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Homework 14 - tgo72 Homework 14 Gompf(58370 This print-out...

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tgo72 – Homework 14 – Gompf – (58370) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 (part 1 of 4) 10.0 points Suppose T 4 ( x ) = 7 - 3( x - 4) + 6( x - 4) 2 - 2( x - 4) 3 + 7( x - 4) 4 is the degree 4 Taylor polynomial centered at x = 4 for some function f . (i) What is the value of f (4)? 1. f (4) = 9 2. f (4) = - 8 3. f (4) = - 7 4. f (4) = 8 5. f (4) = 7 correct Explanation: Since T 4 ( x ) = f (4) + f (4)( x - 4) + f ′′ (4) 2! ( x - 4) 2 + f (3) (4) 3! ( x - 4) 3 + f (4) (4) 4! ( x - 4) 4 , we see that f (4) = 7 . 002 (part 2 of 4) 10.0 points (ii) What is the value of f (3) (4)? 1. f (3) (4) = - 2 3 2. f (3) (4) = 12 3. f (3) (4) = 2 3 4. f (3) (4) = - 2 5. f (3) (4) = - 12 correct Explanation: Since T 4 ( x ) = f (4) + f (4)( x - 4) + f ′′ (4) 2! ( x - 4) 2 + f (3) (4) 3! ( x - 4) 3 + f (4) (4) 4! ( x - 4) 4 , we see that f (3) (4) = - 3! × 2 = - 12 . 003 (part 3 of 4) 10.0 points (iii) Use T 4 to estimate the value of f (4 . 1). 1. f (4 . 1) 6 . 8587 2. f (4 . 1) 6 . 7587 correct 3. f (4 . 1) 6 . 6587 4. f (4 . 1) 6 . 9587 5. f (4 . 1) 7 . 0587 Explanation: Since T 4 ( x ) is an approximation for f ( x ) we see that f (4 . 1) 7 - 3 10 + 6 10 2 - 2 10 3 + 7 10 4 . Consequently, f (4 . 1) 6 . 7587 . 004 (part 4 of 4) 10.0 points
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tgo72 – Homework 14 – Gompf – (58370) 2 (iv) Use T 4 to estimate the value of f (3 . 9). 1. f (3 . 9) ≈ - 4 . 588 2. f (3 . 9) ≈ - 4 . 688 3. f (3 . 9) ≈ - 4 . 288 correct 4. f (3 . 9) ≈ - 4 . 388 5. f (3 . 9) ≈ - 4 . 488 Explanation: The Taylor polynomial centered at x = 4 for f is given by T 3 ( f , x ) = f (4) + f ′′ (4)( x - 4) + f (3) (4) 2! ( x - 4) 2 + f (4) 3! ( x - 4) 3 where f (4) = - 3 , f ′′ (4) = 12 , f (3) (4) = - 12 , f (4) (4) = 168 . Thus T 3 ( f , x ) = - 3 + 12( x - 4) - 6( x - 4) 2 + 28( x - 4) 3 . Now T 3 ( f , x ) is an approximation for f ( x ), so f (3 . 9) ≈ - 3 - 12 10 - 6 10 2 - 28 10 3 . Consequently, f (3 . 9) ≈ - 4 . 288 . 005 10.0 points First find values of A, B so that 21 + 10 x 7 + x = A 7 + x + Bx 7 + x , then use this to determine the Taylor series for 21 + 10 x 7 + x . centered at the origin. 1. 3 - 7 parenleftBig summationdisplay k =1 ( - 1) k 1 x k 7 k parenrightBig 2. 3 + 7 parenleftBig summationdisplay k =0 ( - 1) k 1 x k 7 k parenrightBig 3. 3 + 7 parenleftBig summationdisplay k =1 ( - 1) k 1 x k 7 k parenrightBig correct 4. 3 + 7 parenleftBig summationdisplay k =1 x k 7 k parenrightBig 5. 7 parenleftBig summationdisplay k =1 ( - 1) k 1 x k 7 k parenrightBig Explanation: Set 21 + 10 x 7 + x = A 7 + x + Bx 7 + x = A + Bx 7 + x . Then A = 21 , B = 10 , so 21 + 10 x 7 + x = 3 parenleftBig 1 1 + x/ 7 parenrightBig + 10 7 x parenleftBig 1 1 + x/ 7 parenrightBig . Now by geometric series 1 1 + x/ 7 = 1 - x 7 + x 2 7 2 + . . . = summationdisplay k =0 ( - 1) k x k 7 k for x in the interval ( - 7 , 7). But 3 parenleftBig summationdisplay k =0 ( - 1) k x k 7 k parenrightBig = 3 - 3 parenleftBig summationdisplay k =1 ( - 1) k 1 x k 7 k parenrightBig ,
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tgo72 – Homework 14 – Gompf – (58370) 3 while 10 7 x parenleftBig summationdisplay k =0 ( - 1) k x k 7 k parenrightBig = 10 parenleftBig summationdisplay k =1 ( - 1) k 1 x k 7 k parenrightBig . Thus 21 + 10 x 7 + x = 3 + 7 parenleftBig summationdisplay k =1 ( - 1) k 1 x k 7 k parenrightBig .
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