37 mark for review the function $f$ is given by $f(x)=2sin(4x)+cos(2x)$. using the period of $f$, which of…

Answer:

A. 159

Explanation:

Step1: Find period of $\sin(4x)$

The period of $y = A\sin(Bx)$ is $T_1=\frac{2\pi}{B}$. For $y = 2\sin(4x)$, $B = 4$, so $T_1=\frac{2\pi}{4}=\frac{\pi}{2}$.

Step2: Find period of $\cos(2x)$

The period of $y = A\cos(Bx)$ is $T_2=\frac{2\pi}{B}$. For $y=\cos(2x)$, $B = 2$, so $T_2=\pi$.

Step3: Find period of $f(x)$

The period of $f(x)=2\sin(4x)+\cos(2x)$ is the least - common multiple of $T_1$ and $T_2$. The least - common multiple of $\frac{\pi}{2}$ and $\pi$ is $\pi$.

Step4: Calculate number of cycles

The number of complete cycles of $y = f(x)$ in the interval $0\leq x\leq1000$ is $n=\lfloor\frac{1000}{\pi}\rfloor$. Since $\frac{1000}{\pi}\approx\frac{1000}{3.14159}\approx318.31$, and $n=\lfloor318.31\rfloor = 159$ (because we consider the number of full - cycles and the period is $\pi$ and we are dealing with the sum of the two trigonometric functions where the effective period for counting full - cycles is $\pi$).