Consider the function as representing the value of an ounce of palladium in U.S.
ID: 3152434 • Letter: C
Question
Consider the function as representing the value of an ounce of palladium in U.S. dollars as a function of the time t in days.†
R(t) = 40t 3t2; t = 3
Find the average rate of change of R(t) over the time intervals [t, t + h], where t is as indicated and h = 1, 0.1, and 0.01 days. (Use smaller values of h to check your estimates.) HINT [See Example 1.] (Round your answers to two decimal places.)
Estimate the instantaneous rate of change of R at time t, specifying the units of measurement.
R'(3) = ? $/day
Explanation / Answer
The formula for the average rate of change is:
(f(t + h) - f(t))/h
The t + h in f(t + h) means that you plug in t + h instead of t. Plug in t = 3 and simplify it:
(f(3 + h) - f(3))/h
=> [40(3+h) - 3(3+h)2 - 93 ]/h
=> [120 + 40h - 3(9 +6h + h2) -93 ] / h
=> [120+ 40h -27 -18h -3h2 -93 ] / h
=> [ 93 + 22h - 3h2 -93] / h
=> 22 - 3h
We'll get the instantaneous velocity by differentiating the R(t)
R'(t) = 40 - 6t
substitute here t = 3
=> R'(3) = 40 - 6 * 3
=> R'(3) = 22
h 1 0.1 0.01 Average rate 22-3(1) =19 22-3(0.1) = 21.7 21.97