11X1 T17 07 approximations

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<ul><li> 1. Approximations To Areas (1) Trapezoidal Rule yy = f(x) abx </li></ul> <p> 2. Approximations To Areas (1) Trapezoidal Rule yy = f(x) abx 3. Approximations To Areas (1) Trapezoidal Rule yy = f(x)ba A f a f b 2 abx 4. Approximations To Areas (1) Trapezoidal Rule yy = f(x)ba A f a f b 2yy = f(x) abx a b x 5. Approximations To Areas (1) Trapezoidal Rule yy = f(x)ba A f a f b 2yy = f(x) abx a c b x 6. Approximations To Areas(1) Trapezoidal Rule yy = f(x) baA f a f b 2 yy = f(x)ab xcabc A f a f c f c f b 22a c b x 7. Approximations To Areas(1) Trapezoidal Rule yy = f(x)ba A f a f b 2yy = f(x) a bxcabc A f a f c f c f b 22ca f a 2 f c f b 2 a c b x 8. y y = f(x) a b x 9. y y = f(x) a c d b x 10. y y = f(x)cad c A f a f c f c f d 2 2 bd f d f b 2 a c d b x 11. y y = f(x) cad cA f a f c f c f d 22bd f d f b 2 a c d b x c a f a 2 f c 2 f d f b 2 12. y y = f(x) cad cA f a f c f c f d 22bd f d f b 2ac d b x c a f a 2 f c 2 f d f b 2 In general; 13. y y = f(x) cad cA f a f c f c f d 22bd f d f b 2acdb x c a f a 2 f c 2 f d f b 2 In general; b Area f x dx a 14. y y = f(x) cad cA f a f c f c f d 22bd f d f b 2acdb x c a f a 2 f c 2 f d f b 2 In general; b Area f x dx ah y0 2 yothers yn 2 15. y y = f(x) cad cA f a f c f c f d 22bd f d f b 2acdb x c a f a 2 f c 2 f d f b 2 In general; b Area f x dx ah y0 2 yothers yn 2 ba where h n n number of trapeziums 16. y y = f(x) cad cA f a f c f c f d 22bd f d f b 2acdb x c a f a 2 f c 2 f d f b 2 In general; b Area f x dx ah y0 2 yothers yn NOTE: there is2 ba always one more where h function value n than interval n number of trapeziums 17. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points 18. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba hn204 0.5 19. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba hnx00.5 1 1.5220y21.9365 1.7321 1.3229 04 0.5 20. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba hnx00.5 1 1.5220y21.9365 1.7321 1.3229 0 h4 Area y0 2 yothers yn 0.5 2 21. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba 1 1 hnx00.5 1 1.5220y21.9365 1.7321 1.3229 0 h4 Area y0 2 yothers yn 0.5 2 22. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba 122 2 1 hnx00.5 1 1.5220y21.9365 1.7321 1.3229 0 h4 Area y0 2 yothers yn 0.5 2 23. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba 122 2 1 hnx00.5 1 1.5220y21.9365 1.7321 1.3229 0h4 Area y0 2 yothers yn 0.52 0.5 2 21.9365 1.7321 1.3229 02 2.996 units 2 24. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba 122 2 1 hnx00.5 1 1.5220y21.9365 1.7321 1.3229 0h4 Area y0 2 yothers yn 0.52 0.5 2 21.9365 1.7321 1.3229 02 2.996 units 2 exact value 25. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y 4 x , between x 0 and x 212 2correct to 3 decimal points ba 122 2 1 hnx00.5 1 1.5220y21.9365 1.7321 1.3229 0h4 Area y0 2 yothers yn 0.52 0.5 2 21.9365 1.7321 1.3229 02 2.996 units 2 exact value 3.142 2.996% error 1003.142 4.6% 26. (2) Simpsons Rule 27. (2) Simpsons Rule b Area f x dx a 28. (2) Simpsons Ruleb Area f x dxa h y0 4 yodd 2 yeven yn 3 29. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals 30. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g.x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 31. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g.x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 h Area y0 4 yodd 2 yeven yn 3 32. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g. 11x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 h Area y0 4 yodd 2 yeven yn 3 33. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g. 1 44 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 h Area y0 4 yodd 2 yeven yn 3 34. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g. 1 42 4 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 h Area y0 4 yodd 2 yeven yn 3 35. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g. 1 42 4 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 h Area y0 4 yodd 2 yeven yn 3 0.5 2 41.9365 1.3229 21.7321 03 3.084 units 2 36. (2) Simpsons RulebArea f x dxa h y0 4 yodd 2 yeven yn 3bawhere h nn number of intervals e.g. 1 42 4 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 h Area y0 4 yodd 2 yeven yn 3 0.5 2 41.9365 1.3229 21.7321 0 3.142 3.0843 % error 100 3.084 units 2 3.142 1.8% 37. Alternative working out!!! (1) Trapezoidal Rule 38. Alternative working out!!! (1) Trapezoidal Rule 122 2 1 x 00.5 1 1.52 y 21.9365 1.7321 1.3229 0 39. Alternative working out!!! (1) Trapezoidal Rule 1 22 2 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 02 2 1.9365 1.7321 1.3229 0 Area 2 01 2 2 2 1 2.996 units 2 40. (2) Simpsons Rule 142 4 1 x0 0.5 1 1.52 y2 1.9365 1.7321 1.3229 0 41. (2) Simpsons Rule1 4 2 4 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 02 4 1.9365 1.3229 2 1.7321 0 Area 2 01 4 2 4 1 3.084 units 2 42. (2) Simpsons Rule1 4 2 4 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 02 4 1.9365 1.3229 2 1.7321 0 Area 2 01 4 2 4 1 3.084 units 2 Exercise 11I; oddsExercise 11J; evens </p>