Unformatted text preview: UHIZHIZUUIJ 15:12:! bllﬁb4224lb LILJH Mn'ﬂ'ulN LlHHIﬂuH‘r’ I‘n'ﬂ'ulsilz tilla'riill Department of Mathematics, University of California, Berkeley Math 53
Alan Weinstein, Fall 2005
Second Midterm Exam, Tuesday, November 8th 2005 Instructions. BE SURE TO WRITE YOUR NAME AND YOUR. GSI‘S NAME ON YOUR
BLUE BOOK. Read the problems very carefully to be sure that you understand the statements.
All work should be shown in the blue book; writing should be legible and clear, and there should
be enough work shown to justify your answers. Indicate the ﬁnal answers to problems by
circling them. [Point values of problems are in square brackets. The total point value is 45, for
15% of your course grade] PLEASE HAND IN YOUR SHEET OF NOTES ALONG WITH YOUR BLUE BOOK. YOU
SHOULD N 0T HAND IN THIS EXAM SHEET. 1. [12 points] All that I‘m going to tell you about the differentiable function f on R2 is that its
gradient Vector at (2. 3) is 4i —j, and that f(2, 3) = 7’. AnsWer those of the following questions for
which enough information is given, and explain why not enough information is given for the others.
(a) Is (2, 3) a local maximum or minimum point (or neither) for f? Us) Find the best approximation you can for f (2.04, 2.99). (c) There is one point on the graph of f where you have enough information to ﬁnd the equation
of the tangent plane. Write down the (three) coordinates of that point, and the equation of that
tangent plane. (cl) Is f(2,4) greater or less than 7? (c) Find g’(l), where g{t) = f(2t2,t3). 2. [10 points]
(a) Sketch the region of integration D for the following iterated integral. 1 l
f f V 3:3 +1drody
D W (1:) Evaluate the integral by reversing the order of integration. (c) Find an integer n such that the average value of the function a." + 1 over the region I? lies
between H. and 11+ 1. 3. [13 points] Let A(q) be the area of the region He deﬁned by a: 3 0, y 2 D, and V’E + VIE? 5 q.
[3») Find 9» number 1’01 dEPEnding on q, so that the transformation a: = pa, 3,! = p’v takes the region S
in the (11.,e)plane deﬁned by a E 0, v E O, and «3% + ﬂ 1: .1. to the region R, in the (e, 3;)aplane
(b) Use the transformation in (a) to write A(q) as a double integral over the region S. (c) Using your result in (b), tell what happens to .403) when :3 is doubled.
(d) Use the change of variables a: = r2, 3; = .92 to ﬁnd AM). 4. [10 points] (a) A rectangular box with. no top is to hare a surface area of 48 square meters. Use the method of
Lagrange multipliers to ﬁnd the dimensions which maximize the volume. (h) Extra credit. Using the fact that a cube maximizes volume among all rectangular paral—
lelepipeds with a given surface area, solve part (a) without using any calculus. ...
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