# exam 2 - 3. (6 points) Use diﬀerentials to estimate the...

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Name: Math 211A Fall 2005 Exam # 2 Honor Code Write and sign the honor code here after you have ﬁnished the exam. There will be no calculators allowed during the exam. You may use a 3x5 inch note card. Use full sentences on all explanations and put a box around your ﬁnal answer. Good Luck!! 1. True or False (2 points each) a. The graph of r ( u, v ) = < cos u, cos v, sin( uv ) > is a space curve. b. D ~u f = f · ~u . c. If ρ ( x, y ) is the density of a lamina R at the point ( x, y ), then the center of mass of R is given by Z Z R ρ ( x, y ) dA . d. Diﬀerentials are used to ﬁnd the exact amount that a measurement is oﬀ by. e. The graph of the function f ( x, y, z ) = x 2 + y 2 - z is a 1 dimensional object in 3 dimensional space.

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2. (10 points) Calculate lim ( x,y ) (0 , 0) y 2 x 2 + y 2 + sin 2 x y 2 + sin 2 x . If the limit does not exist, explain why. If the limit does exist, show why it converges to your answer.

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Unformatted text preview: 3. (6 points) Use diﬀerentials to estimate the amount of tin in a closed tin can with diameter 8cm and height 12 cm if the tin is .04 cm thick. 4. (6 points) Calculate the directional derivative of f ( x, y ) = x 2 y in the direction of < 3 , 4 > at the point (1 , 2). 5. (6 points) Show that f ( x, y ) = xy is a solution to the diﬀerential equation ∂ 2 f ∂x 2 + ∂ 2 f ∂y 2 = 0 . Bonus: (3 points) Find a function f so that f xx 6 = 0 and f satisﬁes LaPlace’s equation. 6. (8 points) For Z 1 Z 4 4 x f ( x, y ) dy dx sketch the region of integration and change the order of the integration. Bonus: (4 points) Evaluate the integral Z 1 Z y 3 y e x 2 dx dy....
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## This test prep was uploaded on 04/12/2008 for the course MATH 211 taught by Professor Crow during the Fall '07 term at Gettysburg.

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exam 2 - 3. (6 points) Use diﬀerentials to estimate the...

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