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anonymous

  • one year ago

The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Trapezoidal Sum Approximation, using the intervals between those given points. x 10 12 15 19 20 f(x) –2 –5 –9 –12 –16

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  1. jim_thompson5910
    • one year ago
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    how far did you get?

  2. anonymous
    • one year ago
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    Im stuck on how to start it

  3. anonymous
    • one year ago
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    what trips me up is that it asks to use intervals between those values

  4. jim_thompson5910
    • one year ago
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    I would start it by plotting the points on an xy grid (see attached)

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  5. anonymous
    • one year ago
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    what would you do after plotting the points

  6. jim_thompson5910
    • one year ago
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    then draw trapezoids

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  7. jim_thompson5910
    • one year ago
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    tell me what you get for the area of each trapezoid

  8. jim_thompson5910
    • one year ago
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    |dw:1443395568574:dw| area of trapezoid = h*(B1+B2)/2 notice how the B1 and B2 are parallel

  9. anonymous
    • one year ago
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    17.5 for the first trapezoid

  10. jim_thompson5910
    • one year ago
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    incorrect

  11. anonymous
    • one year ago
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    7

  12. jim_thompson5910
    • one year ago
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    yes for the one on the very left

  13. anonymous
    • one year ago
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    21 for the second

  14. jim_thompson5910
    • one year ago
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    good

  15. anonymous
    • one year ago
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    42 for the third

  16. jim_thompson5910
    • one year ago
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    yep

  17. anonymous
    • one year ago
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    and 14 for the last

  18. jim_thompson5910
    • one year ago
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    very good

  19. jim_thompson5910
    • one year ago
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    now add up those areas to get ???

  20. anonymous
    • one year ago
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    84

  21. anonymous
    • one year ago
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    what would be the next step

  22. jim_thompson5910
    • one year ago
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    |dw:1443396060396:dw|

  23. jim_thompson5910
    • one year ago
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    imagine we have something like this function |dw:1443396074151:dw| some curve below the x axis

  24. jim_thompson5910
    • one year ago
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    points A through E lie on this curve |dw:1443396096733:dw|

  25. jim_thompson5910
    • one year ago
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    we can't get the exact area between the curve and x axis, but we can get the approximate area by using trapezoids |dw:1443396136613:dw|

  26. jim_thompson5910
    • one year ago
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    it's not perfect, but it's the best we can do

  27. jim_thompson5910
    • one year ago
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    what we do is divide the approximate area by the distance between the two endpoints (b-a = 20-10 = 10)

  28. anonymous
    • one year ago
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    so the answer would be -8.4?

  29. jim_thompson5910
    • one year ago
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    you may have seen the formula \[\Large A = \frac{1}{b-a}\int_{a}^{b}f(x)dx\] A = average value of the function f(x) on the interval x = a to x = b

  30. jim_thompson5910
    • one year ago
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    so you just do 84/10 = 8.4

  31. jim_thompson5910
    • one year ago
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    sorry yeah the area is actually negative because we're below the x axis -84/10 = -8.4

  32. anonymous
    • one year ago
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    wow thank you so much! your explanation and steps were extremely helpful!

  33. jim_thompson5910
    • one year ago
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    |dw:1443396331117:dw|

  34. jim_thompson5910
    • one year ago
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    no problem

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