No, and are NOT at all the same thing. If you divide the interval from 0 to 4 in to n pieces, each piece will have length 4/n. The x-values of the endpoints of those sub-intervals are 0, 4/n, 2(4/n), 3(4/n), etc.
e is NOT the correct answer.
In the attachment I have my work scanned and I must find the formula for the upper sum and then take the limit.
My work will show what I have learned in class and I really don't recognize any of the options given to be one of the formulas.
However, by process of elimination I would guess E) would be the correct formula but even then I do not understand this completely.
My main point of confusion is normally I would write it as f(Xk) and not f(k), do these mean the same thing?
I understand how the length of each piece is 1/4 and you can see that at the right of step 5. The left part or what I was shown to be f(Xk) is what I cannot identity.
1. The function is f(x)=x^2 + 3 and this would then become (k/n)^2 + 3
2. You can see in my step (3 k^2 / n^2) + 3.
3. The only thing that could be it then is C since its the function of k over n.
4. If it is C then why is it f(k/n) and not what I wrote in 2. If it was C wouldn't it the 3 also be over n?