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Week1_Suggested Probs

# Week1_Suggested Probs - C/N r for some constant C when N is...

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Mech 221 Math Suggested Problems Week #1 Brian Wetton September 21, 2009 Note: Questions #2 and #3 might be a bit tricky. Feel free to ask me or the TA for help with them. 1. Experimental Measurements determine that a function f ( x ) satisfies f (0) = 0, f (0) = 1 and f (1) = 2. (a) Estimate f (1 / 3) using tangent line (linear) approximation. (b) Estimate f (1 / 3) using linear interpolation. (c) Estimate f (1 / 3) more accurately using all three pieces of informa- tion given. Hint: Construct a quadratic polynomial Q ( x ) that satisfies the data. Approximate f (1 / 3) by Q (1 / 3). 2. A numerical method is used to approximate an improper integral. The method converges as the number of subintervals N gets larger, but con- vergence is slow. The estimates, E N of I for various values of N are given below: N E N 4 1.5250 8 1.3661 16 1.2563 32 1.1799 (a) Estimate the order of convergence. This is the number r such that

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Unformatted text preview: C/N r for some constant C when N is large. Hint: If I-E N ≈ C/N r then E 2 N-E N ≈ C (1-2-r ) /N r 1 and so E 2 N-E N E 4 N-E 2 N ≈ 2 r or r ≈ log 2 ± E 2 N-E N E 4 N-E 2 N ² . Use this last expression to estimate r . (b) Use an appropriate Richardson Extrapolation to get more accurate estimates of I . 3. Suppose that linear interpolation S ( x ) is used to estimate f ( x ) in the interval [0 ,h ] using the values f (0) and f ( h ). Derive an error bound for this interpolation of the form | f ( x )-S ( x ) | ≤ CK 2 h 2 for all x in [0 ,h ] where K 2 is the maximum of f (2) on the interval and C is a constant you must determine. Hint: modify the argument I showed in class for the error bound for tangent line approximation. 2...
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