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Unformatted text preview: 18.02 PRACTICE MIDTERM #2A
BJORN POONEN Please turn cell phones oﬀ completely and put them away.
No books, notes, or electronic devices are permitted during this exam.
The exam runs from 1:05pm to 1:55pm (50 minutes).
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 20 4 out of 15 5 out of 15 6 out of 20 Total out of 100 1) A particle is moving in the plane, and w = w(x, y ) is a function of the position of the
dw
dx
= 6, and
= 8. Given that w
particle. At a particular instant, x = 2, y = 3,
dt
dt
dy
at that same
(computed with respect to x and y ) at the point (2, 3) equals 5, 7 , ﬁnd
dt
instant. 2) Let P be the point on the curve x2 − y 3 = 3.1 such that P is closest to (2, 1). Use a
gradient to estimate the coordinates of P . 3) Given that x, y, t are dependent on u, v , express ∂x
∂t in terms of xu , xv , yu , yv , tu , tv .
v π 4 4) Evaluate r sin θ dr dθ.
0 0 5) A metal plate is in the shape of a quarter of a circle of radius 20 cm. Its density in
g/cm2 at a distance r centimeters from the center of the circle is 10r + 3. Express its total
mass in grams in terms of a double integral but do not evaluate the integral. 6) Use Lagrange multipliers to ﬁnd the minimum and maximum values (if they exist) of
x + y + z , where x, y, z range over positive real numbers satisfying x2 y 3 z 5 = 22 33 55 . 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. ...
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 Fall '08
 Auroux
 Multivariable Calculus

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