What is the simplified form for the Average Rate of Change of the function f(x)= x^2 between x=−1 and x=−1+h?
Average Rate of Change
= (f(-1+h) - f(-1))/(-1+h - (-1))
= ( (h-1)^2 - (-1)^2)/h
= (h^2 - 2h + 1 - 1)/h
= h - 2
could you explain the steps you took?
you really need to review what the average rate of change means. It is the slope of the secant line.
To find the average rate of change of a function, we need to calculate the difference in the function values over the difference in the input values.
In this case, we have the function f(x) = x^2 and we want to find the average rate of change between x = -1 and x = -1 + h.
The formula for the average rate of change is:
Average Rate of Change = (f(x2) - f(x1)) / (x2 - x1),
where f(x2) is the value of the function at the second x-value, f(x1) is the value of the function at the first x-value, x2 is the second x-value, and x1 is the first x-value.
In our case, x1 = -1 and x2 = -1 + h.
So, substituting the values into the formula, we get:
Average Rate of Change = (f(-1 + h) - f(-1)) / ((-1 + h) - (-1)).
Now, let's calculate the function values:
f(-1) = (-1)^2 = 1,
f(-1 + h) = (-1 + h)^2 = (-1 + h)(-1 + h) = 1 - 2h + h^2.
Substituting these values into the average rate of change formula, we have:
Average Rate of Change = (1 - (-1 + h)^2) / ((-1 + h) - (-1)).
Simplifying further, we have:
Average Rate of Change = (1 - (1 - 2h + h^2)) / (h).
Simplifying the expression inside the parentheses:
Average Rate of Change = (1 - 1 + 2h - h^2) / h.
Further simplifying:
Average Rate of Change = (2h - h^2) / h.
So, the simplified form for the average rate of change of the function f(x) = x^2 between x = -1 and x = -1 + h is (2h - h^2) / h.