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3sol - Assignment 3 Solution 1 a 0.12 x T x = b 0.16 x...

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Assignment 3 Solution 1. 000 , 20 000 , 20 ) 20000 ( 16 . 0 12 . 0 ) ( > + + = x x x b x a x T i) We want 0 ) ( lim 0 = + x T x 0 12 . 0 lim 0 = + x a x 0 ) 0 ( 12 . 0 = + a 0 = a ii) 000 , 20 000 , 20 ) 20000 ( 16 . 0 12 . 0 ) ( > + = x x x b x x T For T ( x ) to be continuous at x = 20,000 we must have: ) 000 , 20 ( ) ( lim ) ( lim 000 , 20 000 , 20 T x T x T x x = = + Now, 2400 ) 000 , 20 ( 12 . 0 12 . 0 lim ) ( lim 000 , 20 000 , 20 = = = x x T x x b b x b x T x x = + = + = + ) 20000 20000 ( 16 . 0 ) 20000 ( 16 . 0 lim ) ( lim 000 , 20 000 , 20 2400 ) 000 , 20 ( 12 . 0 12 . 0 lim ) 000 , 20 ( ) ( lim 000 , 20 000 , 20 = = = = x T x T x x So, b = 2400 and 000 , 20 000 , 20 ) 20000 ( 16 . 0 2400 12 . 0 ) ( > + = x x x x x T Why is it important for these two conditions to hold? 0 ) ( lim 0 = + x T x tells us that people with very little to no income should be paying small amounts to no taxes. If a person has zero income, his/her taxes should be zero too.
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A person who makes 20,000 should not be exempted from taxes so T ( 20000 ) has to be defined. Secondly as income approaches 20, 000, the tax liability should be close to the tax liability for an income of 20,000. This is same as saying ) 000 , 20 ( ) ( lim ) ( lim 000 , 20 000 , 20 T x T x T x x = = + or that T ( x ) should be continuous at x = 20,000.
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