https://file.aiquickdraw.com/mathbot/file/1726955131343pnptex08.jpg

This exercise involves estimating the slope of the tangent line at the point \( (0, 8) \) on the graph of the function \( f(x) = \frac{16}{x+2} \), by using the slopes of several secant lines.

To find the slope of a secant line passing through the points \( (0, 8) \) and \( (x, f(x)) \), we use the formula for the slope between two points:

\[
\text{slope of secant line} = \frac{f(x) - f(0)}{x - 0}
\]

Given that \( f(0) = 8 \) (since \( f(x) = \frac{16}{x+2} \), and \( f(0) = \frac{16}{0+2} = 8 \)):

\[
\text{slope of secant line} = \frac{f(x) - 8}{x}
\]

Now, for each \( x \) value provided in the table, we substitute into \( f(x) = \frac{16}{x+2} \) to compute the slope.

Let me calculate the slopes for each \( x \) value in the table.Here are the slopes of the secant lines for each \( x \) value in the table:

\[
\begin{array}{|c|c|}
\hline
\text{Value of } x & \text{Slope} \\
\hline
-0.2 & -4.4444 \\
-0.1 & -4.2105 \\
-0.01 & -4.0201 \\
-0.001 & -4.0020 \\
0.001 & -3.9980 \\
0.01 & -3.9801 \\
0.1 & -3.8095 \\
0.2 & -3.6364 \\
\hline
\end{array}
\]

These slopes give an estimate of the slope of the tangent line at \( (0, 8) \), which appears to be approaching a value near \(-4\).

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How does the slope of the secant line change as \( x \) approaches zero?
2. What is the mathematical definition of a tangent line in calculus?
3. Can you explain why the slopes approach a specific value as \( x \to 0 \)?
4. How would you use limits to find the exact slope of the tangent line at \( (0, 8) \)?
5. What is the derivative of \( f(x) = \frac{16}{x+2} \), and how does it relate to these secant slopes?

**Tip**: The slope of the tangent line at a point can be found using the derivative of the function at that point!

For each value of x given in the table below, find the slope of the secant line passing through (0, 8) and (x, f(x)). Do not round intermediate computations, and round your answers to 4 decimal places if necessary.

This problem involves finding the slope of secant lines and estimating the slope of the tangent line at (0, 8) for the function f(x) = 16 / (x + 2). The solution uses calculus concepts including limits and derivatives. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation