homework1sol - Math 4650 Homework #1 Solutions 1. 1.1 #2....

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Math 4650 Homework #1 Solutions 1. 1.1 #2. Find intervals containing solutions to the following equations. Solution: For all of these problems, we need to use the intermediate value theorem. So we want to find numbers a and b such that f ( a ) > 0 and f ( b ) < 0; then there is a solution to f ( x ) = 0 for some x between a and b . (a) x - 3 - x = 0. The function is f ( x ) = x - 3 - x . Use trial and error: f (0) = - 1, f (1) = 1 - 1 3 = 2 3 . Since f (0) and f (1) have opposite signs, there is a solution in the interval (0 , 1). (b) 4 x 2 - e x = 0. The function is f ( x ) = 4 x 2 - e x . f (0) = - 1, f (1) = 4 - e = 1 . 3 . . . , so again there is a solution in the interval (0 , 1). (c) x 3 - 2 x 2 - 4 x +3 = 0. The function is f ( x ) = x 3 - 2 x 2 - 4 x +3. We have f (0) = 3, f (1) = - 2, so again there is a solution in (0 , 1). (d) x 3 +4 . 001 x 2 +4 . 002 x +1 . 101 = 0. We have f ( x ) = x 3 +4 . 001 x 2 +4 . 002 x +1 . 101, so that f (0) = 1 . 101, f (1) = 10 . 104, f (2) > 33, etc. We’re going in the wrong direction. In the other direction, we have
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homework1sol - Math 4650 Homework #1 Solutions 1. 1.1 #2....

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