Question

Let f be a continuous function on the interval [a, b]. Determine if the following statement is true or false. The expression lim f(xi") ax may lead to different limits if we choose the x * to be the left endpoints instead of midpoints. ni=1 Choose the correct answer below. O A. The statement is true. Each sum will be smaller when picking a left endpoint, so the limit will be smaller. OB. The statement is false. Any sample point can be chosen within the interval without affecting the limit. O C. The statement is true. The value of the function at the left endpoint is not the same as the value at the midpoint of the interval D. The statement is false. The value of the function at the left endpoint is the same as the value at the midpoint of the interval. w Click to select your answer​

Answer 1

Answer:

The correct answer is B.

Step-by-step explanation:

We have the expression:

$\displaystyle \lim_{n\to\infty}\sum_{i=1}^{n}f(x_i\, ^\star)\Delta x$

In this case, B is the correct answer.

B states that any sample point can be chosen without affecting the limit.

This is a true statement. Regardless of which point we choose for our sample point, since n approaches infinity, we will have infinite partitions.

Hence, this will not affect the sum of our expression.

A and C are false because we have infinite partitions, so the statements cannot be true. D is not true simply because the left-hand values do not have to be the same as the midpoint-values depending on the behavior of the function for [a, b].