126_Sp08_Taylor - sin( x ) 1+ x 2 near b = 0. 3. Working...

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Practice Questions on Taylor Series 1. Finding Taylor polynomials using Taylor’s formula. You are going to need f ( x ), n , b to use in the formula T n ( x ) = f ( b ) + f p ( b )( x - b ) + 1 2! f pp ( b )( x - b ) 2 + 1 3! f ppp ( b )( x - b ) 3 + ... + 1 n ! f ( n ) ( b )( x - b ) n . (a) Find the quadratic approximation for f ( x ) = ln(1 + x ) near b = 0. (b) Find the third Taylor polynomial for f ( x ) = 2 x 3 - 2 x + 1 near b = 0. (c) Find the third Taylor polynomial for f ( x ) = 2 x 3 - 2 x + 1 near b = 1. (d) Find T 3 ( x ) for f ( x ) = x + x near b = 1 and use it to approximate 0 . 9 + 0 . 9. 2. Finding Taylors series or polynomials by manipulating series for sin x , cos x , e x and 1 1 - x . Remember the series for 1 1 - x converges when | x | < 1. Your answers should be of the form n =0 a n ( x - b ) n except for the last one. Also, give the radius of convergence. (a) Find the Taylor series for f ( x ) = 3 5+9 x 3 near b = 0. (b) Find the Taylor series for f ( x ) = sin( x 2 ) cos( x 2 ) near b = 0. (c) Find the ±rst 3 non-zero terms for the Taylor series for f ( x ) =
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Unformatted text preview: sin( x ) 1+ x 2 near b = 0. 3. Working with Taylors Inequality. i. e. the error formula | T n ( x )-f ( x ) | M | x-b | n +1 ( n + 1)! (a) Find an upper bound for the error in approximating e x using the 4 th Taylor polynomial about 0 for x in the interval [-. 2 , . 2]. (b) Find an upper bound for your error in 1(d). (c) Find the second Taylor polynomial for f ( x ) = x 3 + x based at b = 1. Find an interval J around b = 1 such that the error in approximating f ( x ) using T 2 ( x ) is less than 0 . 001 for all x in the interval J . (d) Let f ( x ) = 1 5-x and b = 0. Find an integer n so that the error in approximating f ( x ) with T n ( x ) for x in the interval [-2 , 2] is smaller than 0 . 009. 1...
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