The function \( f(x) \) in this context is determined by the expression inside the summation in the given limit. The Riemann sum provided is:

\[
\sum_{k=1}^n \frac{\pi}{4n} \cdot \tan\left(\frac{k\pi}{16n}\right)
\]

This sum suggests that the function \( f(x) \) is the expression that corresponds to \( \tan\left(\frac{k\pi}{16n}\right) \) when transformed from the summation index \( k \) to the continuous variable \( x \).

By comparing this with the standard form of a Riemann sum, where \( x_k = \frac{k}{n} \) and \( \Delta x = \frac{1}{n} \), we can see that the argument of the tangent function is dependent on \( k/n \).

To express this in terms of a continuous function \( f(x) \) over an interval \( [A, B] \):

1. The summation index \( k \) in the Riemann sum translates to \( x \) in the integral.
2. The factor \( \frac{k\pi}{16n} \) becomes \( \frac{\pi x}{16} \) as \( n \) becomes large and \( k/n \) approaches a continuous variable \( x \) in the interval.

Thus, the function \( f(x) \) is:

\[
f(x) = \tan\left(\frac{\pi x}{16}\right)
\]

This function \( f(x) \) represents the integrand that, when integrated over the interval \([A, B]\), gives the same result as the original limit.

Do you need more details, or do you have further questions? Here are some related questions:

1. How does the tangent function affect the shape of the graph for \( f(x) \)?
2. What is the significance of the factor \( \frac{\pi}{16} \) in the argument of the tangent function?
3. How would the function \( f(x) \) change if the argument were \( \frac{\pi x}{8} \) instead of \( \frac{\pi x}{16} \)?
4. Can \( f(x) \) be integrated directly, or does it require special techniques?
5. How does the behavior of \( f(x) \) near the endpoints \( A = \frac{1}{16} \) and \( B = 1 \) influence the integral?

**Tip:** When dealing with Riemann sums, identifying the continuous function \( f(x) \) from the discrete summation is a key step in translating the problem into an integral.

Explore the function f(x) defined by the Riemann sum \( \sum_{k=1}^n \frac{\pi}{4n} \cdot \tan\left(\frac{k\pi}{16n}\right) \) and its significance in integral calculus. Learn about the tangent function's role in shaping f(x) and its behavior near the endpoints. Suitable for college-level students and calculus enthusiasts.

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