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Mathematical Approximation and Error Analysis for College Students

Learn about differentiation, integration, significant digits, errors, and binary representation in mathematics for college-level studies. Understand sources of error like round-off and truncation errors. Study floating-point representation and more.

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Mathematical Approximation and Error Analysis for College Students

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  1. NCR Register http://nm.mathforcollege.com

  2. INTRODUCTION, APPROXIMATION AND ERRORS http://nm.mathforcollege.com

  3. Identify the picture of your instructor 7 http://nm.mathforcollege.com

  4. 01.01INTRODUCTION http://nm.mathforcollege.com

  5. To find velocity from location vs time data of the body, the mathematical procedure used is • Differentiation • Integration http://nm.mathforcollege.com

  6. To find velocity from acceleration vs time data, the mathematical procedure used is • Differentiation • Integration 30 http://nm.mathforcollege.com

  7. The form of the exact solution to is 30 http://nm.mathforcollege.com

  8. y 5 a b 2 c x 7 Given the f (x) vs x curve, and the magnitude of the areas as shown, the value of • -2 • 2 • 12 • Cannot be determined http://nm.mathforcollege.com

  9. Given y= sin(2x), dy/dx at x=3 is • 0.9600 • 0.9945 • 1.920 • 1.989 http://nm.mathforcollege.com

  10. A steel cylindrical shaft at room temperature is immersed in a dry-ice/alcohol bath. A layman estimates the reduction in diameter by using while using the value of the thermal expansion coefficient at -108oF. Seeing the graph below, the magnitude of contraction you as a USF educated engineer would calculate would be ______________than the layman’s estimate. • Less • More • Same 30 http://nm.mathforcollege.com

  11. END http://nm.mathforcollege.com

  12. 01.02MEASURINGERRORS http://nm.mathforcollege.com

  13. The number of significant digits in 2.30500 is • 3 • 4 • 5 • 6 7 http://nm.mathforcollege.com

  14. The number of significant digits in 0.0023406 is • 4 • 5 • 6 • 7 http://nm.mathforcollege.com

  15. The number of significant digits in 2350 is • 3 • 4 • 5 • 3 or 4 http://nm.mathforcollege.com

  16. The absolute relative approximate error in an iterative process at the end of the tenth iteration is 0.007%. The least number of significant digits correct in the answer is • 2 • 3 • 4 • 5 7 http://nm.mathforcollege.com

  17. Three significant digits are expected to be correct after an iterative process. The pre-specified tolerance in this case needs to be less than or equal to • 0.5% • 0.05% • 0.005% • 0.0005% http://nm.mathforcollege.com

  18. 01.03SOURCES OF ERROR http://nm.mathforcollege.com

  19. Dreaming http://nm.mathforcollege.com

  20. The error caused by representing numbers such as 1/3 approximately is called • Round-off error • Truncation error http://nm.mathforcollege.com

  21. The number 6.749832 with 3 significant digits with rounding is • 6.74 • 6.75 • 6.749 • 6.750 http://nm.mathforcollege.com

  22. The error caused by using only a few terms of the Maclaurin series to calculate ex results mostly in • Truncation Error • Round off Error http://nm.mathforcollege.com

  23. The number 6.749832 with 3 significant digits with chopping is • 6.74 • 6.75 • 6.749 • 6.750 http://nm.mathforcollege.com

  24. END http://nm.mathforcollege.com

  25. 01.04BINARY REPRESENTATION http://nm.mathforcollege.com

  26. This is what you have been saying about your TI-30Xa • I don't care what people say The rush is worth the priceI pay I get so high when you're with meBut crash and crave you when you are away • Give me back now my TI89Before I start to drink and whine TI30Xa calculators make me cryIncarnation of of Jason will you ever die • TI30Xa – you make me forget the high maintenance TI89. • I never thought I will fall in love again! http://nm.mathforcollege.com

  27. (8)10=(?)2 • 1110 • 1011 • 0100 • 1000 http://nm.mathforcollege.com

  28. (01011)2 =(?)10 • 7 • 11 • 15 • 22 http://nm.mathforcollege.com

  29. The binary representation of (0.3)10 is • (0.01001……...)2 • (0.10100……...)2 • (0.01010……...)2 • (0.01100……...)2 http://nm.mathforcollege.com

  30. In a five bit fixed representation, (0.1)10 is represented as (0.00011)2. The true error in this representation most nearly is • 0.00625 • 0.053125 • 0.09375 • 9.5x10-8 http://nm.mathforcollege.com

  31. 01.05FLOATING POINT REPRESENTATION http://nm.mathforcollege.com

  32. Using fixed point representation in a computer puts a upper bound on the ________________ in representing a number. • Absolute true error • Relative absolute true error http://nm.mathforcollege.com

  33. Using floating point representation in a computer puts a upper bound on the ________________ in representing a number. • Absolute true error • Relative absolute true error http://nm.mathforcollege.com

  34. The absolute relative true error in a floating point representation using chopping for a number is less than • Machine epsilon • 2-bits used for exponent http://nm.mathforcollege.com

  35. The smallest positive number in a 7 bit word where 1st bit is used for the sign of number, 2nd bit is used for the sign of exponent, next 3 bits are used for the magnitude of the mantissa and the last 2 bits are used for the magnitude of the exponent is __________. • 0.000 • 0.125 • 0.250 • 1.000 http://nm.mathforcollege.com

  36. Five bits are used for the biased exponent. To convert a biased exponent to an unbiased exponent, you would • add 7 • subtract 7 • add 15 • subtract 15 http://nm.mathforcollege.com

  37. The largest positive number in a 7 bit word where 1st bit is used for the sign of number, 2nd bit is used for the sign of exponent, next 3 bits are used for the magnitude of the mantissa and the last 2 bits are used for the magnitude of the exponent is _________ • 1.875 • 4 • 7 • 15 http://nm.mathforcollege.com

  38. The machine epsilon in a 7 bit word where 1st bit is used for the sign of number, 2nd bit is used for the sign of exponent, next 3 bits are used for the magnitude of the mantissa and the last 2 bits are used for the magnitude of the exponent is _________ • 0.125 • 0.25 • 0.5 • 1.0 http://nm.mathforcollege.com

  39. END http://nm.mathforcollege.com

  40. 01.07TAYLOR SERIES http://nm.mathforcollege.com

  41. Taylor series (given below) is only valid • if values of h are small • if function and all its derivatives are defined and continuous at x • if function and all its derivatives are defined and continuous in [x,x+h] http://nm.mathforcollege.com

  42. END http://nm.mathforcollege.com

  43. What will Maury Povich say when he dies, goes to heaven, and sees God? • Why am I here; I should be with the lawyers. • Is Connie here? • You are the Father. • You are not the father. http://nm.mathforcollege.com

  44. I walk like a pimp – Jeremy Reed You know it's hard out here for a pimp, When he tryin’ to get this money for the rent, For the Cadillacs and gas money spent

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