuate J F: dr where
A C /\ > /‘
F = (ycos(x)  cos(y)) i + (sin(x) + xsin(y)) j and C is the curve ’ .A A A
r(t)=7ft2i+ﬂt3jwith0_<_t_<_1. "A * ’1‘ 41‘ .
8. Evaluate Finds whereF=yI+x1+kandSIs 8 surface of the paraboloid z = 1 '$(2  y2, z _>_ 0 and is the
upward pointing unit normal. r k .2:_/; .pS ‘A /‘ A ,
9. Evaluate #F'ndSwhereF=xi+yj+zkandS S is the surface of the solid bounded by z = x2 + y2 and the
plane 2 = 1. we 2 —s A /\ A
10. Evaluate F: dr where F = 22 i + 4x j + 5y k and C is . C _
the intersection of the plane 2 = x + 4 and the cylinder x2 + y2 = 4. C is oriented counterclockwise when viewed from
above. . // ...
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This note was uploaded on 09/15/2009 for the course MATH 234 taught by Professor Dickey during the Spring '08 term at Wisconsin.
 Spring '08
 DICKEY
 Multivariable Calculus

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