Given that \( P(x) = \left( x, \frac{x^3}{1 + 4x} \right) \), we want to find the slope of the secant line \( PQ \) when \( x = 1.2 \) and compare it with a nearby point, typically \( x_1 = 1 \) and \( x_2 = 1.2 \).

The slope of the secant line between two points \( x_1 \) and \( x_2 \) is given by:

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

Where \( P(x) = \frac{x^3}{1 + 4x} \).

Let's calculate this for \( x_1 = 1 \) and \( x_2 = 1.2 \).The slope of the secant line \( PQ \) when \( x_1 = 1 \) and \( x_2 = 1.2 \) is approximately \( 0.4897 \) (rounded to four decimal places).

Do you have any other questions, or would you like further details on this process? Here are some related questions you might find helpful:

1. How would the slope change if \( x_2 \) were closer to \( x_1 \)?
2. Can you find the derivative of \( P(x) \) at a specific point to compare with the secant slope?
3. What is the interpretation of the secant line in terms of average rate of change?
4. How does the slope of the tangent line compare to the slope of the secant line?
5. How does the function \( P(x) = \frac{x^3}{1 + 4x} \) behave as \( x \) becomes very large?

**Tip:** When comparing slopes of secant lines, try calculating the derivative at a midpoint to see how closely it approximates the slope for small intervals.

Learn how to calculate the slope of the secant line for the function P(x) = x^3 / (1 + 4x) at x = 1.2. Understand the concept of secant lines and their application in calculus. Suitable for advanced high school students and those interested in calculus problems.

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