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Unformatted text preview: MA1a HW 3 Solutions 10/21/2007 Question 1 Problem 23 Let f be defined as follows: f ( x ) = 2 cos x, x c ax 2 + b, x > c where a, b, c are constants. It b and c are given, find all values of a for which f is continuous at the point x = c . Problem 28 For x = 0, let f ( x ) = [1 /x ], where [ t ] denotes the greatest integer function t . Sketch the graph of f over the intervals [- 2 ,- 1 5 ] and [- 2 ,- 1 5 ]. What happens to f ( x ) as 0 through positive values? through negative val- ues? Can you define f (0) so that f becomes continuous at 0? Problem 30 Same as Problem 28, when f ( x ) = x (- 1) [1 /x ] for x = 0. Solution 1 In Problem 23, clearly as x c from the right, f ( x ) ac 2 + b . On the other hand, both the left limit and the value of the function at c equal 2 cos c . For the function to be continuous, the three quantities must be equal, which happens iff 2 cos c = ac 2 + b . Let us assume c = 0. Then we get a = 2 cos c- b c 2 . If c = 0 then the function is continuous iff b = 2. In that case, a can be any real number. So the values of a for which f is continuous at c are: 2 cos c- b c 2 , c = 0 R , c = 0 and b = 2 , c = 0 and b = 2 For Problem 28 and 30, the image files of the plots are attached. In problem 28,For Problem 28 and 30, the image files of the plots are attached....
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