MATH SOLVE

3 months ago

Q:
# Given the function f(x) = 2(3)x, Section A is from x = 0 to x = 1 and Section B is from x = 2 to x = 3.Part A: Find the average rate of change of each section. (4 points)Part B: How many times greater is the average rate of change of Section B than Section A? Explain why one rate of change is greater than the other. (6 points)

Accepted Solution

A:

Part A:
To find the average rate of change, let us first write out the equation to find it.
Δy/Δx = average rate of change.

Finding average rate of change for Section A Δy = f(1) - f(0) = 2(3)^1 - 2(3)^0 = 6 - 1 = 5 Δx = 1- 0 = 1

Plug the numbers in: Δy/Δx = 5/1 = 5 Therefore, the average rate of change for Section A is 5.

Finding average rate of change for Section B Δy = f(3) - f(2) = 2(3)^3 - 2(3)^2 = 2(27) - 2(9) = 54 - 18 = 36 Δx = 3 - 2 = 1

Plug the numbers in: Δy/Δx = 36/1 = 36 Therefore, the average rate of change for Section B is 36.

Part B:

(a) How many times greater is the average rate of change of Section B than Section A?

If Section B is on the interval [2,3] and Section A is on the interval [0,1]. For the function f(x) = 2(3)^x, the average rate of change of Section B is 7.2 times greater than the average rate of change of Section A.

(b) Explain why one rate of change is greater than the other.

Since f(x) = 2(3)^x is an exponential function the y values do not increase linearly, instead increase exponentially. In an interval with smaller x values the rate of change is lower than an interval with larger x values.

Finding average rate of change for Section A Δy = f(1) - f(0) = 2(3)^1 - 2(3)^0 = 6 - 1 = 5 Δx = 1- 0 = 1

Plug the numbers in: Δy/Δx = 5/1 = 5 Therefore, the average rate of change for Section A is 5.

Finding average rate of change for Section B Δy = f(3) - f(2) = 2(3)^3 - 2(3)^2 = 2(27) - 2(9) = 54 - 18 = 36 Δx = 3 - 2 = 1

Plug the numbers in: Δy/Δx = 36/1 = 36 Therefore, the average rate of change for Section B is 36.

Part B:

(a) How many times greater is the average rate of change of Section B than Section A?

If Section B is on the interval [2,3] and Section A is on the interval [0,1]. For the function f(x) = 2(3)^x, the average rate of change of Section B is 7.2 times greater than the average rate of change of Section A.

(b) Explain why one rate of change is greater than the other.

Since f(x) = 2(3)^x is an exponential function the y values do not increase linearly, instead increase exponentially. In an interval with smaller x values the rate of change is lower than an interval with larger x values.