sheet4 - In each case below, decide whether it is possible...

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CM221A ANALYSIS I Exercise Sheet 4 1. Evaluate lim x π/ 2 - 0 tan x 1 + tan x and explain why you answer is correct. 2. Prove that the function f ( x ) = tan( x + x 2 ) 1 + x + x 2 is continuous at every x [0 , 1] except for one point, which you should identify. 3. Use the intermediate value theorem to prove that the equation (sin x )(tan x ) = 1 has a solution in the range π/ 4 < x < π/ 3. Prove that the exact solution is x = arccos 5 - 1 2 ! . 4. Sketch the graphs of the functions on the two sides of the equation below. Use the intermediate value theorem to prove that the equation sin x = 1 1 + x 2 has an infinite number of real solutions.
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Unformatted text preview: In each case below, decide whether it is possible for a function f to be 5. continuous and unbounded on [0 , 1); 6. continuous and unbounded on [0 , 1]; 7. continuous on [0 , 1] with the range { f ( x ) : x ∈ [0 , 1] } = (0 , 1); 8. continuous on (0 , 1) with the range { f ( x ) : x ∈ (0 , 1) } = [0 , 1]; 9. continuous on (0 , 1) with the range { f ( x ) : x ∈ (0 , 1) } = [0 , 1] ∪ [3 , 4]. In each case justify your answer by giving an example or by quoting a theorem that shows that no example can exist. 1...
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This document was uploaded on 01/31/2011.

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