Let be a differentiable function on with for all in

MATHEMATICS · Single correct · A Level · Mathematics · Integration · Integration

Problem

Let

h

be a differentiable function on

[0,6]

with

h''(x)>0

for all

x

[0,6]

. Let

E_M

and

E_T

denote the errors in the midpoint and trapezoidal approximations, respectively, when four equal subintervals are used to approximate

\displaystyle \int_{0}^{6} h(x)\,dx

; that is,

E_M=M-\int_{0}^{6} h(x)\,dx

and

E_T=T-\int_{0}^{6} h(x)\,dx

. Which of the following must be true? (a)

E_T<0<E_M

and

|E_T|<|E_M|

(b)

E_M<0<E_T

and

|E_M|<|E_T|

(c)

E_M<0<E_T

and

|E_M|>|E_T|

(d)

E_M>0

and

E_T>0

with

E_M<E_T