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# 11 x1 t16 07 approximations (2013)

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### 11 x1 t16 07 approximations (2013)

1. 1. Approximations To Areas (1) Trapezoidal Rule y y = f(x) a b x
2. 2. Approximations To Areas (1) Trapezoidal Rule y y = f(x) a b x
3. 3. Approximations To Areas (1) Trapezoidal Rule y y = f(x) a b x ba A  f a   f b  2
4. 4. Approximations To Areas (1) Trapezoidal Rule y y = f(x) ba A  f a   f b  2 y a b y = f(x) x a b x
5. 5. Approximations To Areas (1) Trapezoidal Rule y y = f(x) ba A  f a   f b  2 y a b y = f(x) x a c b x
6. 6. Approximations To Areas (1) Trapezoidal Rule y y = f(x) ba A  f a   f b  2 y a b y = f(x) x ca bc A  f a   f c    f c   f b  2 2 a c b x
7. 7. Approximations To Areas (1) Trapezoidal Rule y y = f(x) ba A  f a   f b  2 y a b y = f(x) x ca bc A  f a   f c    f c   f b  2 2 ca   f a   2 f c   f b  2 a c b x
8. 8. y y = f(x) a b x
9. 9. y y = f(x) a c d b x
10. 10. y y = f(x) ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 a c d b x
11. 11. y y = f(x) a c d b ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 x  c  a  f a   2 f c   2 f d   f b  2
12. 12. y y = f(x) a c In general; d b ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 x  c  a  f a   2 f c   2 f d   f b  2
13. 13. y y = f(x) a c In general; d ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 x  c  a  f a   2 f c   2 f d   f b  2 b b Area   f  x dx a
14. 14. y y = f(x) a c In general; d ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 x  c  a  f a   2 f c   2 f d   f b  2 b b Area   f  x dx a h  y0  2 yothers  yn  2
15. 15. y y = f(x) a c In general; d ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 x  c  a  f a   2 f c   2 f d   f b  2 b b Area   f  x dx a h  y0  2 yothers  yn  2 ba n n  number of trapeziums where h 
16. 16. y y = f(x) a c In general; d ca d c A  f a   f c    f c   f d  2 2 bd   f d   f b  2 x  c  a  f a   2 f c   2 f d   f b  2 b b Area   f  x dx a h  y0  2 yothers  yn  2 ba n n  number of trapeziums where h  NOTE: there is always one more function value than interval
17. 17. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points   , between x  0 and x  2 1 2 2
18. 18. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5  , between x  0 and x  2 1 2 2
19. 19. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5 x y 0 2  , between x  0 and x  2 1 2 2 0.5 1.9365 1 1.7321 1.5 1.3229 2 0
20. 20. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5 x y 0 2  , between x  0 and x  2 1 2 2 0.5 1.9365 1 1.7321 h Area  y0  2 yothers  yn  2 1.5 1.3229 2 0
21. 21. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5  , between x  0 and x  2 1 2 2 1 x y 0 2 1 0.5 1.9365 1 1.7321 h Area  y0  2 yothers  yn  2 1.5 1.3229 2 0
22. 22. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5 1 x y 0 2  , between x  0 and x  2 1 2 2 2 2 2 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  2 yothers  yn  2
23. 23. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5 1 x y 0 2  , between x  0 and x  2 1 2 2 2 2 2 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  2 yothers  yn  2 0.5  2  21.9365  1.7321  1.3229  0 2  2.996 units 2
24. 24. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5 1 x y 0 2  , between x  0 and x  2 1 2 2 2 2 2 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  2 yothers  yn  2 0.5  2  21.9365  1.7321  1.3229  0 2 exact value  π   2.996 units 2
25. 25. e.g. Use the Trapezoida l Rule with 4 intervals to estimate the area under the curve y  4  x correct to 3 decimal points  ba h n 20  4  0.5 1 x y 0 2  , between x  0 and x  2 1 2 2 2 2 2 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  2 yothers  yn  2 0.5  2  21.9365  1.7321  1.3229  0 2 exact value  π   2.996 units 2 3.142  2.996 100 3.142  4.6% % error 
26. 26. (2) Simpson’s Rule
27. 27. (2) Simpson’s Rule b Area   f  x dx a
28. 28. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3
29. 29. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h 
30. 30. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  e.g. x y 0 2 0.5 1.9365 1 1.7321 1.5 1.3229 2 0
31. 31. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  e.g. x y 0 2 0.5 1.9365 1 1.7321 h Area  y0  4 yodd  2 yeven  yn  3 1.5 1.3229 2 0
32. 32. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  1 e.g. x y 0 2 1 0.5 1.9365 1 1.7321 h Area  y0  4 yodd  2 yeven  yn  3 1.5 1.3229 2 0
33. 33. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  1 e.g. x y 4 0 2 0.5 1.9365 4 1 1.7321 h Area  y0  4 yodd  2 yeven  yn  3 1 1.5 1.3229 2 0
34. 34. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  1 e.g. x y 4 2 4 1 0 2 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  4 yodd  2 yeven  yn  3
35. 35. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  1 e.g. x y 4 2 4 1 0 2 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  4 yodd  2 yeven  yn  3 0.5  2  41.9365  1.3229  21.7321  0 3  3.084 units 2
36. 36. (2) Simpson’s Rule b Area   f  x dx a h  y0  4 yodd  2 yeven  yn  3 ba n n  number of intervals where h  1 e.g. x y 4 2 4 1 0 2 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 h Area  y0  4 yodd  2 yeven  yn  3 0.5  2  41.9365  1.3229  21.7321  0 3.142  3.084 3 % error  100 3.142  3.084 units 2  1.8%
37. 37. Alternative working out!!! (1) Trapezoidal Rule
38. 38. Alternative working out!!! (1) Trapezoidal Rule 1 x y 0 2 2 2 2 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0
39. 39. Alternative working out!!! (1) Trapezoidal Rule 1 x y Area  0 2 2 2 2 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 2  2 1.9365  1.7321  1.3229   0 1 2  2  2 1  2.996 units 2   2  0
40. 40. (2) Simpson’s Rule 1 x y 0 2 4 2 4 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0
41. 41. (2) Simpson’s Rule 1 x y Area  0 2 4 2 4 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 2  4 1.9365  1.3229   2 1.7321  0 1 4  2  4 1  3.084 units 2   2  0
42. 42. (2) Simpson’s Rule 1 x y Area  0 2 4 2 4 1 0.5 1.9365 1 1.7321 1.5 1.3229 2 0 2  4 1.9365  1.3229   2 1.7321  0 1 4  2  4 1  3.084 units 2 Exercise 11I; odds Exercise 11J; evens   2  0