A twice–differentiable function is positive on the closed interval and satisfies for…

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

Problem

A twice–differentiable function

g

is positive on the closed interval

[2,8]

and satisfies

g''(x)>0

for all

x

in that interval. A student partitions

[2,8]

into

n

equal subintervals of width

\Delta x

and, on each subinterval

[x_k,x_{k+1}]

, replaces the graph of

g

with the straight-line segment joining the points

(x_k,g(x_k))

and

(x_{k+1},g(x_{k+1}))

. The student then adds the areas bounded by each segment, the

x

-axis, and the lines

x=x_k

and

x=x_{k+1}

to obtain an estimate

T_n

for

\displaystyle\int_{2}^{8} g(x)\,dx

. (1) Part 1: (a) Determine whether

T_n

is an overestimate or an underestimate of

\displaystyle\int_{2}^{8} g(x)\,dx

. Clearly explain your reasoning. (2) Part 2: (b) Describe how the absolute percent error

\dfrac{|T_n-\int_{2}^{8}g(x)\,dx|}{\int_{2}^{8}g(x)\,dx}\times100\%

changes as

n

increases. Justify your conclusion without performing any explicit numerical calculation.