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Unformatted text preview: hernandez (ah29758) – M408D Quest Homework 5 – pascaleff – (54550) 1 This printout should have 16 questions. Multiplechoice 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 Taylor’s 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: Taylor’s 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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This note was uploaded on 11/15/2011 for the course M 408 D taught by Professor Textbookanswers during the Spring '07 term at University of Texas.
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