Let be twice–differentiable on with and for all in that interval

MATHEMATICS · Multi-part · A Level · Mathematics · Integration · Integration

Problem

Let

f

be twice–differentiable on

[1,4]

with

f'(x)>0

and

f''(x)<0

for all

x

in that interval. Consider four different numerical approximations of

\displaystyle \int_{1}^{4} f(x)\,dx

that each use

n=4

equal subintervals: the Left–Riemann sum

L_4

, the Right–Riemann sum

R_4

, the Midpoint sum

M_4

, and the Trapezoidal sum

T_4

. (a) Rank the four approximations

L_4,\;R_4,\;M_4,

and

T_4

from least to greatest. If any are equal, indicate that explicitly. Provide a clear justification that cites the given information about

f'(x)

and

f''(x)

. (b) Explain why

M_4

must be closer to the exact integral than

L_4

and

R_4

. Support your explanation with geometric or analytic reasoning related to concavity. (c) Suppose the number of subintervals is doubled to

n=8

. Discuss qualitatively how the absolute error in

T_8

compares to the absolute error in

L_8

. No calculation is required, but your answer must reference properties of

f

[1,4]