Rate of change versus frequency: A 15 kHz sine wave changes faster (has larger dv/dt) than which of the following sine-wave frequencies?

Introduction / Context:The instantaneous rate of change of a sine wave (slope or dv/dt) scales with angular frequency. Higher frequency signals swing through their cycles faster and therefore exhibit greater maximum slope. This concept underpins bandwidth limits, slew-rate considerations, and filter behavior.

Given Data / Assumptions:

• Reference signal: f_ref = 15 kHz.
• Candidates: 25 kHz, 12 kHz, 18 kHz, 1.3 MHz.
• Equal amplitudes assumed for fair comparison.

Concept / Approach:For v(t) = Vp sin(2π f t), the maximum slope is dv/dt|max = 2π f Vp. With fixed amplitude, slope increases linearly with f. Therefore, any frequency lower than 15 kHz has a lower maximum rate of change.

Step-by-Step Solution:

Verification / Alternative check:Compute the ratio of slopes: dv/dt|max(15k)/dv/dt|max(12k) = 15/12 = 1.25 > 1, confirming higher rate at 15 kHz for equal amplitudes.

Why Other Options Are Wrong:

• 25 kHz, 18 kHz, 1.3 MHz: All exceed 15 kHz and thus have equal or greater rate of change.

Common Pitfalls:

• Comparing periods without recognizing inverse relationship to frequency.
• Ignoring amplitude; here we assume equal Vp for a fair slope comparison.

Final Answer:12 kHz