Mathematics

Problem

A continuous function

f

is defined on the closed interval

[0,5]

f(x)=e^{0.2x}

. The interval is partitioned into four equal subintervals. (a) Let

L_{4}

denote the left–endpoint Riemann sum and

R_{4}

denote the right–endpoint Riemann sum that approximate

\int_{0}^{5} f(x)\,dx

using this partition. Write each sum explicitly as a numerical expression (do not evaluate). (b) Without calculating either sum, determine which of

L_{4}

R_{4}

gives the larger approximation. Provide a reasoned explanation based on the behavior of

f

[0,5]

. (c) The Mean Value Theorem for Integrals guarantees the existence of

c\in[0,5]

such that

\int_{0}^{5} f(x)\,dx=f(c)(5-0)

. Explain how the conclusion of part (b) is consistent with this theorem in light of the fact that

f

is increasing on

[0,5]

. (a) — see stem