Chapter 11.1, Problem 57E

Solution

To sketch: a graph of the derivative $y=f\text{'}\left(x\right)$ using geometrical interpretation.

  

Given information:

The graph of $f\left(x\right)=x^2{e}^{-x}$ is:

  

Calculation:

Tabulate the values for the slopes, as derivative of the graph at a specific point is the slope of tangent of the curve at that point. The points are then plotted to draw the graph. It is evident from the graph that the tangent lines are horizontal at $0$ and $x=2$ . It can be observed from the graph that slopes are positive when the values of $x$ lie between $0$ and $2$ . The slopes are negative when $x$ becomes greater than $2$ . The slopes approach a horizonal line which implies that $f\text{'}\left(x\right)$ approaches $0$ .

$x$	$f\text{'}\left(x\right)$
$0$	$0$
$0.5$	$0.45$
$1$	$0.37$
$1.5$	0.16
$2$	$0$
$2.5$	$-0.10$
$3$	$-0.15$
$4$	$-0.15$
$6$	$-0.06$
$8$	$-0.02$
10	0