The function f(x) = x2 − 5 is defined over the interval [0, 5]. If the interval is divided into n equal parts, what is the area of the kth rectangle from the right?
A) [(2+k(5/n))^2+5](5/n)
B) [(k(3/n))^2−5](5/n)
C) [(k(5/n))^2+5](5/n)
D) [(k(3/n))^2+3](5/n)
E) [(k(5/n))^2−5](5/n)
To find the area of the kth rectangle from the right, we first need to determine the width of each rectangle, which is the width of the interval divided by the number of parts, or (5/n).
Next, we need to find the height of the kth rectangle. To do this, we substitute the x-value of the right endpoint of the kth rectangle into the function f(x) = x^2 - 5.
Since the interval is divided into equal parts, the width of each rectangle is the same, and the x-value of the right endpoint of the kth rectangle can be found by multiplying the width by k. So, the x-value of the right endpoint is k*(5/n).
Substituting x = k*(5/n) into the function f(x) = x^2 - 5, we get the height of the kth rectangle as (k*(5/n))^2 - 5.
Finally, we calculate the area of the kth rectangle by multiplying the width and height together: (height)*(width) = [(k*(5/n))^2 - 5](5/n).
Therefore, the correct answer is option E) [(k(5/n))^2 - 5](5/n).