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hw11_solutions

hw11_solutions - Homework#1 Section 1.1 2 Computing first 6...

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Homework #1 Section 1.1 2. Computing first 6 derivatives and evaluating them at we get the answer from the general formula for Taylor series 5. . 9. There are 2 possible ways of solving this problem : one can approximate and use or and use . For the first choice the remainder would be , then . ,
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. Therefore one would get ( ) ( ) ( ) . 10a. We have , so ( ) ( ) ( ) 11a. and , . , . Such remainder could be bounded for all 11b. and , . .
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Remainder cannot be bounded for all because of the division by zero. 12c. and , . Remainder ( ) ( ) , so that . To get desired accuracy one needs to consider large amount of terms which is very impractical. However, change of the interval would allow to get accuracy with much less number of terms. 15. ( ) . Define as the approximation generated by using an m term Gregory series to approximate ( ) and n term Gregory series for . Then we have ( ) ( ) where is the remainder in the Gregory series. Therefore . If we require both expansions to be equally accurate, then we have that
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and
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