M408D Quest Homework 5-solutions

M408D Quest Homework 5-solutions - hernandez (ah29758)...

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Unformatted text preview: hernandez (ah29758) M408D Quest Homework 5 pascaleff (54550) 1 This print-out should have 16 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. 001 10.0 points Determine the degree 2 Taylor polynomial T 2 ( x ) centered at x = 1 for the function f when f ( x ) = radicalbig 8 + x 2 . 1. T 2 ( x ) = 3- 1 3 ( x + 1) + 8 27 ( x + 1) 2 2. T 2 ( x ) = 3 + 1 3 ( x- 1) + 8 27 ( x- 1) 2 3. T 2 ( x ) = 3- 1 3 ( x- 1) + 4 27 ( x- 1) 2 4. T 2 ( x ) = 3 + 1 3 ( x + 1) + 8 27 ( x + 1) 2 5. T 2 ( x ) = 3 + 1 3 ( x- 1) + 4 27 ( x- 1) 2 correct 6. T 2 ( x ) = 3- 1 3 ( x + 1) + 4 27 ( x + 1) 2 Explanation: For a function f the degree 2 Taylor poly- nomial centered at x = 1 is given by T 2 ( x ) = f (1) + f (1)( x- 1) + 1 2 f (1)( x- 1) 2 . Now when f ( x ) = radicalbig 8 + x 2 , f ( x ) = x 8 + x 2 , while f ( x ) = 8 + x 2- x 2 8 + x 2 8 + x 2 = 8 (8 + x 2 ) 3 / 2 . But then f (1) = 3 , f (1) = 1 3 , f (1) = 8 27 . Consequently, T 2 ( x ) = 3 + 1 3 ( x- 1) + 4 27 ( x- 1) 2 . 002 (part 1 of 2) 10.0 points (i) Compute the degree 2 Taylor polynomial for f centered at x = 1 when f ( x ) = x. 1. T 2 ( x ) = 1- 1 4 ( x- 1)- 1 4 ( x- 1) 2 2. T 2 ( x ) = 1- 1 2 ( x- 1) + 1 8 ( x- 1) 2 3. T 2 ( x ) = 1- 1 4 ( x- 1) + 1 8 ( x- 1) 2 4. T 2 ( x ) = 1 + 1 4 ( x- 1) + 1 4 ( x- 1) 2 5. T 2 ( x ) = 1 + 1 2 ( x- 1)- 1 4 ( x- 1) 2 6. T 2 ( x ) = 1 + 1 2 ( x- 1)- 1 8 ( x- 1) 2 correct Explanation: The degree 2 Taylor polynomial centered at x = 1 for a general f is given by T 2 ( x ) = f (1)+ f (1) ( x- 1)+ f (1) 2! ( x- 1) 2 . Now when f ( x ) = x , f ( x ) = 1 2 x , f ( x ) =- 1 4 x x , in which case, f (1) = 1 , f (1) = 1 2 , hernandez (ah29758) M408D Quest Homework 5 pascaleff (54550) 2 while f (1) 2! =- 1 8 . Consequently, T 2 ( x ) = 1 + 1 2 ( x- 1)- 1 8 ( x- 1) 2 . 003 (part 2 of 2) 10.0 points (ii) What estimate does Taylors Inequality provide for the error R 2 ( x ) = x- T 2 ( x ) in using the degree 2 Taylor polynomial T 2 ( x ) you derived in part (i) as an approximation to x on the interval [1 , 1 . 15]? 1. | R 2 ( x ) | 23 . 193 10 5 2. | R 2 ( x ) | 16 . 893 10 5 3. | R 2 ( x ) | 18 . 993 10 5 4. | R 2 ( x ) | 21 . 093 10 5 correct 5. | R 2 ( x ) | 25 . 293 10 5 Explanation: Taylors Inequality says that if T 2 ( x ) is the degree 2 Taylor polynomial for f centered at x = a and if | f (3) ( x ) | M , then the error R 2 ( x ) = f ( x )- T 2 ( x ) satisfies the inequality | R 2 ( x ) | 1 3! M | x- a | 3 . We apply this estimate with f ( x ) = x, a = 1 ....
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M408D Quest Homework 5-solutions - hernandez (ah29758)...

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