Which of these has the greatest average rate of change between 2 and 3 : f(x)=x ;g(x)=X^2 ;h(x)=x^3?
To find the average rate of change between two points on a function, we need to calculate the difference in the values of the function at those two points and divide it by the difference in the corresponding x-values.
Let's calculate the average rate of change for each function between x = 2 and x = 3:
1. For f(x) = x:
- f(2) = 2
- f(3) = 3
- Average rate of change = (f(3) - f(2)) / (3 - 2) = (3 - 2) / (1) = 1
2. For g(x) = x^2:
- g(2) = 2^2 = 4
- g(3) = 3^2 = 9
- Average rate of change = (g(3) - g(2)) / (3 - 2) = (9 - 4) / (1) = 5
3. For h(x) = x^3:
- h(2) = 2^3 = 8
- h(3) = 3^3 = 27
- Average rate of change = (h(3) - h(2)) / (3 - 2) = (27 - 8) / (1) = 19
Comparing the average rates of change we found:
- f(x) = x has an average rate of change of 1 between x = 2 and x = 3.
- g(x) = x^2 has an average rate of change of 5 between x = 2 and x = 3.
- h(x) = x^3 has an average rate of change of 19 between x = 2 and x = 3.
Therefore, h(x) = x^3 has the greatest average rate of change between 2 and 3.