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Unformatted text preview: 18.02 PRACTICE MIDTERM #1B
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 10 2 out of 10 3 out of 20 4 out of 20 5 out of 15 6 out of 25 Total out of 100 1) Let A, B , and P be points in space such that P is on the line segment AB and the
distance AP is twice the distance BP . Find the position vector P in terms of the position
vectors A and B . T
0 1 −2
654
3 0 −4 ?
321
5 −6 0 2) What is the upper right entry of 3) Let a be a nonzero vector. What is the geometric shape formed by the set of points P
in space whose position vector P satisﬁes P × a = 1? 4) Show that the line given in parametric form r(t) = 3 + 2t, 5t, 7 − 6t and the plane
2x − 2y − z = 6 are parallel, and ﬁnd the distance between them. 5) Estimate the value of ln(x2 + y ) at the point (0.4999, 0.7502). 6) Find all critical points of the function f (x, y ) := x4 + xy + y 4 and decide whether they
are local maxima, local minima, or neither. 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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