A function is given.
f(t)=3t square, t=5,t=5+h
(a) Determine the net change between the given values of the variable.
(b) Determine the average rate of change between the given values of the variable.
To determine the net change between the given values of the variable, we need to find the difference between f(t=5) and f(t=5+h).
(a) Net change: f(t=5+h) - f(t=5)
= (3(5+h)^2) - (3(5)^2)
= (3(25+10h+h^2)) - (3(25))
= 75 + 30h + 3h^2 - 75
= 30h + 3h^2
Therefore, the net change between the given values is 30h + 3h^2.
(b) To find the average rate of change between the given values, we need to divide the net change by the change in t.
Average rate of change = (30h + 3h^2)/h
= 30 + 3h
Therefore, the average rate of change between the given values is 30 + 3h.
To determine the net change between the given values of the variable, we need to find the difference between the function values at those points.
(a)
Given function: f(t) = 3t^2
Let's calculate the function values at t = 5 and t = 5+h.
For t = 5:
f(5) = 3(5)^2
= 3(25)
= 75
For t = 5+h:
f(5+h) = 3(5+h)^2
= 3(25 + 10h + h^2)
= 75 + 30h + 3h^2
The net change between the given values of the variable is the difference between f(5+h) and f(5):
Net Change = f(5+h) - f(5)
= (75 + 30h + 3h^2) - 75
= 30h + 3h^2
(b)
To determine the average rate of change between the given values of the variable, we need to find the slope of the line connecting the two points.
The average rate of change (or slope) between the given values is given by:
Average Rate of Change = Net Change / Change in t
In this case, the change in t is (5+h) - 5 = h.
Average Rate of Change = (30h + 3h^2) / h
= 30 + 3h
Therefore, the average rate of change between the given values of the variable is 30 + 3h.