For the function f(x) = x^3+3x, determine the average rate of change of f(x) with respect to x over the interval 2<x<4
1. 31
2. -36
3. -40
4. 36
To find the average rate of change of a function over an interval, we need to calculate the difference in function values and divide it by the difference in x-values.
The function is f(x) = x^3 + 3x.
First, we find f(4):
f(4) = 4^3 + 3(4) = 64 + 12 = 76
Next, we find f(2):
f(2) = 2^3 + 3(2) = 8 + 6 = 14
The difference in function values is:
f(4) - f(2) = 76 - 14 = 62
The difference in x-values is:
4 - 2 = 2
Now, we calculate the average rate of change:
Average rate of change = (f(4) - f(2))/(4 - 2) = 62/2 = 31
Therefore, the correct answer is 1. 31.
To determine the average rate of change of the function f(x) = x^3 + 3x over the interval 2 < x < 4, we need to find the difference in the function values at the endpoints of the interval divided by the difference in the x-values.
First, let's find the value of f(x) at the lower endpoint of the interval (x = 2):
f(2) = (2)^3 + 3(2) = 8 + 6 = 14
Next, let's find the value of f(x) at the upper endpoint of the interval (x = 4):
f(4) = (4)^3 + 3(4) = 64 + 12 = 76
Now, we can find the average rate of change by taking the difference in function values divided by the difference in x-values:
Average Rate of Change = (f(4) - f(2)) / (4 - 2) = (76 - 14) / 2 = 62 / 2 = 31
Therefore, the average rate of change of f(x) with respect to x over the interval 2 < x < 4 is 31. Hence, the correct answer is option 1.