The maximum gain (with respect to frequency) of second-order filters such as low-pass, high-pass, band-pass, low-pass notch and high-pass notch filters is derived without using calculus. Our method uses the fact that the square of the magnitude response (gain) for these filters can be written as a positive constant divided by a quadratic function of frequency. Hence, the gain is maximized when the denominator of the gain is minimized, which is easily achieved without calculus, as the denominator is parabolic.
Latin American Journal of Physics Education
Cartwright, K., P. Russell and E. Kaminsky, “Finding the maximum magnitude response (gain) of second-order filters without calculus,” Latin-American J. Phys. Educ., vol. 6, no. 4, Dec. 2012, pp. 559-565.