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# Exam_exam_2_ - C such that f x = Ce kx 4(20 points[7 points...

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MIDTERM 2 — MATH 141 — FALL 2009 — BOYLE No books, no notes, no calculators. Show work. Use exactly one page for each question (use the back of the page if needed). Put your name, your TA’s name and the question number on each page. Put a box around the ﬁnal answer to a question. 1. (20 points) Compute the following integral Z - 2 x = - 5 1 x 2 + 4 x + 13 dx . 2. (a) (7 points) Suppose f ( x ) = x 4 + 3 x + 2sin( x ) and f - 1 denotes the inverse function of f . What is ( f - 1 ) 0 (0)? (b) (10 points) Give a formula for tan(arcsin( x )) in terms of x (without using trig or inverse trig functions). (c) (8 points) Given g ( x ) = log 5 ( x 2 ), compute g 0 (3). 3. (10) Suppose y = f ( x ) is a solution to the diﬀerential equation dy/dx = ky , where k is a constant. Prove that there exists a constant
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Unformatted text preview: C such that f ( x ) = Ce kx . 4. (20 points) [7 points for (a), 7 points for (b), 6 points for (c).] Compute the following limits. The possible answers are a real number, + ∞ ,-∞ , or DNE (does not exist). ( a ) lim x → + ∞ arcsec( x ) ( b ) lim x → + x ln( x ) ( c ) lim x → + cos( x ) 1-sin( x ) 5. (15 points) Solve the following initial value problem. Express y as a function of x with an appropriately simple formula. dy dx = 2 x p 1-y 2 , y (0) = 1 6. (15 points) Solve the following initial value problem. Express y as a function of x with an appropriately simple formula. x dy dx + y = 2 , y (1) = 5 . Typeset by A M S-T E X 1...
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