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homework3sol

# homework3sol - Math 4650 Homework#3 Solutions 1.3#7 Find...

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Math 4650 Homework #3 Solutions 1.3 #7. Find the rates of convergence of the following functions as h 0. 1. lim h 0 sin h h = 1. Use sin h = h - h 3 6 + · · · , and we get sin h h - 1 = sin h - h h = - h 3 6 h + · · · = - h 2 6 + · · · . So it’s O ( h 2 ). 2. lim h 0 1 - cos h h = 0. Use cos h = 1 - h 2 2 + · · · to get 1 - cos h h = 1 - 1 - h 2 2 + · · · h = h 2 + · · · . So the order is O ( h ). 3. lim h 0 sin h - h cos h h = 0. Use sin h = h - h 3 6 + · · · and cos h = 1 - h 2 2 + · · · , and we find sin h - h cos h = h - h 3 6 - h + h 3 2 + · · · = h 3 3 + · · · . Therefore sin h - h cos h h = h 2 3 + · · · , and the order is O ( h 2 ). 4. lim h 0 1 - e h h = - 1. Use e h = 1 + h + h 2 2 + · · · , and get 1 - e h h + 1 = 1 + h - e h h = 1 + h - 1 - h - h 2 2 - · · · h = - h 2 + · · · so that the order is O ( h ). 1.3, #8. 1. How many multiplications and additions are required to determine a sum of the form n i =1 i j =1 a i b j ? 1

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Solution: We clearly have one multiplication for each summand, so there are n i =1 i j =1 (1) = n i =1 i = n ( n +1) 2 multiplications. We then have ( i - 1) additions
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homework3sol - Math 4650 Homework#3 Solutions 1.3#7 Find...

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