This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 ....
View Full
Document
 Spring '07
 TextbookAnswers

Click to edit the document details