Question

for the function \\(f(x) = -6x^2\\), make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at \\(x = 2\\).

complete the table.
(do not round until the final answer. then round to the nearest thousandth as needed.)

\

$$\begin{tabular}{|c|c|} \\hline interval & slope of the secant line \\\\ \\hline \\(1, 2\\) & \\\\ \\hline \\(1.5, 2\\) & \\\\ \\hline \\(1.9, 2\\) & \\\\ \\hline \\(1.99, 2\\) & \\\\ \\hline \\(1.999, 2\\) & \\\\ \\hline \\end{tabular}$$

it can be conjectured that the slope of the tangent line at \\(x = 2\\) is .
(round to the nearest integer as needed.)

Explanation:

Calculate secant line slopes

$$
LATEXBLOCK0
$$

Conjecture tangent slope

$$ \lim_{x_1 \to 2^-} m_{\sec} = \lim_{x_1 \to 2^-} -6(2 + x_1) = -24 $$

Answer:

Complete the table:

Interval	Slope of the Secant Line
\([1.5, 2]\)	<blank>-21</blank>
\([1.9, 2]\)	<blank>-23.4</blank>
\([1.99, 2]\)	<blank>-23.94</blank>
\([1.999, 2]\)	<blank>-23.994</blank>

It can be conjectured that the slope of the tangent line at \(x = 2\) is <blank>-24</blank>.