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### Part (a)
To calculate the slope \( m_{PQ} \) of the secant line between the points \( P(4, -3) \) and \( Q(x, \frac{3}{3 - x}) \), we use the formula for the slope between two points:

\[
m_{PQ} = \frac{y_Q - y_P}{x_Q - x_P}
\]

where:
- \( P(4, -3) \) is a fixed point on the curve,
- \( Q(x, \frac{3}{3 - x}) \) is another point on the curve.

Thus, the slope becomes:

\[
m_{PQ} = \frac{\frac{3}{3 - x} - (-3)}{x - 4} = \frac{\frac{3}{3 - x} + 3}{x - 4}
\]

Now, I will calculate this for the given values of \( x \) from 3.9 to 4.0001 and fill in the table.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, if you'd like, I can guide you step-by-step on how to calculate the slopes manually, or feel free to provide another file or ask a different question.

Would you like assistance with a detailed guide?

The point P(4, -3) lies on the curve y = 3/(3 - x). (a) If Q is the point (x, 3/(3 - x)), use your calculator to find the slope mPQ of the secant line PQ (correct to six decimal places) for the following values of x. (i) 3.9, (ii) 3.99, (iii) 3.999, (iv) 3.9999, (v) 4.1, (vi) 4.01, (vii) 4.001, (viii) 4.0001. (b) Using the results of part (a), guess the value of the slope m of the tangent line to the curve at P(4, -3). (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(4, -3).

Learn how to find the slope of the tangent line to a curve at a given point by calculating the slopes of secant lines and using limits. This problem covers key calculus concepts including derivatives and the limit definition of a derivative. Suitable for high school students in Grades 11-12.

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