11X1 T16 07 approximations (2011)

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<p> 1. Approximations To Areas(1) Trapezoidal Ruley y = f(x)abx 2. Approximations To Areas(1) Trapezoidal Ruley y = f(x)abx 3. Approximations To Areas(1) Trapezoidal Ruley y = f(x)baA f a f b 2abx 4. Approximations To Areas(1) Trapezoidal Ruley y = f(x)baA f a f b 2 yy = f(x)abxa b x 5. Approximations To Areas(1) Trapezoidal Ruley y = f(x)baA f a f b 2 yy = f(x)abxa c b x 6. Approximations To Areas (1) Trapezoidal Ruley y = f(x) ba A f a f b 2yy = f(x) ab x cabcA f a f c f c f b 22 a c b x 7. Approximations To Areas (1) Trapezoidal Ruley y = f(x)baA f a f b 2 yy = f(x)a bx cabcA f a f c f c f b 22 ca f a 2 f c f b 2a c b x 8. yy = f(x)a b x 9. yy = f(x)a c d b x 10. yy = f(x) cad cA f a f c f c f d 2 2bd f d f b 2a c d b x 11. yy = f(x)cad c A f a f c f c f d 22 bd f d f b 2a c d b x c a f a 2 f c 2 f d f b 2 12. yy = f(x)cad c A f a f c f c f d 22 bd f d f b 2 ac d b x c a f a 2 f c 2 f d f b 2In general; 13. yy = f(x)cad c A f a f c f c f d 22 bd f d f b 2 acdb x c a f a 2 f c 2 f d f b 2In general; bArea f x dxa 14. yy = f(x)cad c A f a f c f c f d 22 bd f d f b 2 acdb x c a f a 2 f c 2 f d f b 2In general; bArea f x dxa h y0 2 yothers yn 2 15. yy = f(x)cad c A f a f c f c f d 22 bd f d f b 2 acdb x c a f a 2 f c 2 f d f b 2In general; bArea f x dxa h y0 2 yothers yn 2bawhere h nn number of trapeziums 16. yy = f(x)cad c A f a f c f c f d 22 bd f d f b 2 acdb x c a f a 2 f c 2 f d f b 2In general; bArea f x dxa h y0 2 yothers yn NOTE: there is 2ba always one morewhere h function valuenthan intervaln 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 2 1 2 2 correct 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 2 1 2 2 correct to 3 decimal points bah n 20 4 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 2 1 2 2 correct to 3 decimal points bah nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 4 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 2 1 2 2 correct to 3 decimal points bah nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 h 4 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 2 1 2 2 correct to 3 decimal points ba 1 1h nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 h 4 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 2 1 2 2 correct to 3 decimal points ba 122 2 1h nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 h 4 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 2 1 2 2 correct to 3 decimal points ba 122 2 1h nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 h 4 Area y0 2 yothers yn 0.520.5 2 21.9365 1.7321 1.3229 0 2 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 2 1 2 2 correct to 3 decimal points ba 122 2 1h nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 h 4 Area y0 2 yothers yn 0.520.5 2 21.9365 1.7321 1.3229 0 2 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 2 1 2 2 correct to 3 decimal points ba 122 2 1h nx00.5 1 1.52 20y21.9365 1.7321 1.3229 0 h 4 Area y0 2 yothers yn 0.520.5 2 21.9365 1.7321 1.3229 0 2 2.996 units 2 exact value 3.142 2.996 % error 100 3.142 4.6% 26. (2) Simpsons Rule 27. (2) Simpsons RulebArea f x dxa 28. (2) Simpsons Rule bArea f x dx ah y0 4 yodd 2 yeven yn 3 29. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervals 30. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0 31. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0hArea y0 4 yodd 2 yeven yn 3 32. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. 11 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0hArea y0 4 yodd 2 yeven yn 3 33. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. 1 44 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0hArea y0 4 yodd 2 yeven yn 3 34. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. 1 42 4 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0hArea y0 4 yodd 2 yeven yn 3 35. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. 1 42 4 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0hArea y0 4 yodd 2 yeven yn 30.5 2 41.9365 1.3229 21.7321 0 3 3.084 units 2 36. (2) Simpsons Rule b Area f x dx ah y0 4 yodd 2 yeven yn 3 ba where h n n number of intervalse.g. 1 42 4 1 x 0 0.5 1 1.52 y 2 1.9365 1.7321 1.3229 0hArea y0 4 yodd 2 yeven yn 30.5 2 41.9365 1.3229 21.7321 0 3.142 3.084 3 % error 100 3.084 units 2 3.142 1.8% 37. Alternative working out!!!(1) Trapezoidal Rule 38. Alternative working out!!!(1) Trapezoidal Rule122 2 1x 00.5 1 1.52y 21.9365 1.7321 1.3229 0 39. Alternative working out!!!(1) Trapezoidal Rule1 22 2 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 2 2 1.9365 1.7321 1.3229 0Area 2 0 1 2 2 2 1 2.996 units 2 40. (2) Simpsons Rule142 4 1x0 0.5 1 1.52y2 1.9365 1.7321 1.3229 0 41. (2) Simpsons Rule 1 4 2 4 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 2 4 1.9365 1.3229 2 1.7321 0Area 2 0 1 4 2 4 1 3.084 units 2 42. (2) Simpsons Rule 1 4 2 4 1x 0 0.5 1 1.52y 2 1.9365 1.7321 1.3229 0 2 4 1.9365 1.3229 2 1.7321 0Area 2 0 1 4 2 4 1 3.084 units 2Exercise 11I; oddsExercise 11J; evens</p>