Using the exponential function 2^x + 3, what is the average rate of change between the values 1 ≤ x ≤ 5?
f(1) = 5
f(5) = 35
(35 - 5) / (5 - 1) = ?
that would, of course, be ∆y/∆x =
f(5)-f(1)
------------- = ((2^5+3)-(2^1+3))/4 = (35-5)/4 = 15/2
5-1
To find the average rate of change between the values 1 ≤ x ≤ 5 for the exponential function 2^x + 3, we need to calculate the difference in function values divided by the difference in x values.
Step 1: Calculate the function values for x = 1 and x = 5.
For x = 1: 2^1 + 3 = 2 + 3 = 5
For x = 5: 2^5 + 3 = 32 + 3 = 35
Step 2: Calculate the difference in function values.
Difference in function values = 35 - 5 = 30
Step 3: Calculate the difference in x values.
Difference in x values = 5 - 1 = 4
Step 4: Calculate the average rate of change.
Average rate of change = Difference in function values / Difference in x values
Average rate of change = 30 / 4 = 7.5
Therefore, the average rate of change between the values 1 ≤ x ≤ 5 for the exponential function 2^x + 3 is 7.5.
To find the average rate of change of a function between two points, we need to calculate the difference in function values over the difference in the independent variable. In this case, the function is given as 2^x + 3, and we are looking for the average rate of change between x = 1 and x = 5.
First, let's find the function values at those two points:
At x = 1:
f(1) = 2^1 + 3 = 2 + 3 = 5
At x = 5:
f(5) = 2^5 + 3 = 32 + 3 = 35
Now, we can calculate the average rate of change:
Average Rate of Change = (f(5) - f(1)) / (5 - 1)
Substituting the values we found:
Average Rate of Change = (35 - 5) / (5 - 1)
= 30 / 4
= 7.5
Therefore, the average rate of change between x = 1 and x = 5 for the function 2^x + 3 is 7.5.