anonymous
  • anonymous
Given f(x) > 0 with f ′(x) < 0, and f ′′(x) > 0 for all x in the interval [0, 2] with f(0) = 1 and f(2) = 0.2, the left, right, trapezoidal, and midpoint rule approximations were used to estimate the integral from 0 to 2 of f of x, dx. The estimates were 0.7811, 0.8675, 0.8650, 0.8632 and 0.9540, and the same number of subintervals were used in each case. Match the rule to its estimate.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
https://gyazo.com/1e7e62cfc6ca6ef45e70b557bc662245
anonymous
  • anonymous
How would this be solved without knowing f(x)
anonymous
  • anonymous
@jim_thompson5910

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jim_thompson5910
  • jim_thompson5910
That's a good question. Let me think it over
anonymous
  • anonymous
Ok
anonymous
  • anonymous
You still there right?
jim_thompson5910
  • jim_thompson5910
yes, I'm drawing out the 4 cases right now
jim_thompson5910
  • jim_thompson5910
ok there's actually a faster way look at this pdf http://math.arizona.edu/~calc/Text/Section7.5.pdf you'll see on page 4 that they write what you see that I'm attaching as an image file
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jim_thompson5910
  • jim_thompson5910
so you first need to ask yourself: is f increasing? or decreasing?
anonymous
  • anonymous
Increasing
jim_thompson5910
  • jim_thompson5910
you sure?
anonymous
  • anonymous
Oh f' is less than 0
anonymous
  • anonymous
means it's decreasing
jim_thompson5910
  • jim_thompson5910
yes, so f is decreasing on the interval (0,2)
jim_thompson5910
  • jim_thompson5910
so based on this rule (attached) we know \[\Large \text{RIGHT}(n) \le \int_{a}^{b}f(x)dx\le \text{LEFT}(n)\]
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jim_thompson5910
  • jim_thompson5910
is f concave up? or concave down?
anonymous
  • anonymous
concave up
jim_thompson5910
  • jim_thompson5910
f is concave up, so \[\Large \text{MID}(n) \le \int_{a}^{b}f(x)dx \le \text{TRAP}(n)\] see page 6 of that pdf
jim_thompson5910
  • jim_thompson5910
the midpoint and trapezoidal approximations are much closer than the left and right endpoints, so we can write this \[\large \text{RIGHT}(n) \le \text{MID}(n) \le \int_{a}^{b}f(x)dx \le \text{TRAP}(n) \le \text{LEFT}(n)\]
jim_thompson5910
  • jim_thompson5910
Sort out the five given decimal approximations (0.7811,0.8632,0.8650,0.8675,0.9540) and you'll find that \[\large \text{RIGHT}(n) \le \text{MID}(n) \le \int_{a}^{b}f(x)dx \le \text{TRAP}(n) \le \text{LEFT}(n)\] \[\large 0.7811 \le 0.8632 \le 0.8650 \le 0.8675 \le 0.9540\]
anonymous
  • anonymous
Thanks but I don't know where all this came from, like my course never had any of this
jim_thompson5910
  • jim_thompson5910
so it never mentioned about under-estimates and over-estimates ?
anonymous
  • anonymous
It did but not in this manner.
anonymous
  • anonymous
Thanks for the explanation and help, now I'll be able to do something my course hasn't covered.
jim_thompson5910
  • jim_thompson5910
no problem

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