Let's break this down into manageable steps to complete your request.

Steps:

1. Choose a transcendental function: A transcendental function is one that cannot be expressed as a finite combination of algebraic operations (e.g., exponential, logarithmic, or trigonometric functions). For this example, I'll use the function $f(x) = \sin(x)$.

2. Obtain 8 equally spaced data points: We need 8 data points of the function $f(x) = \sin(x)$ over a chosen interval. The data points will be equidistant, and we can choose an appropriate step size to space the points.

3. Apply interpolation methods:

   • Newton’s Forward Difference Interpolation: This method is used to interpolate when the interpolation point is near the beginning of the table.
   • Newton’s Backward Difference Interpolation: This is applied when the interpolation point is near the end of the table.
   • Stirling's Interpolation: This method is particularly suited when the interpolation point lies near the center of the table of values, making it a good candidate for our case.

4. Compare with the exact solution: Using $f(x) = \sin(x)$, we can compute the exact value at the point of interest (the midpoint of the interval) and compare it with the interpolated values.

5. Conclusion: After obtaining the results from the three interpolation methods and comparing them with the exact solution, we will analyze the accuracy and behavior of each method.

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Step-by-Step Solution:

1. Choose the interval and step size: Let's take the interval from $x = 0$ to $x = \frac{\pi}{2}$ (i.e., 0 to 1.5708 radians). We will choose 8 equally spaced points, which gives us a step size $h = \frac{\pi}{14}$.

   So the points are: $x_0 = 0, \quad x_1 = h, \quad x_2 = 2h, \dots, \quad x_7 = 7h$ where $h = \frac{\pi}{14} \approx 0.2244$.

2. Data Points Calculation: The function values at these points will be: $f(x_0), f(x_1), \dots, f(x_7)$ where $f(x) = \sin(x)$.

   Let's calculate the values of $\sin(x)$ at these points and then proceed to construct the forward and backward difference tables for interpolation.

I will now calculate these values to proceed with the interpolation techniques.We have calculated 8 equally spaced data points for $f(x) = \sin(x)$ on the interval $[0, \frac{\pi}{2}]$ with a step size of $h = \frac{\pi}{14} \approx 0.2244$. Here are the $x$-values and their corresponding $f(x) = \sin(x)$ values (truncated to 5 decimal places):

x_0 &= 0, & f(x_0) &= 0.00000 \\ x_1 &= 0.22440, & f(x_1) &= 0.22252 \\ x_2 &= 0.44880, & f(x_2) &= 0.43388 \\ x_3 &= 0.67320, & f(x_3) &= 0.62348 \\ x_4 &= 0.89760, & f(x_4) &= 0.78183 \\ x_5 &= 1.12200, & f(x_5) &= 0.90096 \\ x_6 &= 1.34640, & f(x_6) &= 0.97492 \\ x_7 &= 1.57080, & f(x_7) &= 1.00000 \\ \end{aligned}$$ Next, we'll calculate the interpolations at the midpoint (after the fourth data point) using the following methods: 1. Newton's Forward Difference Interpolation 2. Newton's Backward Difference Interpolation 3. Stirling Interpolation I will begin by constructing the forward difference table for the first method.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have any other questions or need assistance with something else, feel free to ask!