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.
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.
f(x) = 2*3^x
on [0,1] the average rate is
(f(1)-f(0)))/(1-0) = (2*3^1 - 2*3^0)/1 = 6-2 = 4
on [2,3] the average rate is (f(3)-f(2))/(3-2) = (2*3^3 - 2*3^2)/1 = 54-18 = 36
the ratio of the rates is 36/4 = 9.
The reason the rate increases is that exponentials grow ever faster. The larger x gets, the steeper the curve is. The rate of change is proportional to the function value.
Thank you so much Steve :)
Part A: To find the average rate of change of each section, we will use the formula:
Average rate of change = (f(b) - f(a)) / (b - a)
For Section A, a = 0 and b = 1:
Average rate of change (Section A) = (f(1) - f(0)) / (1 - 0)
Substituting the values into the function:
Average rate of change (Section A) = (2(3)^1 - 2(3)^0) / (1 - 0)
Average rate of change (Section A) = (6 - 2) / 1
Average rate of change (Section A) = 4
For Section B, a = 2 and b = 3:
Average rate of change (Section B) = (f(3) - f(2)) / (3 - 2)
Substituting the values into the function:
Average rate of change (Section B) = (2(3)^3 - 2(3)^2) / (3 - 2)
Average rate of change (Section B) = (54 - 18) / 1
Average rate of change (Section B) = 36
Therefore, the average rate of change for Section A is 4, and for Section B is 36.
Part B: To find out how many times greater the average rate of change of Section B is than Section A, we can divide the rate of change of Section B by the rate of change of Section A:
Times greater = Rate of change (Section B) / Rate of change (Section A)
Times greater = 36 / 4
Times greater = 9
Therefore, the average rate of change of Section B is 9 times greater than the average rate of change of Section A.
The reason one rate of change is greater than the other is due to the exponential nature of the function f(x) = 2(3)^x. As the value of x increases, the function grows at an exponential rate, resulting in a higher rate of change. The greater base of the exponential function in Section B (3^x) compared to Section A (3^x) leads to a significantly larger average rate of change in Section B.
Part A: To find the average rate of change of a function over a specific interval, we need to calculate the change in y divided by the change in x.
For Section A, the interval is from x = 0 to x = 1. Therefore, we need to calculate the change in y and the change in x.
Change in y = f(1) - f(0) = 2(3)^1 - 2(3)^0 = 6 - 2 = 4.
Change in x = 1 - 0 = 1.
Average rate of change for Section A = Change in y / Change in x = 4 / 1 = 4.
For Section B, the interval is from x = 2 to x = 3. Similarly, we calculate the change in y and the change in x.
Change in y = f(3) - f(2) = 2(3)^3 - 2(3)^2 = 54 - 18 = 36.
Change in x = 3 - 2 = 1.
Average rate of change for Section B = Change in y / Change in x = 36 / 1 = 36.
Part B: To find how many times greater the average rate of change of Section B is than Section A, we can simply calculate the ratio of the average rates of change.
Ratio of average rates of change = Average rate of change of Section B / Average rate of change of Section A.
In this case, the ratio = 36 / 4 = 9.
The average rate of change of Section B is 9 times greater than the average rate of change of Section A.
The reason one rate of change is greater than the other is because the exponential function grows at an increasing rate. As x increases, the values of 3^x become larger and larger, causing the function to grow more rapidly. Therefore, the average rate of change for Section B is higher than that of Section A.