Unformatted text preview: h4aﬂ1241-—anﬂiExaniJub/25,2008 Directions: Please write your answers and steps neatly. You are allowed 90 minutes to ﬁnish them.
No calculator is allowed.
(1) Let L1 be the line with equation “—gl- = % = z~§ and let L2 be the line with equation
:c=1-—2t, y=t, z=-1+3t.
(a) (10) Find the angle between L1 and L2.
(b) (10) In fact these two lines lie in the same plane, ﬁnd the equation of the plane.
(2) Suppose the position function of a particle at any time t is given by .. n t3~
ﬂnzmmaﬁ+§h (a) (15) Find the velocity, acceleration, and speed of particle at any time t.
(b) (10) Find the total distance travelled by the particle in the time interval 0 g t S 1.
(c) (10) Find the curvature n of the trajectory at any time t. (3) Suppose that the differentiable function f (x, y, z) = 3:2 + 4mg + 3/2 + 3z,
(a) (10) Find the directional derivative of f at (1,1,2) in the direction of the vector a=f+i+2E
(b) (10) Suppose that x = (5+t)/2, y = ——s+2t, z = s2+t2. Find 8f/85 and Bf/Bt
when (s,t) = (1,1).
(4) (20) Compute
‘ / 2y3dx + ($4 + 6y2x)dy,
C where C is the boundary of the region in the ﬁrst quadrant bounded by y = 0, as = 0
and x4 + y4 = 1, oriented counterclockwise. (5) (15) Evaluate f0 yzdx + (1 — x)3dy+ zdz Where C is parametrized by F(t) = (1 — t);+
t;+ t2]; on the interval [0, l]. (6) (20) Evaluate ffg curl F - fidS', where F(x,y,z) = y5— xj+ z]: and where Z is the
portion of paraboloid z = —1 + x2 + 3/2 below the plane 2 = 0. (E oriented by the
normal directed downward) (7) Let D be the solid region bounded above by the sphere x2 + y2 + 22 = 9 and below by
the nay—plane. Let 2 be the boundary of D.
(a) (12) Evaluate fffD de.
(b) (8) Compute ffz F ~ fidS, where F(:I;,y, z) = 6yi— 12xzi+ 4221—5 and the normal
vector 75’ is directed outward. ...
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- Spring '08