Derivatives of Trig Functions: Applications

4. A weight which is connected to a spring moves so that its displacement is:

x = sin 2t + 3^{1/2}cos 2t in centimeters.

What is its maximum displacement?

Take the derivative and set it equal to zero.

dx/dt = 2 cos 2t + 3^{1/2} * 2(-sin 2t)

0 = 2 cos 2t – 2*3^{1/2} sin 2t

sin 2t/cos 2t = 1/3^{1/2}

tan 2t = 1/3^{1/2}

2t = 30^{o}

Plug this into the original equation for x:

x_{max} = sin (30) + 3^{1/2}cos (30) = 2 cm