practiceFinal1solutions

practiceFinal1solutions - Math 1A Spring 2008 Wilkening...

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Unformatted text preview: 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 (c) f ( x ) has an absolute maximum at x 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 ) is defined and lim x → x f ( x ) = f ( x ) (c) For every x , if a ≤ x ≤ b then f ( x ) ≤ f ( x ) (Assuming that f ( x ) is defined on [ a, b ] and x is in [ a, b ]) 2. (5 points) Evaluate the integral: Z sinh- 1 (4 / 3) e cosh x sinh x dx Answer: u = cosh x , du = sinh x dx , cosh(0) = 1, cosh(sinh- 1 (4 / 3)) = q 1 + sinh 2 (sinh- 1 (4 / 3)) = p 1 + (4 / 3) 2 = p 25 / 9 = 5 / 3 Z sinh- 1 (4 / 3) e cosh x sinh x dx = Z 5 / 3 1 e u du = e u 5 / 3 1 = e 5 / 3- e 3. (6 points) Let f ( x ) = sin x x x 6 = 0 , 1 x = 0 ....
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This note was uploaded on 09/24/2009 for the course MATH 1A taught by Professor Wilkening during the Spring '08 term at Berkeley.

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practiceFinal1solutions - Math 1A Spring 2008 Wilkening...

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