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Unformatted text preview: Math 1920, Prelim ||
November 18, 2010, 7:30 PM to 9:00 PM You are NOT allowed calculators, the teat or any other book or notes. SHOW ALL WORK!
Writing clearly and legibly will improve your chances of receiving the maximum credit that
your solution deserves. Please label the questions in you answer booklet clearly. Write your name and section number on each boohlet you use. You may leave when you
haue ﬁnished, but you have not handed in your erram booklet and left the room by 8:45pm,
please remain in your seat so as not to disturb others who are still working. 1. Consider the function f (3:,y,z) = eyln (3:2), and the point P (1,0,1). (a) (5 pts) Find the direction of greatest increase of f at P. 1
(b) (10 pts) Find the linear approximation of the change in f for a step size of — 10
unit length in direction (1, x/E, 3) from P. 2. (10 pts) Find the equation of the tangent plane to the Surface 5' at the point closest
‘ to the plane P, where P and .S' are given by 23: + 4y + z = —8 and z = $2 + yg,
respectively. Note that the plane P and the surface 3 do not intersect. 3. (a) (10 pts) The equation 2:32 + y2 = 3 deﬁnes an ellipse, and for a non-zero constant
k the equation 9:2 — 2y2 : it deﬁnes a hyperbole. Find the value of k such that for each point in the intersection, the tangent to the ellipse and the tangent to
the hyperbola are perpendicular. (b) (5 pts) Determine whether the the foci for the hyperbola you found are the same as
the foci for the ellipse. (Recall that the fool for an ellipse given by a)2+(y/b)2 =
1 are at (to, 0), where c = Va? -— b2 for a 2 b > 0, and for a hyperbola given by
(m/a)2 — (y/b)2 = 1, they are are at (icﬁ), where c = \/a2 + b2.) 4. (10 pts) Take a ball of radius 2 centered at the origin, and remove a smaller ball of
radius 1 centered at (0, O, 1). Set up the integral (in spherical coordinates as an iterated
integral) that computes the ﬁrst moment of this solid about the ryeplane for uniform
mass density 5 = 1. (You don‘t need to evaluate it.) Note that the smaller ball has
the equation p = 2 cos qt). Extra Credit (5 pts) Find the coordinates of the center of mass, the centroid, of
this solid, i. e. not just in terms of an integral. 5. Consider the vector ﬁeld given by F = (1:2 — y)i + (y2 — (a) (10 pts) Find a function f such that F is the gradient vector ﬁeld of f . (b) (10 pts) Find all the critical points of f, and for each one determine whether it is
a local maximum, local minimum or a saddle point. (c) (5 pts) Calculate the work done by F along any path from one critical point to
’ another. (over) 6. Consider the vector ﬁeld where r : M332 + yg. (a) (5 pts) Calculate the divergence of F, div F. (b) (10 pts) For a closed curve C that does not go around the origin, evaluate the
outward ﬂux 3901? - nds. Can Green’s Theorem be used here? Explain your
answer. ((2) (10 pts) For the unit circle C around the origin, ﬁnd the outward ﬂux 3% F - n d8.
Can Green’s Theorem be used here? Explain'your answer. ...
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This note was uploaded on 09/29/2011 for the course MATH 1920 at Cornell University (Engineering School).