Understanding the behavior of the function y = sin 2x can provide valuable insights into its properties, particularly in identifying intervals where the function is decreasing. This function, being a sinusoidal wave, oscillates between -1 and 1, and its behavior is influenced by the frequency factor.
Function Analysis
The function y = sin 2x can be analyzed by observing its derivative. The derivative of y = sin 2x is y’ = 2 cos 2x. The function y = sin 2x is decreasing where the derivative is negative, which happens when cos 2x is negative. This occurs within intervals where 2x lies between π/2 and 3π/2 plus any integer multiples of π.
Interval Calculation
To find the specific intervals, solve the inequality cos 2x < 0. This simplifies to 2x ∈ (π/2 + nπ, 3π/2 + nπ), where n is an integer. Dividing by 2, the function y = sin 2x is decreasing in the intervals (π/4 + nπ/2, 3π/4 + nπ/2).
Summary
In summary, the function y = sin 2x decreases in intervals determined by the cosine function being negative. By understanding these intervals, we can better predict the behavior of the function over its periodic cycles. This analysis is crucial for applications requiring precise knowledge of the function’s behavior.
