Trapezoidal Rule - Confused?

2015-03-18 6:18 am
I found this question in my exam paper about the trapezoidal rule which I cannot understand the answer.

In the question there is a function that has a graph which closely resembles y=x^2-4x, so I will state my problem using y=x^2-4x.

This is the graph of y=x^2-4x.
https://www.google.com/search?q=y%3Dx%5E2-4x&rct=j

If I use the trapezoidal rule to find the integral for the curve in the range of 0<x<4, I get a negative number as the answer, which is normal since the curve is below the x-axis in that range.

If I find the second derivative of the function, f ' ' (x) > 0, so it proves the curve is pointing upwards, just like the graph shown in the link given.

Since I know that the curve is pointing upwards AND the graph is below the x-axis in that range, I answered that the trapezoidal rule estimate was an under-estimate. (Just like graphs that are concave downwards and above the x-axis)

However the answer guide says that it is an over-estimate because f ' ' (x) > 0.

So is it necessary to take into account the fact that the graph is below the x-axis when I determine whether the estimate is an under-estimate or over-estimate? If not, why?

Or was the answer guide wrong?

Thank you for your guidance.

回答 (1)

2015-03-18 6:45 am
✔ 最佳答案
Yes, you want to take into account 'where' the graph is when you would be drawing your trapezoids.

For your graph between x=0 and x=4.... yes, the graph is below the x-axis
this means that ALL the trapezoids you draw will be ABOVE the curve
this means that any negative value you would get for them would be closer to zero than the actual area/integral calculation
.... so, yes, you will get a larger value than needed - thus 'over-estimate'... even though it's still negative
Ex: true calc: -10... while the trapezoidal calc: -8


On the reverse side... if you had a Concave Down parabola ABOVE the x-axis
-all the trapezoids would be BELOW the curve, thus giving an under-estimate
Ex: true calc: +10... trapezoidal calc: +8


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