Given f(x) 0 with 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.

Question

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.

anonymous · Answer

https://gyazo.com/1e7e62cfc6ca6ef45e70b557bc662245

anonymous · Answer

How would this be solved without knowing f(x)

anonymous · Answer

@jim_thompson5910

jim_thompson5910 · Answer

That's a good question. Let me think it over

anonymous · Answer

Ok

anonymous · Answer

You still there right?

jim_thompson5910 · Answer

yes, I'm drawing out the 4 cases right now

jim_thompson5910 · Answer

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

jim_thompson5910 · Answer

so you first need to ask yourself: is f increasing? or decreasing?

anonymous · Answer

Increasing

jim_thompson5910 · Answer

you sure?

anonymous · Answer

Oh f' is less than 0

anonymous · Answer

means it's decreasing

jim_thompson5910 · Answer

yes, so f is decreasing on the interval (0,2)

jim_thompson5910 · Answer

so based on this rule (attached) we know 

$$\Large \text{RIGHT}(n) \le \int_{a}^{b}f(x)dx\le \text{LEFT}(n)$$

jim_thompson5910 · Answer

is f concave up? or concave down?

anonymous · Answer

concave up

jim_thompson5910 · Answer

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 · Answer

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 · Answer

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 · Answer

Thanks but I don't know where all this came from, like my course never had any of this

jim_thompson5910 · Answer

so it never mentioned about under-estimates and over-estimates ?

anonymous · Answer

It did but not in this manner.

anonymous · Answer

Thanks for the explanation and help, now I'll be able to do something my course hasn't covered.

jim_thompson5910 · Answer

no problem