A function is given. Determine the average rate of change of the function between the given values of the variable.
f(t) =
6/t
; t = a, t = a + h
f(t)=6/t
t1=a, t2=a+h
average rate of change
= Δf(t)/Δt
=(f(a+h)-f(a))/(a+h-a)
=(6/(a+h)-6/a))/(h)
=(6/h)(1/(a+h)-1/a)
...
simplify to get final answer.
To determine the average rate of change of a function between two points, we need to find the difference in the function values at those points divided by the difference in the variable values.
In this case, the function is f(t) = 6/t, and we want to find the average rate of change between t = a and t = a + h.
Step 1: Find the function value at t = a.
Substitute a into the function: f(a) = 6/a.
Step 2: Find the function value at t = a + h.
Substitute a + h into the function: f(a + h) = 6/(a + h).
Step 3: Calculate the difference in function values.
Subtract the function value at t = a from the function value at t = a + h: f(a + h) - f(a) = (6/(a + h)) - (6/a).
Step 4: Calculate the difference in variable values.
The difference in variable values is h, since we are finding the average rate of change between t = a and t = a + h.
Step 5: Calculate the average rate of change.
Divide the difference in function values by the difference in variable values: (f(a + h) - f(a))/h = ((6/(a + h)) - (6/a))/h.
Therefore, the average rate of change of the function f(t) = 6/t between t = a and t = a + h is ((6/(a + h)) - (6/a))/h.