Oil pumped into a tanker truck

Name: ______________________ Date: ____________ Oil pumped into a tanker truck (9 points)

Table

Oil is pumped into a tanker truck at a rate

R(t)

gallons per minute, where

t

is measured in minutes.

R

is differentiable and increasing on

0 \le t \le 12

. Selected values of

R(t)

are shown in the table.

Rate R(t), in gallons per minute, at selected times t, in minutes
$t$ (minutes)	0	3	5	9	12
$R(t)$ (gallons per minute)	12	18	21	29	30

(a) Use the data in the table to estimate $R'(4)$ . Show the computations that lead to your answer. Indicate units of measure. [2 points]
(b) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate $\int_0^{12} R(t)\,dt$ . Show the computations that lead to your answer. [2 points]
(c) Using correct units, explain the meaning of $\int_0^{12} R(t)\,dt$ in the context of the problem. [1 point]
(d) A left Riemann sum with the four subintervals indicated by the table is used to approximate $\int_0^{12} R(t)\,dt$ . Find its value, and explain whether it is an overestimate or an underestimate. [2 points]
(e) Must there be a value $t$ , with $5 < t < 9$ , for which $R'(t) = 2$ ? Justify your answer. [2 points]

The question