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Third page 4 3 12 pts use eulers method with n 3 to

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Third Exam–April 14, 2009 Page 4 3. [12 pts] Use Euler’s method with n = 3 to obtain an approximate solution to the initial value problem y = cos ( x - y ) , y (0) = 0 over the interval [0 , π ].
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Math 122 Third Exam–April 14, 2009 Page 5 4. [10 pts] Find the 3rd Taylor polynomial of f ( x ) = e - x at x = 1 and use it to approximate the value of e - 1 2 .
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Math 122 Third Exam–April 14, 2009 Page 6 5. Let f ( x ) = 3 e - 3 x . (a) [8 pts] Verify that f ( x ) is a probability density function on [0 , ). (b) [8 pts] Find P ( x 1). (c) [8 pts] Find the expected value of the random variable x associated with this probability density function.
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Math 122 Third Exam–April 14, 2009 Page 7 6. [10 pts] Suppose that the function f ( x, y ) = kx y is a joint probability density function on the region D = { 0 x 2; 0 y 4 } where k is a constant. (a) Find the value of k . (b) Find the probability that both x and y are less than 1.
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Math 122 Third Exam–April 14, 2009 Page 8 7. [8 pts] Suppose that f ( x ) is a continuous probability density function on the interval [0 , ), 0 xf ( x ) dx = 10, and 0 x 2 f ( x ) dx = 125. What are the variance and the standard deviation of the random variable x ? 8. [6 pts] Find the general term of the sequence 2 2 , 4 5 , 6 10 , 8 17 , 10 26 , · · · .
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