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Unformatted text preview: Math 226
Practice Final Exam I Problem 1. Consider the curve C parametrized by'?’(t) = t
t. 6 [0,2]. (a) (7%) Compute its arc length. (b) (8%) Give the vector parametric equation for the line tangent to C at the point
(1, a? i). Problem 2. Consider the two lines of vector parametric equations "1310) I *5? +
—} —+ —> > > 91: +t(“z:"+ 3‘ “2m and ‘i"2(u):4i —4j +43: +u(2:’— j’— k). (a) (10%) Show that these two lines intersect, ﬁnd their intersection point, and determine the angle they form at their intersection point. (b) (10%) Give the equation of the plane containing these two lines. Problem 3 (15%). Find the point on the paraboloid 1:2 —— y2 — 22 = 0 that is closest to
the point (3, 2, 1). (Hint: Consider the square of the distance from y, z) to (3, 2, 1).) Problem 4 (10%). Consider the surface of equation $33,: — scyz + 22 : 16. At which
points of this surface is the tangent plane horizontal? ' Problem 5 (10%). Find the centroid of the inside of the cardioid, whose equation in
polar coordinates is r z 1 + cos 6'. Problem 6. Let C be the circle of equation 112 + y2 : 9, oriented clockwise. Compute
the line integral fC 1,12 d3: + (2a m 3y)dy (a) (8%) directly, using the deﬁnition of line integrais. (b) (7%) using Green’s theorem. Problem 7. Compute the ﬂux of the vector ﬁeld y, z) z +22?> across the 'spl'iere Sr of radius 2 centered at the origin, w'ith'ﬁaspect to the outer"finit—normal vector— '1??? (a) (5%) using the divergence theorem. (b) (10%) directly, without using the divergence theorem. (Hint: Consider the upper
hemisphere 3+ of equation 2 2 «J4 — 332 — y? and the lower hemisphere 8— of equation 2: : —x/4 — 31:2 * yz.) Math 226
Practice Final Exam II Problem 1 (15% total) Let C be the curve in space parametrized by F(t) 2 (t cos t) 3+
(t sin t) 5+ 25352 W2 E, where 0 g t g 7r. a) (7%) Compute the arc length of C.
b) (8%) Give the vector parametric equation for the tangent line to C at the point
(0, %7r, 34;??3/2). Problem 2 (20% total) Consider the points P 2 (1,0,1), Q 2 (—1,0, —2), R 3 (2,1,1)
in space, let (31 denote the line containing P and Q, and let £2 denote the line containing
P and R. a) (10%) Give parametric equations for the lines 61 and £2} and determine the angle they
make at P. b) (10%) Give the equation of the plane containing 61 and £2. Problem 3 (10%) Consider the surface S of equation yz — in + z3 = 6. At which points
of S is the tangent plane horizontal? Problem 4 (15%) Find the absolute maximum and minimum of the function f(:1:, y) =
2:122 — 43: + :92 + 43; — 1 on the triangular region in the ﬁrst quadrant bounded by the lines
m:0,y=2,y:2m. Problem 5 (10%) Find the centroid of the planar region inside the cardioid r = a(l+cos S)
and outside the circle 1" z a. Problem 6 (15% total) Let C be the circle of equation 3:2 + y2 z 16 oriented counter
clockwise. Compute the circulation IC 13  (1?, where F(:c: y) : (:1: —— y)i+ difggtly as anhﬂgdgltggral‘. _._._.._._—_— . . . b) (7%) using Green’s theorem. Problem 7' (15% total) Compute the ﬂux of F(w,y, z) = 2yf— 2mjf—l— 2mg}; across the
surface bounded above by the sphere of radius 2 centered at the origin and below by the plane 2 z 0
a) (5%) using the divergence theorem. b) (10%) directly as a surface integral. ...
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 Spring '07
 KAMIENNY
 Calculus

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