Final answer:
The fundamental period of the function f₁(x) = sin 7x + cos 14x is found by determining the periods of the individual sine and cosine components, which are 2π/7 and π/7 respectively. The smallest common multiple of these periods is 2π/7, thus the fundamental period of f₁(x) is 2π/7.
Step-by-step explanation:
Calculating the Fundamental Period of Sin 7x + Cos 14x
The fundamental period of a function is the smallest positive interval over which the function's graph repeats. For sinusoidal functions such as sin(x) and cos(x), the fundamental period is the length of one complete cycle. To find the fundamental period of f₁(x) = sin 7x + cos 14x, we should identify the periods of the individual sine and cosine functions and then find a common multiple.
The general form of a sine or cosine function is A sin(Bx + C) or A cos(Bx + C), where 2π/B represents the period of the wave. In the given function f₁(x), the period of sin 7x is 2π/7 and the period of cos 14x is 2π/14 which simplifies to π/7. Now, the smallest number that both 2π/7 and π/7 divide into evenly is 2π/7. Therefore, the fundamental period of f₁(x) is 2π/7.
The function f₁(x) oscillates between its maximum and minimum values over this interval, completing one full cycle before repeating. By understanding the basic properties of sine and cosine waves, we can determine the characteristics of more complex wave functions, such as their amplitude, wave number, angular frequency, initial phase shift, and fundamental period.