s(t)= -4.9t^2+15t+1 - write an expression to represent the average rate of change over the interval '1' less than or equal to 't' less than or equal to '1+h'
To find the average rate of change of a function over an interval, you need to calculate the difference in function values at the endpoints of the interval and then divide it by the difference in the independent variable (in this case, t).
Let's calculate the average rate of change of the function s(t) = -4.9t^2 + 15t + 1 over the interval 1 ≤ t ≤ 1 + h.
First, we need to find the value of s(t) at the endpoint t = 1. We can substitute this value into the function and simplify:
s(1) = -4.9(1)^2 + 15(1) + 1
s(1) = -4.9 + 15 + 1
s(1) = 11.1
Next, we need to find the value of s(t) at the endpoint t = 1 + h. Similarly, we substitute this value into the function:
s(1 + h) = -4.9(1 + h)^2 + 15(1 + h) + 1
s(1 + h) = -4.9(1 + 2h + h^2) + 15 + 15h + 1
s(1 + h) = -4.9 - 9.8h - 4.9h^2 + 15 + 15h + 1
s(1 + h) = -4.9h^2 + 5.2h + 11.1
Now, we can calculate the average rate of change (AROC) by subtracting the value at t = 1 from the value at t = 1 + h and then dividing it by h:
AROC = (s(1 + h) - s(1)) / h
= (-4.9h^2 + 5.2h + 11.1 - 11.1) / h
= (-4.9h^2 + 5.2h) / h
= -4.9h + 5.2
Therefore, the expression representing the average rate of change over the interval 1 ≤ t ≤ 1 + h for the function s(t) = -4.9t^2 + 15t + 1 is -4.9h + 5.2.