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Unformatted text preview: 3:7;PE -50} MATHEMATICS 182 FINAL EXAM 1) Given the points P1 = (1,1,1), P2 = (2, 3, 1), and P3 = (2, —1, 2) ﬁnd 2) 3) a) The equation of the line through P1 and P2,
b) The equation of the plane through P1, P2, and P3. 0) The intersection point of the line of part a) with the y — 2 plane. a) If 2 is deﬁned as a a function of ac and y by the equation asz + yz3 — 2503/ = 0 ﬁnd
82 g at (1, 1, 1).
_1 82 Bz
b) If z = ln(w) and w = (V?) +3)(tan u) ﬁnd 51—; and the a when u = 1 and
v = ~2.
a) Show that the vector ﬁeld F = (z2 + cos y)z' + (—x sin y)j + (2562 + cos 7%) k is
conservative. b) Find f such that 13" = V f . .. t .
c) What is/F'dRifC'is the curvexzcost y: E z=smt, ogtgzm?
o 4) If a curve is parametrized by at 2 cost y = sint z 2 4t, —00 < t < 00, ﬁnd _, a) The unit tangent vector, T, as a function of t, b) E as a function of t,
c) K = d— as a function of t.
s d) N7 as a function of t. e) E as a function of t. 5) If f(m, y, z) = $23; + xyez ﬁnd b) the direction in which f changes most rapidly at (1,2,0),
c) the tangent plane and normal line to $023; + xyez = 4 at (1,2,0). 3
6) a) Find all critical points of 9m3 + 3% — 4mg. b) Which of the critical points are maxima, minima, saddle points. 7) If f(a:,y) = ysinx ﬁnd
a) the values of fan, fmy, fyy at (0,0)
b) the quadratic approximation of f. 8) a) Parametrize the part of the spherical surface :32 + 3/2 + z2 = 9 over the cone
2 = (/5132 + 3/2. b) What is the surface area of the surface of a) 9) Set up but do not evaluate integrals for the following:
a) The area inside the cordiod r = 2(1 + sin 0) and outside the circle r = 1, b) The mass of the tetrahedron with corners (0,0,0), (1,0,0), (0,2,0), (0,0,2) if
the density 6 = my. c) The volume between the cylinders 962 + 3/2 = 4 and $2 + y2 = 1 inside the sphere
x2 + y2 + z2 = 9. 10) a) Use Stoke’s theorem to evaluate /(V X 13“) - Nda if 1-7l : 2z2' + 3117' + 53119 and S S
is the part of the paraboloid z = 4 — 3:2 ~ y2 above the a: — 3/ plane. b) Use Green’s Theorem to ﬁnd the area inside the ellipse 2 2
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- Fall '09