I was given a decreasing concave down graph of the integral F(x) from 0-8.
i have to say whether the following are greater then, less then,equal to, or unable to determine each other
1. T (greater then) M
2. T (greater then) R
3. M (less then) S
4. S ? actual integral
where
t=trapazoidal rule
m=midpoint
s=simpson
r=right end points
are these right?
I said you weren't sure with simpsons whether it was greater than or less than the integral so it would be a ?
the others were determined based on increasing/decreasing and concavity
I assume by concave down you mean it holds rather than sheds water.
1. agree
2. agree
3. agree
4. Unless they say it is a parabola, You can not say Simpson's rule (the parabolic one -second order) is exact. So agree, can not tell.
To determine the relationships between the different methods of approximation (T, M, S, R) and the actual integral, we need to consider the characteristics of each method and the given information about the graph.
1. T (greater than) M:
The Trapezoidal Rule (T) is known to be an overestimate of the integral when the function is decreasing. Since the graph is decreasing and concave down, the Trapezoidal Rule will consistently overestimate the integral. On the other hand, the Midpoint Rule (M) provides a better approximation of the integral since it uses the midpoint of each subinterval instead of the line segment connecting the endpoints. Therefore, T > M.
2. T (greater than) R:
Similar to the explanation above, the Trapezoidal Rule tends to overestimate the integral when the graph is decreasing. The Right Endpoint Rule (R) uses the right endpoint of each subinterval to approximate the integral. Since the graph is decreasing, the Trapezoidal Rule will likely produce a larger overestimate than the Right Endpoint Rule. Therefore, T > R.
3. M (less than) S:
The Midpoint Rule (M) and Simpson's Rule (S) both provide better estimates than the Trapezoidal Rule when the graph is decreasing (which is the case here). However, Simpson's Rule is known to provide a more accurate approximation for concave-down functions. Therefore, S will likely provide a smaller overestimate compared to M. Hence, M < S.
4. S (unable to determine) actual integral:
Since we are given only the graph of the integrand (F(x)) and not the actual function itself, it is challenging to determine whether Simpson's Rule (S) will be greater than or less than the actual integral. This estimation depends on the specific shape of the curve that is not presented in the given information. Therefore, we cannot establish a reliable relationship between S and the actual integral.
In summary:
1. T > M
2. T > R
3. M < S
4. S ? actual integral