Answer
(a) We can see a sketch of the graph below.
(b) The average rate of change is larger on the interval $[2,3]$
(c) The instantaneous rate of change is larger at $x = 2$
(d) $f'(2) \gt f'(5)$
Work Step by Step
$f(x) = x - 2sin~x$
(a) We can see a sketch of the graph below.
(b) On the interval $[1,2]$, the graph increases by less than 1 unit.
On the interval $[2,3]$, the graph increases by more than 1 unit.
Therefore, the average rate of change is larger on the interval $[2,3]$
(c) The instantaneous rate of change is the slope of the graph at any point $x$.
The slope at $x = 2$ seems to be larger than the slope at $x = 5$
Therefore, the instantaneous rate of change is larger at $x = 2$
(d) $f'(x) = 1-2cos~x$
$f'(2) = 1-2cos~(2) = 1.8$
$f'(5) = 1-2cos~(5) = 0.4$
Therefore, $f'(2) \gt f'(5)$
