8 3 4 3 4 3 4 3 4 4 3 4 4 4 4 4 3 4 1 4 16 1 2 3 2 x

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Nature of Mathematics
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Chapter 11 / Exercise 4
Nature of Mathematics
Smith
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= 8 @ 3 4 0 3 4 3 4 3 4 4 3 4 4 4 4 4 3 4 1 4 16 0 0 1 2 3 2 , , ... x R 3 64 4 0 4 ! 1 2 3 4 5 6 = ! ! = ... 2. f x x c a b f x x x x x x x x x x x R n n n n ( ) = ! = = = ( ) = ! = ! % & ( ) * = ! ( ) ! ( ) + ( ) + ( ) + ( ) + [ ] ! + + + + [ ] = ! = 8 = 8 @ @ 5 2 1 0 5 2 1 2 5 2 1 5 5 5 2 5 2 2 2 2 5 1 2 4 8 5 0 0 0 1 2 3 2 4 , , . ... ... . ! = 0 5 . 3. f x x c a b c f x x x x x x x n n ( ) = ! ! = = ! = = ( ) = ! ! = ! ! % & ( ) * ! ! % & ( ) * + ! ! % & ( ) * + ! ! % & ( ) * + ! ! % & ( ) * + 1 2 3 3 = 8 @ 1 3 6 1 3 6 1 3 1 3 6 3 1 3 6 3 6 3 6 3 6 3 0 0 1 2 3 , , , ... 4 5 6 6 ! ! ( ) + ! ( ) ! ! ( ) ! 1 2 3 3 4 5 6 6 = ! = 3 4 1 6 3 6 9 6 27 3 6 3 2 3 x x x R ... 4. f x x c x x a b c f x x x x x x x n n ( ) = ! = ! ! = ! ! = ! = = ! ( ) = ! ! = + % & ( ) * + % & ( ) * + + % & ( ) * + + % & ( ) * + + % & ( = 8 @ 3 4 1 3 4 3 4 3 4 1 3 4 3 4 1 5 3 4 1 5 1 5 1 5 1 5 0 0 1 2 , , , Hint : ) * + 1 2 3 3 4 5 6 6 + + ( ) + + ( ) + + ( ) ! 1 2 3 3 4 5 6 6 = + = 3 2 3 3 4 1 1 5 1 25 1 125 4 1 5 ... ... x x x R 5. f x x x x c x x x x x a b x x x x x x R a b x x n n n ( ) = + ! = + ! = + + ! = = ! + = ! % & ( ) * + = ! + ! + = = = ! = ! % & ( ) * = 8 = @ 3 2 0 3 2 2 2 1 1 2 2 2 2 2 2 2 1 2 4 8 2 1 1 1 1 1 2 2 0 2 3 , , : ... , : Hint :use partial fractions 0 2 3 2 2 3 2 1 1 1 3 2 3 2 3 4 9 8 8 @ ! = ! ! ! ! = + ! = ! ! ! + n x x x x R x x x x x x ... ... 6. f x x c f x x a b c f x x x x x x x f n n ( ) = + = ( ) = + = = ! = ( ) = + = ! % & ( ) * ! % & ( ) * + ! % & ( ) * + ! % & ( ) * + ! % & ( ) * + = 8 @ 4 4 0 4 4 4 4 0 4 4 4 4 4 4 4 4 4 2 0 0 1 2 3 , , , ... :Hint : Start with x x x x R f x x x x ( ) = ! + ! + = ! ! = ( ) = ! + ! + 1 4 16 64 4 0 4 1 4 4 16 64 2 3 2 2 6 ... ...
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Nature of Mathematics
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Chapter 11 / Exercise 4
Nature of Mathematics
Smith
Expert Verified
© BC Solutions - 217 - !""#$%" to post on Internet J%G")* %.> L%’"%6*-. I#*-#+ 1 2"%++3)*4 We now go back to where this section started. We can now generate a procedure for deriving a power series for a function that has derivatives of all orders. =#?-.-(-). )? J%G")* %.> L%’"%6*-. I#*-#+ If a function f has derivatives of all orders at x = c , then the series f c n x c f c f c x c f c x c f c x c f c x c n n n n n n ( ) = 8 ( ) ( ) ! ( ) = ( ) + $ ( ) ! ( ) + $$ ( ) ! ( ) + $$$ ( ) ! ( ) + + ( ) ! ( ) @ ! ! ! ! ... ! 0 2 3 1 2 3 is called the Taylor series for f x ( ) . If c = 0, then the series is called the Maclaurin series for f . In reality, this is the same result we reached in our study of Taylor polynomials, but we can extend the pattern indefinitely. Example 1) dse the function f x x ( ) = sin to find the Maclaurin series. Dirst, we want to write f x x ( ) = sin using the formula above: f n x f f x f x f x f x n n n ( ) = 8 ( ) ( ) = ( ) + $ ( ) + $$ ( ) + $$$ ( ) + ( ) + @ 0 0 0 1 0 2 0 3 0 4 0 2 3 4 4 ! ! ! ! ! ... ( ) We need successive differentiation of f x x ( ) = 1#" : f x x f f x x f f x x f f x x f f x x f f x x f ( ) = ( ) = $$$ ( ) = ! $$$ ( ) = ! $ ( ) = $ ( ) = ( ) = ( ) = $$ ( ) = ! $$ ( ) = ( ) = ( ) = ( ) ( ) ( ) ( ) sin cos cos sin sin cos 0 0 0 1 0 1 0 0 0 0 0 4 4 5 5 1 So we can write f x x x x x x ( ) = = ! + ! +… sin 3 5 7 3 5 7 ! ! ! We can tell by the ratio test that this series converges for all values of x . Ketgs do so. f x x x n x n n x x n n n n n n n n ( ) = = ! ( ) ! ( ) + ( ) " ! ( ) = + ( )( ) = + ! = 8 .8 + ! @ sin lim 1 2 1 2 1 2 1 2 1 2 0 1 2 1 1 2 1 2 1 2 ! ! ! y x = sin ! x x x = ! 3 3 sin ! ! x x x x = ! + 3 5 3 5 Note how as n increases, the graph of P n more closely resembles the sine function.
© BC Solutions - 218 - !""#$%" to post on Internet R6->#"-.#+ ?)* ?-.>-.$ % J%G")* I#*-#+ 1. Differentiate f x ( ) several times and evaluate each derivative at c . f c f c f c ( ) $ ( ) $$ ( ) , , , etc. and try to recogniCe a pattern to these numbers. 2. dse the se>uence developed in step 1 to form the Taylor coefficients f c n n ( ) ( ) ! and determine the interval of convergence for power series f c f c x c f c x c f c x c f c x c n n n ( ) + $ ( ) ! ( ) + $$ ( ) ! ( ) + $$$ ( ) ! ( ) + + ( ) ! ( ) + ( ) 1 2 3 2 3 ! ! ! ... ! ... Example 2) Dind the Maclaurin series for f x x ( ) = sin 3 . To take successive derivatives is a royal pain. Try it.

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