{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

151Ahw1solns

# 151Ahw1solns - Math 151A HW1 Solutions Will Feldman 1 We...

This preview shows pages 1–2. Sign up to view the full content.

Math 151A HW1 Solutions Will Feldman 1. We want to find the third order Taylor polynomial for the function f ( x ) = 1 + x about x 0 = 0. Recall the formula for the Taylor polynomial of order n centered at x 0 is given by P n ( x ) = n X k =0 f ( k ) ( x 0 ) k ! ( x - x 0 ) k . Calculating some derivatives of f you will find that P 3 ( x ) = 1 + x 2 - x 2 8 + x 3 16 . Now we want to use this to approximate some values of the square root. Note that y = f ( y - 1), so that (for example) 0 . 5 = f ( - 0 . 5) P 3 ( - 0 . 5) 0 . 711. The same method applies to the other values of square root. 2. We can calculate the Taylor series for f ( x ) = (1 - x ) - 1 by calculating that f ( n ) ( x ) = n !(1 - x ) - ( n +1) . Then we have P n ( x ) = n X k =0 x k . In order to calculate the error of using P n ( x ) to approximate f on x [0 , . 5] note that we can calculate P n ( x ) exactly since xP n ( x ) = x n +1 + P n ( x ) - 1 P n ( x ) = x n +1 - 1 x - 1 . So that we have, | f ( x ) - P n ( x ) | = x n +1 1 - x 1 2 n . Then just choose n > 6 log 10 / log 2 so that (1 / 2) n < 10 - 6 . 3. Solution can be found in Burden and Faires.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

151Ahw1solns - Math 151A HW1 Solutions Will Feldman 1 We...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online