Chapter 11.3, Problem 52E

To determine

The table values for the given function and the compare the value of first derivative with a visual approximation of the slope of the graph.

Answer to Problem 52E

Using Desmos graphing utility, the graph of the given function is

  

And the table values are as follows:

$x$	$-2$	$-1.5$	$-1$	$-0.5$	0	0.5	1	1.5	2
$f(x)$	$-2$	$-0.84375$	$-0.25$	$-0.03125$	0	$0.03125$	0.25	$0.84375$	2
$f\text{'}(x)$	3	$1.6875$	$0.75$	$0.1875$	0	$0.1875$	0.75	1.6875	3

When comparing the values of the first derivative with a visual approximation of the slope of the graph, they appear to the same.

Explanation of Solution

Giveninformation:

The given function is $f(x)=\frac{1}{4}{x}^3$ and the table is as follows:

$x$	$-2$	$-1.5$	$-1$	$-0.5$	0	0.5	1	1.5	2
$f(x)$
$f\text{'}(x)$

Calculations:

Using Desmos graphing utility, the graph of the given function is

  

And the table values are as follows:

$x$	$-2$	$-1.5$	$-1$	$-0.5$	0	0.5	1	1.5	2
$f(x)$	$-2$	$-0.84375$	$-0.25$	$-0.03125$	0	$0.03125$	0.25	$0.84375$	2
$f\text{'}(x)$	3	$1.6875$	$0.75$	$0.1875$	0	$0.1875$	0.75	1.6875	3

Conclusion:

When comparing the values of the first derivative with a visual approximation of the slope of the graph, they appear to the same.