how many times do the graphs of y=3sinx and y=3sin3x intersect for x in the interval [0, 2012]
To determine the number of intersections between the graphs of y = 3sin(x) and y = 3sin(3x) in the interval [0, 2012], you need to compare their graphs and identify the points of intersection.
Here's how you can approach the problem step-by-step:
1. Start by considering the given interval [0, 2012]. This means you need to look for intersections in the range of x values from 0 to 2012.
2. Notice that the period of y = 3sin(x) is 2π because sin(x) repeats itself every 2π radians. In other words, the graph of y = 3sin(x) repeats itself every multiple of 2π. Similarly, the period of y = 3sin(3x) is 2π/3 because when you take x to be x/3, the graph repeats itself every multiple of (2π/3).
3. To find the number of intersections, consider the smallest common multiple of the periods. The smallest common multiple of 2π and 2π/3 is 2π. This means that both graphs will complete a full cycle, 360 degrees or 2π radians, before they intersect again.
4. Take the interval [0, 2012] and divide it by the period (2π): [0, 2012] / 2π ≈ [0, 320.038].
5. Count the number of full cycles within this interval. Since each cycle corresponds to 2π or 360 degrees, you simply need to divide the interval by 2π: 320.038 / (2π) ≈ 50.957.
6. Round down the result to the nearest whole number to account for the fact that we are only interested in full cycles: 50.
Therefore, the graphs of y = 3sin(x) and y = 3sin(3x) intersect approximately 50 times in the interval [0, 2012].
well, how many times do they intersect on [0,2π]?
Then they intersect 2012/2π times that many times on the larger interval.