find the average rate of the function over the given interval
y = x ^ 2 + 4x, [3, 8]
To find the average rate of the function over the interval [3, 8], we need to calculate the average rate of change, also known as the slope of the line connecting the endpoints.
The formula for the average rate of change is given by:
Average Rate of Change = (f(b) - f(a)) / (b - a)
Where f(b) represents the value of the function at the endpoint b, f(a) represents the value of the function at the endpoint a, and b and a are the endpoints of the interval.
In this case, the endpoints are a = 3 and b = 8.
Let's calculate the average rate of change:
f(3) = (3^2) + 4(3) = 9 + 12 = 21
f(8) = (8^2) + 4(8) = 64 + 32 = 96
Average Rate of Change = (f(8) - f(3)) / (8 - 3)
= (96 - 21) / (8 - 3)
= 75 / 5
= 15
Therefore, the average rate of the function over the interval [3, 8] is 15.
To find the average rate of the function over the interval [3, 8], we need to calculate the slope of the function between these two points and divide it by the difference in x-values.
1. Calculate the y-values for the given x-values:
When x = 3: y = (3^2) + (4*3) = 9 + 12 = 21
When x = 8: y = (8^2) + (4*8) = 64 + 32 = 96
2. Calculate the difference in y-values (Δy):
Δy = 96 - 21 = 75
3. Calculate the difference in x-values (Δx):
Δx = 8 - 3 = 5
4. Calculate the average rate of the function:
Average rate = Δy / Δx = 75 / 5 = 15
Therefore, the average rate of the function y = x^2 + 4x over the interval [3, 8] is 15.