A twice-differentiable function is defined on the closed interval by f(x)=x^ 2 +3x+1

MATHEMATICS · Multi-part · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

A twice-differentiable function

f

is defined on the closed interval

[0,4]

f(x)=x^{2}+3x+1.

Students wish to approximate the value of the definite integral

\displaystyle\int_{0}^{4} f(x)\,dx

using two subintervals of equal length. (a) Using the endpoints

x=0,2,4

, calculate the trapezoidal sum

T_{2}

that approximates

\displaystyle\int_{0}^{4} f(x)\,dx

. Show all numerical work. (b) Using the midpoints

x=1

and

x=3

, calculate the midpoint Riemann sum

M_{2}

that approximates

\displaystyle\int_{0}^{4} f(x)\,dx

. Show all numerical work. (c) Without evaluating the definite integral exactly, determine which of the two approximations

T_{2}

M_{2}

is closer to the exact value of the integral. Justify your conclusion by referencing appropriate features of

f

on the interval. (d) Let

g

be any twice-differentiable function on

[a,b]

for which

g''(x)>0

for all

x

(a,b)

. Prove that when the interval

[a,b]

is partitioned into a single subinterval, the trapezoidal approximation overestimates

\displaystyle\int_{a}^{b} g(x)\,dx

while the midpoint approximation underestimates it. Your proof must employ a calculus-based argument relying on properties of

g''