Mathematics

Problem

Let

N

and

D

be differentiable functions. The function

R

is defined by

R(x)=\dfrac{N(x)}{D(x)}

. Certain values related to

N

and

D

are shown in the table below.

$x$	$N(x)$	$N'(x)$	$D(x)$	$D'(x)$
$2$	$7$	$-3$	$4$	$5$

(a) Using only the information in the table, compute

R'(2)

. Show all reasoning. (b) Determine whether

R

is increasing or decreasing at

x=2

. Justify your conclusion using your result from part (a). (c) Without performing any additional differentiation, decide which is larger,

R(2.01)

R(2)

, and explain your reasoning.