Answer
b) (i)$R_4\approx 4.4252$
(ii) $M_4\approx 3.8426$
c) (i) $R_8 \approx 4.1339$
(ii) $M_8\approx 3.8833$
Work Step by Step
a) $f(x) = x - 2 \ln(x)$
We graph the function.
b)
$R_4 = 1 \times [(2 - 2\ln(2)) + (3 - 2\ln(3)) + (4 - 2\ln(4)) + (5 - 2\ln(5))]$
$R_4 \approx 4.4252$
$M_4 = 1 \times [(1.5 - 2\ln(1.5)) + (2.5 - 2\ln(2.5)) + (3.5 - 2\ln(3.5)) + (4.5 - 2\ln(4.5))]$
$M_4 \approx 3.8426$
c) $R_8 = 0.5 \times [(1.5 - 2\ln(1.5)) + (2 - 2\ln(2)) + (2.5 - 2\ln(2.5)) + (3 - 2\ln(3)) + (3.5 - 2\ln(3.5)) + (4 - 2\ln(4)) + (4.5 - 2\ln(4.5)) + (5 - 2\ln(5))]$
$R_8 \approx 4.1339$
$M_8 = 0.5 \times [(1.25 - 2\ln(1.25)) + (1.75 - 2\ln(1.75)) + (2.25 - 2\ln(2.25)) + (2.75 - 2\ln(2.75)) + (3.25 - 2\ln(3.25)) + (3.75 - 2\ln(3.75)) + (4.25 - 2\ln(4.25)) + (4.75 - 2\ln(4.75))]$
$M_8 \approx 3.8833$
