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1271practicemidterm2F09

# 1271practicemidterm2F09 - x = 0 a(15 points If y(0 =-1...

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Name: Signature: Section: Math 1271. Lecture 030 Practice Midterm Exam II There are a total of 100 points on this exam. To get full credit for a problem you must show the details of your work. Answers unsupported by an argu- ment will get little credit. No books, notes, calculators, cell phones or other elecronic devices are allowed. Do all of your calculations on this test paper. Problem Score 1. 2. 3. 4. 5. Total: 1

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Problem 1 Compute the indicated derivatives of the functions y = f ( x ) . It is not necessary to simplify. a. (5 points) f 0 ( x ) where f ( x ) = sin(1 - e x ) x . b.(5 points) y 0 where y = ( x + 3) 10 ( x 2 - 9) 6 ( x - 1) 7 . 2
c.(5 points) f 0 ( x ) where f ( x ) = x arcsin( x ) . d.(5 points) f (7) ( x ) where f ( x ) = 1 2 x + 1 . . 3

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Problem 2 (20 points) Sand falling at the rate of 3 ft 3 / min forms a conical pile whose radius r always equals twice the height h . Find the rate at which the height is changing at the instant when the height is 10 feet. Recall that the volume V of a right circular cone is V = 1 3 πr 2 h . 4
Problem 3 Let y be a function of x such that x 2 y - y 3 = 1 and the deriva-

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Unformatted text preview: x = 0 . a. (15 points) If y (0) =-1 , compute y (0) . b. (5 points) Also compute y 00 (0) . 5 Problem 4 (20 points) Use diﬀerentials to ﬁnd an approximation to (26 . 96) 1 3 . Problem 5 A radioactive material of initial mass 20 milligrams decays to 5 milligrams after 5 years. a. 15 points Find an expression for the radioactive mass remaining after t years. 6 b. 5 points What is the half-life of the material? (Very ) Brief Solutions: 1a. f ( x ) =-xe x cos(1-e x )-sin(1-e x ) x 2 1b. y = ( x + 3) 10 ( x 2-9) 6 ( x-1) 7 ± 10 x + 3 + 12 x x 2-9 + 7 x-1 ² 1c. f ( x ) = arcsin( x ) + x √ 1-x 2 1d. f (7) ( x ) =-2 7 (7!) (2 x + 1) 8 2 dh dt | h =10 = 3 400 π ft ./ min . 3a. y (0) = 0 3b. y 00 (0) =-2 / 3 4. 3-1 / 675 7 5a. m ( t ) = 20( 1 4 ) t/ 5 milligrams, or m ( t ) = 20 e-ln 4 5 t milligrams. 5b. 2.5 years 8...
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