practiceFinal1solutions

# practiceFinal1solutions - Math 1A Spring 2008 Wilkening...

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Math 1A, Spring 2008, Wilkening Sample Final Exam 1 You are allowed one 8 . 5 × 11 sheet of notes with writing on both sides. This sheet must be turned in with your exam. Calculators are not allowed. 0. (1 point) write your name, section number, and GSI’s name on your exam. 1. (3 points) give precise definitions of the following statements: (a) lim x 3 - f ( x ) = 17. ( δ - definition) (b) f ( x ) is continuous at x 0 (c) f ( x ) has an absolute maximum at x 0 over the interval [ a, b ] Answer: (a) For every > 0 there exists a δ > 0 such that: if 0 < 3 - x < δ then | f ( x ) - 17 | < (b) f ( x 0 ) is defined and lim x x 0 f ( x ) = f ( x 0 ) (c) For every x , if a x b then f ( x ) f ( x 0 ) (Assuming that f ( x ) is defined on [ a, b ] and x 0 is in [ a, b ]) 2. (5 points) Evaluate the integral: sinh - 1 (4 / 3) 0 e cosh x sinh x dx Answer: u = cosh x , du = sinh x dx , cosh(0) = 1, cosh(sinh - 1 (4 / 3)) = 1 + sinh 2 (sinh - 1 (4 / 3)) = 1 + (4 / 3) 2 = 25 / 9 = 5 / 3 sinh - 1 (4 / 3) 0 e cosh x sinh x dx = 5 / 3 1 e u du = e u 5 / 3 1 = e 5 / 3 - e

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3. (6 points) Let f ( x ) = sin x x x = 0 , 1 x = 0 . (a) Use the definition of the derivative to evaluate f (0).
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