A function is given.
f(t)=6/t,t=a,t=a+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 find the net change between the given values of the variable, we need to find the difference between the function values at those points. In this case, we are given f(t)=6/t and the values of t are a and a+h.
So, the net change is given by:
Change = f(a+h) - f(a)
Substituting the given function values, we get:
Change = (6/(a+h)) - (6/a)
To find the average rate of change between the given values of the variable, we need to find the slope of the secant line passing through the two points. This is given by the formula:
Average rate of change = (f(a+h) - f(a))/(a+h - a)
Simplifying this equation, we get:
Average rate of change = (f(a+h) - f(a))/h
Substituting the given function values, we get:
Average rate of change = ((6/(a+h)) - (6/a))/h
To determine the net change between the given values of the variable, we need to evaluate f(a+h) - f(a).
(a) Net change:
f(a+h) = 6 / (a + h)
f(a) = 6 / a
Therefore, the net change between f(a+h) and f(a) is:
Net change = f(a+h) - f(a)
Net change = (6 / (a + h)) - (6 / a)
To determine the average rate of change between the given values of the variable, we need to evaluate the slope of the line connecting the two points (a, f(a)) and (a+h, f(a+h)).
(b) Average rate of change:
Average rate of change = (f(a+h) - f(a)) / (a+h - a)
Average rate of change = (f(a+h) - f(a)) / h
From part (a), we know that f(a+h) - f(a) = (6 / (a + h)) - (6 / a)
Therefore, the average rate of change is:
Average rate of change = [(6 / (a + h)) - (6 / a)] / h