Let's break down the problem step by step.

We are given:

• Point P(16, 4) and Point Q(x, y) where $f(x) = x$.
• The equation of the function is $f(x) = x$, so $y = x$ for any given point on the graph.

Step 1: Find the y-coordinate of Q for each given x-coordinate

From the equation of the function $f(x) = x$, the y-coordinate of point Q will be the same as its x-coordinate:

• For $x = 16.1$, the y-coordinate of Q is $y = 16.1$.
• For $x = 16.01$, the y-coordinate of Q is $y = 16.01$.
• For $x = 16.001$, the y-coordinate of Q is $y = 16.001$.
• For $x = 16.0001$, the y-coordinate of Q is $y = 16.0001$.

Step 2: Calculate the slope of the secant line ($m_{\text{sec}}$)

The slope of the secant line passing through points $P(16, 4)$ and $Q(x, y)$ is given by the formula:

$m_{\text{sec}} = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1) = (16, 4)$ and $(x_2, y_2) = (x, y)$.

Let's calculate $m_{\text{sec}}$ for each value of $x$:

1. For $x = 16.1$: $m_{\text{sec}} = \frac{16.1 - 4}{16.1 - 16} = \frac{12.1}{0.1} = 121$

2. For $x = 16.01$: $m_{\text{sec}} = \frac{16.01 - 4}{16.01 - 16} = \frac{12.01}{0.01} = 120.9999999$

3. For $x = 16.001$: $m_{\text{sec}} = \frac{16.001 - 4}{16.001 - 16} = \frac{12.001}{0.001} = 120.999999999$

4. For $x = 16.0001$: $m_{\text{sec}} = \frac{16.0001 - 4}{16.0001 - 16} = \frac{12.0001}{0.0001} = 120.9999999999$

Step 3: Use the values from the secant slopes to estimate the tangent slope at $x = 16$

Looking at the values of the secant slopes, they approach 121 as $x$ gets closer to 16. Thus, the slope of the tangent line at $x = 16$ is approximately 121.

Step 4: Find the equation of the tangent line

The equation of a tangent line at a point $(x_1, y_1)$ on a function is given by:

$y - y_1 = m_{\text{tangent}} (x - x_1)$ where $m_{\text{tangent}}$ is the slope of the tangent line, and $(x_1, y_1) = (16, 4)$.

Substituting the values:

• $m_{\text{tangent}} = 121$
• $x_1 = 16$, $y_1 = 4$

$y - 4 = 121(x - 16)$ Expanding: $y - 4 = 121x - 1936$ $y = 121x - 1932$

Final Answer

The equation of the tangent line at $x = 16$ is:

$y = 121x - 1932$

Summary Table:

x	y	$m_{\text{sec}}$
16.1	16.1	121.0000000
16.01	16.01	120.9999999
16.001	16.001	120.999999999
16.0001	16.0001	120.9999999999

Would you like more details or clarification on any step?