The function f(x) = 2x + 1 is defined over the interval [2, 5]. If the interval is divided into n equal parts, what is the value of the function at the right endpoint of the kth rectangle?

A) 2+3k/n
B) 4+3k/n
C) 4+6k/n
D) 5+6k/n
E) 5+3k/n

spread = 5-2 = 3

if cut into n pieces, each base = 3/n

endpoint of 1st rectangle = 2 + 3/n
endpoint of 2nd rectangle = 2 + 2(3/n)
...
endpoint of kth rectangle = 2 + k(3/n)

= 2 + 3k/n

f(2+3k/n)
= 2(2 + 3k/n) + 1
= 5 +6k/2

I see you have two other questions above that follow this reasoning.
Follow the same analytic thinking by setting up a pattern.

Why you bully me

To find the value of the function at the right endpoint of the kth rectangle, we first need to determine the width of each rectangle.

The interval [2, 5] is divided into n equal parts, so each rectangle has a width of (5 - 2)/n = 3/n.

The right endpoint of the kth rectangle can be represented as xk = 2 + (k * 3/n), where k ranges from 1 to n.

To find the value of the function at xk, we substitute xk into the function f(x) = 2x + 1:

f(xk) = 2(xk) + 1

Substituting xk = 2 + (k * 3/n):

f(xk) = 2(2 + (k * 3/n)) + 1
= 4 + (6k/n) + 1
= 5 + (6k/n)

Therefore, the value of the function at the right endpoint of the kth rectangle is 5 + (6k/n).

The correct answer is option D) 5 + (6k/n).

To find the value of the function at the right endpoint of the kth rectangle, we need to determine the x-coordinate of the right endpoint of the kth rectangle first.

The interval [2, 5] is divided into n equal parts. The length of each part, denoted as Δx, can be calculated by taking the difference between the endpoints of the interval and dividing it by the number of parts: Δx = (5 - 2) / n = 3/n.

Since we start at x = 2, the x-coordinate of the right endpoint of the kth rectangle can be expressed as x = 2 + kΔx.

Now, let's substitute the expression for x into the function f(x) = 2x + 1:

f(2 + kΔx) = 2(2 + kΔx) + 1 = 4 + 2kΔx + 1 = 5 + 2k(3/n) + 1.

Simplifying the above expression, we get:

f(2 + kΔx) = 6k/n + 5.

Therefore, the value of the function at the right endpoint of the kth rectangle is 6k/n + 5.

Comparing this expression to the given answer choices, we can see that the correct answer is D) 5 + 6k/n.