What is the average rate of change of the function f, given by f(x) = x2 + 1, on the interval [2,5]?
Calculus: rate of change=2x
avg rate= (final+initial)/2=(10+4)/2= 7
Now, pre cal answer>
F(5)=26
F(2)=5
avg change=(26-5)/(5-2)=7
To find the average rate of change of a function on an interval, you need to calculate the difference in function values divided by the difference in the input values on that interval.
In this case, the function f(x) = x^2 + 1 is defined, and you want to find the average rate of change on the interval [2,5].
Step 1: Calculate the function values at the endpoints of the interval.
To find the function value at x = 2, substitute x = 2 into the function equation:
f(2) = (2^2) + 1
f(2) = 4 + 1
f(2) = 5
To find the function value at x = 5, substitute x = 5 into the function equation:
f(5) = (5^2) + 1
f(5) = 25 + 1
f(5) = 26
Step 2: Calculate the difference in function values.
The difference in function values is: f(5) - f(2).
Therefore, 26 - 5 = 21.
Step 3: Calculate the difference in input values.
The difference in input values is: 5 - 2 = 3.
Step 4: Calculate the average rate of change.
The average rate of change is the difference in function values divided by the difference in input values:
Average Rate of Change = (f(5) - f(2)) / (5 - 2)
Average Rate of Change = 21 / 3
Average Rate of Change = 7.
Therefore, the average rate of change of the function f(x) = x^2 + 1 on the interval [2,5] is 7.