Finding maximum value of periodic function.

Question

In my question, It is asked to find the maximum value of the equation of motion such as u_t=(11/210)sin(10t)-(1/21)sin(t11). I have found that when t=2pi*n/21 it reached its maximum value. But it is obvious from its graph and calculations that the value of function increases from n=1 to n=10 and equals to 0.0997 when n=11. After that, it decreases. My question is there any way to calculate it more easily or it is what it is?

Answer 1

The graph of a function has slope 0 at it's maximum point.

Slope of a function can be determined by dy/dx that is its derivative.

So, by equating it's derivative to 0 we get,

dy/dx = {(11/210) * 10 * cos(10t)} - {(1/21) * 11 * cos(11t)} = 0

= (11/21) * [cos(10t) - cos(11t)] = 0 (since, cos(c)-cos(d) we know)

0 = 2 * sin(21t/2) * sin(t/2)

so, we get two cases

sin(21t/2) = 0 or sin(t/2) = 0

Since we know, sin^-1(0) = n*pi

therefore t = 2 * n * pi /21 or t = 2 * n * pi.

and the maximum value by putting the above t value of the function will be 0.099717665

Check this graph solution for the problem