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Unformatted text preview: 18.02 PRACTICE MIDTERM #2B
BJORN POONEN Please turn cell phones oﬀ completely and put them away.
No books, notes, or electronic devices are permitted during this exam.
Generally, you must show your work to receive credit.
Name:
Student ID number:
Recitation instructor’s last name:
Recitation time (e.g., 10am): (Do not write below this line.) 1 out of 15 2 out of 15 3 out of 15 4 out of 15 5 out of 20 6 out of 20 Total out of 100 1) A particle is moving in the xy plane so that relative to the origin it is rotating counterclockwise at a rate of 2 radians per second while its distance to the origin is increasing at
dy
?
a rate of 10 meters per second. At a time when the particle is at (−4, 3), what is
dt 2) Let f (x, y, z ) = 2xy 2 + z 3 .
(a) (10 pts.) Find the unit vector u in R3 that minimizes the directional derivative Du f
√
at the point (0, 2, −1). √
(b) (5 pts.) Find the value of Du f at (0, 2, −1) in that direction u. 3) Let w = 2x + 3y + 5z where x2 + 2y 2 = z 2 . Evaluate ∂w
∂x at (x, y, z ) = (1, 2, 3).
y π /2 4 (r2 + cos θ) dr dθ. 4) Evaluate
0 0 5) Set up an iterated integral that computes the volume of the bounded region between
the surfaces z = 9x2 and z = 9 − y 2 . (Your answer should include the integrand and the
limits of integration, but do not evaluate the iterated integral.) 6) Use Lagrange multipliers to ﬁnd the maximum value of x3 + y 3 , where x and y range
over nonnegative real numbers satisfying x2 + 2y 2 = 36. This is the end! If you ﬁnish during the last 5 minutes of the exam, please remain in your
seat until the end, and then pass your exams towards the aisles, and continue to wait
silently in your seat until the proctors have collected all exams. Scratch work, or continuation of work on problem number
(Detach and recycle this page if it is not part of your solutions.) Scratch work, or continuation of work on problem number
(Detach and recycle this page if it is not part of your solutions.) ...
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This note was uploaded on 09/14/2011 for the course MATH 18.02 taught by Professor Auroux during the Fall '08 term at MIT.
 Fall '08
 Auroux
 Multivariable Calculus

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