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Unformatted text preview: MATHEMATICS 182 FINAL EXAM (20 pts) 1) Let f(x, 3/, z) = m2 cosy + 22.
a) Find Vf at (1,0,1).
b) Find the directional derivative of f in the direction of l7 1: 2k. c) In What direction does f change most rapidly? (20 pts) 2) Given the points P1 = (1,2,1), P2(2, 2, 3), and P3 = (2, —1, 1) ﬁnd
a) the equations of a line through P1 and P2,
b) the equation of a plane through P1, P2, and P3. 82w 82w
(15PtS) 3) Ifw=m2+y2, x=r—s,andy=r+sﬁndw+w (25 pts) 4) a) Show that the vector ﬁeld
F = (z + cosy)z' + [—xsiny)j + (x + z2)k
is conservative. b) Find f such that F = V f . c) What isfFOdRifC’isthecurvemztB‘, y=7rt2, z=t, Ogtgl.
C (20 pts) 5) a) Find parametric equations for the cone
3(m2+y2)=22 , ogzgﬁ b) Find the surface area of the cone. (20 pts) 6) a) Use the divergence theorem to evaluate / o N)d0 where l7 = 311713 —— yj — zk s
and S is the sphere m2 + 1/2 + z2 = 9. b) Use Stoke’s Theorem to evaluate /(V X o Nda if I7 = mzyzz' and S is the 3
upper half of x2 + y2 + 22 = 9. (20 pts) 7) Find the mass of the paraboloid z = 1 —- x2 — yz, z 2 0, if the density f = 332. (25 pts) 8) If x 2 cost, y = sin t, and z 2 4t ﬁnd a) the unit tangent vector as a function of t, ds b) a as a function of t, dT‘ as a function of t.
ds c) the curvature If, 2 (20 pts) 9) a) Find all critical points of
f(% y) = 9:63 + 213/3 - 4962/ b) Decide which of the critical points are maxima, minima, at saddle points. (157Pts) 10) a) If a twice differentiable vector ﬁeld 17 = v, 2' +712 j +123]: is the gradient of a scalar
function f, l7 = Vf, Show V X l7 =' 0. —l _) —;
b) If / V - dR depends on the endpoints only and if V is twice differentiable show
0 l7 = V f for some scalar function f. ...
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This note was uploaded on 09/15/2011 for the course MATH 182 taught by Professor staff during the Fall '09 term at Purdue.
- Fall '09