Question

which of the rectangles below are both left and right rectangles?
select all of the rectangles which are both left and right rectangles below.
a. rectangle a
b. rectangle e
c. rectangle b
d. rectangle c
e. rectangle d
f. none of the rectangles are both left and ri

Explanation:

Step1: Understand left - and right - rectangle concepts

In a left - rectangle approximation, the height of the rectangle is determined by the function value at the left - endpoint of the sub - interval. In a right - rectangle approximation, the height is determined by the function value at the right - endpoint of the sub - interval. A rectangle that is both a left and a right rectangle must have the same height when calculated using the left - endpoint and the right - endpoint of its sub - interval.

Step2: Analyze each rectangle

• Rectangle A:

The left - endpoint of its sub - interval is \(x = 0\) and the right - endpoint is \(x=2\). The function value at \(x = 0\) (left - endpoint) and \(x = 2\) (right - endpoint) are different.

• Rectangle B:

The left - endpoint of its sub - interval is \(x = 2\) and the right - endpoint is \(x = 4\). The function value at \(x = 2\) (left - endpoint) and \(x = 4\) (right - endpoint) are different.

• Rectangle C:

The left - endpoint of its sub - interval is \(x = 4\) and the right - endpoint is \(x = 6\). The function value at \(x = 4\) (left - endpoint) and \(x = 6\) (right - endpoint) are different.

• Rectangle D:

The left - endpoint of its sub - interval is \(x = 6\) and the right - endpoint is \(x = 8\). The function value at \(x = 6\) (left - endpoint) and \(x = 8\) (right - endpoint) are the same (since the function is constant over this sub - interval).

• Rectangle E:

The left - endpoint of its sub - interval is \(x = 8\) and the right - endpoint is \(x = 10\). The function value at \(x = 8\) (left - endpoint) and \(x = 10\) (right - endpoint) are different.

Answer:

E. Rectangle D