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Unformatted text preview: MATH 241, 241H Fall 2008
Uniform Final Examination
Monday, 12/15/08 Instructions: Number the answer sheets from 1 to 8. Fill out all the information on the top of
@311 sheet. Answer problem 71 on page n, n z 1, . . . ,8. Do not answer one problem on more than
one sheet. If you need more space use the back of the correct sheet. Please write out and sign the
Honor Pledge on page 1 only. SHOW ALL WORK
You May Not Use Calculators, Notes, Or Any Other Form Of Assistance On This Exam 1. (a) (10 pts) Let a = i + 2j — 2k and b 2 2i + 3j — 5k. Find the projection of b onto a. (b)(10 pts) Find the symmetric equations for the line that passes through the point (1,0, 2)
and is perpendicular to the plane 393 + y = 7. 2. The trajectory of a particle is given by 2
r(t) = (sin 275) i + (cos 2t)j + 5 t3/2 k for t2 0. (a) (10 pts) Find the velocity and speed of this particle.
(b) (10 pts) Find the total distance L traveled by the particle for 0 g t S 5. 3. Consider the curve C parametrized by r(t) = (t2 + 2)i + (3t — 3)j + (t2 — 6t)k. (a) (5 pts) Show that the point P = (3, —6, 7) lies on C.
(b) (10 pts) Find an equation of the line through P in the direction of the tangent to C at P. 4. (10 pts each) Consider the function f(:c, y, z) : xyez. (a) Find the gradient of f. (b) Compute the directional derivative Du f at the point (1, 1, 0), where u is the unit vector in
the direction i -— j + 2k. (0) Find a point on the level surface f (x, y, z) = 1 where the tangent plane is parallel to the
p1anex+y+22=3. EXAM CONTINUES ON THE OTHER SIDE CI! Consider the function f(:r, y) : m2y ~ 2mg + 23/2 — 15y.
(a) (15 pts) Find all the critical points of f. (b) (10 pts) Use the second partials test to classify the critical points found in (5a) above as
relative maxima, relative minima, or saddle points. (25 pts) Compute the double integral Iz/R/ydA, where R is the region of the ﬁrst quadrant that is bounded above by the circle :62 + y2 = 1 and
below by the parabola y = 1 — $2. (15 pts each) Consider the vector ﬁeld
F = 2xyzi+ z$2j + ($231+ 1)k. (a) Is it possible to write F = grad f for some function f ? EXPLAIN. If the answer is yes,
ﬁnd this function f.
/ F ' dr ,
c (b) Compute the line integral
k 0 < t < 1 . 1+t2 ’ — — where C is the curve parametrized by r(t) = cos(7rt3) i + t5/3 j + Consider the solid region D that lies above the any—plane and is bounded above by the sphere
m2+y2+z2 = 4, below by the sphere x2 +y2+z2 = 1 and on the sides by the cone 322 = :62 +342. (a) (20 pts) Compute the triple integral
D (b) (15 pts) Compute the ﬂux integral
/ F . n dS ,
2 where F is the vector ﬁeld F = m3 i + y3 j + z3 k, E is the boundary of region D and n is the
unit outward normal vector on E. Hint: You might ﬁnd part (8a) helpful. END OF EXAM - GOOD LUCK! ...
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- Spring '08