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Unformatted text preview: 18.02 PRACTICE MIDTERM #1A
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 25 4 out of 20 5 out of 25 Total out of 100 1) Find the equation of the tangent plane to the graph of f (x, y ) := x2 y − 3y 2 at the point
where x = 2 and y = 1. 2) Given that x1 , x2 , x3 , y1 , y2 , y3 are real numbers satisfying the conditions
x2 + x2 + x2 = 4
1
2
3
2
2
2
y1 + y2 + y3 = 9, what is the range of possible values of
x1 y1 + x2 y2 + x3 y3 ? 3) Let f (x, y ) = x3 + 3xy − y 3 .
(a) (15 pts.) Find all points (if any) at which f (x, y ) has a local minimum.
(b) (10 pts.) Find all points (if any) at which f (x, y ) is at its global minimum. (Explain
how you know that your answer is correct.) 4) Given that a, b, c, d, e, f, g, h, i are real numbers such that 120
abc
100
3 0 4 d e f = 0 1 0 ,
056
ghi
001
ﬁnd b. 5) What is the distance between the two lines in R3 given in parametric form by the
equations r1 (t) = ti + (3t + 8)j and r2 (t) = (t − 5)i + 2tk? 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
 Real Numbers, Multivariable Calculus, student ID number, BJORN POONEN

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