Questions: For the function (f(x)=4 x^2), make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at (x=3). Complete the table. (Do not round until the final answer. Then round to the nearest thousandth as needed.) Interval Slope of the Secant Line ------ [2,3] 20 [2.5,3]

For the function (f(x)=4 x^2), make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at (x=3).

Complete the table.
(Do not round until the final answer. Then round to the nearest thousandth as needed.)

Interval Slope of the Secant Line
------
[2,3] 20
[2.5,3]

Transcript text: For the function $f(x)=4 x^{2}$, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at $x=3$. Complete the table. (Do not round until the final answer. Then round to the nearest thousandth as needed.) \begin{tabular}{|c|c|} \hline Interval & \begin{tabular}{c} Slope of the \\ Secant Line \end{tabular} \\ \hline$[2,3]$ & 20 \\ \hline$[2.5,3]$ & $\square$ \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Define the Function and Calculate Secant Slope

The function given is $ f(x) = 4x^2 $. To find the slope of the secant line over the interval $[2.5, 3]$, we use the formula for the slope of a secant line:

\[ \text{Slope of secant line} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]

Substituting $ x_1 = 2.5 $ and $ x_2 = 3 $:

\[ f(2.5) = 4(2.5)^2 = 25 \] \[ f(3) = 4(3)^2 = 36 \]

The slope of the secant line is:

\[ \frac{36 - 25}{3 - 2.5} = \frac{11}{0.5} = 22 \]

Step 2: Conjecture the Slope of the Tangent Line at $ x = 3 $

To conjecture the slope of the tangent line at $ x = 3 $, we observe the pattern of the secant slopes as the intervals approach $ x = 3 $. The slope of the secant line over $[2, 3]$ is 20, and over $[2.5, 3]$ is 22. As the interval gets smaller, the slope approaches a certain value.

For a quadratic function $ f(x) = ax^2 $, the derivative $ f'(x) = 2ax $ gives the slope of the tangent line. Here, $ a = 4 $, so:

\[ f'(x) = 8x \]

At $ x = 3 $:

\[ f'(3) = 8 \times 3 = 24 \]