midterm2

# midterm2 - 1(6 1 and(4 3 in the xy-plane The density of the...

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18.02 MIDTERM #2 BJORN POONEN October 21, 2010, 1:05pm to 1:55pm (50 minutes). 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 leader’s last name: Recitation time (e.g., 10am): (Do not write below this line.) 1 out of 20 2 out of 15 3 out of 20 4 out of 20 5 out of 25 Total out of 100

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1) Suppose that the function T = x 2 + 2 xy + 2 y 2 gives the temperature at each point of the plane, and that one starts at the point (2 , 1). (a) (10 pts.) In which direction should one go to obtain the most rapid increase in T ? Express your answer as a unit vector. (b) (10 pts.) Approximately how far in that direction should one go to get an increase of 0 . 1 in T ?
2) Let R be the circle of radius 2 centered at (0 , 0). Evaluate ZZ R ( x 2 + y 2 ) 2 dA . (Write your answer in simplest form.)

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3) A metal plate is in the shape of a triangle T with vertices at (2

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Unformatted text preview: , 1), (6 , 1), and (4 , 3) in the xy-plane. The density of the plate at each point is given by δ ( x,y ) = x + y . Write down an iterated integral whose value is the moment of inertia of the plate when it is rotated around the y-axis. Do not evaluate the integral! 4) Given that variables w,x,y,z satisfy the constraints w = xyz and w 2 + z 2 = 13, ﬁnd the value of ± ∂w ∂x ² y when w = 3, x = 3, y = 1 / 2, and z = 2. 5) Use Lagrange multipliers to ﬁnd the point(s) on the curve x 4 + y 4 = 32 closest to the origin. (Make sure to show that you have checked everything that you need to check.) This is the end! 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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midterm2 - 1(6 1 and(4 3 in the xy-plane The density of the...

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