## University Calculus: Early Transcendentals (3rd Edition)

Doubling the frequency multiples the velocity by $2$, the acceleration by $4$ and the jerk by $8$ times.
$$s=A\cos(2\pi bt)$$ Since we concern ourselves only with the change in frequency $b$, we take $A$ and $t$ to be constants, which means their derivatives equal $0$. a) Without doubling the frequency: $$s=A\cos(2\pi bt)$$ - Velocity: $$\frac{ds}{db}=A(-\sin(2\pi bt))\frac{d}{db}(2\pi bt)=-A\sin(2\pi bt)\times2\pi t=-2A\pi t\sin(2\pi bt)$$ - Acceleration: $$\frac{d^2s}{db^2}=-2A\pi t\cos(2\pi bt)\frac{d}{db}(2\pi bt)=-2A\pi t\cos(2\pi bt)\times2\pi t$$ $$=-4A\pi^2t^2\cos(2\pi bt)$$ - Jerk: $$\frac{d^3s}{db^3}=-4A\pi^2t^2(-\sin(2\pi bt))\frac{d}{db}(2\pi bt)=4A\pi^2 t^2\sin(2\pi bt)\times2\pi t$$ $$=8A\pi^3t^3\sin(2\pi bt)$$ b) Doubling the frequency: $$s=A\cos(2\pi 2bt)=A\cos(4\pi bt)$$ - Velocity: $$\frac{ds}{db}=A(-\sin(4\pi bt))\frac{d}{db}(4\pi bt)=-A\sin(4\pi bt)\times4\pi t=-4A\pi t\sin(4\pi bt)$$ - Acceleration: $$\frac{d^2s}{db^2}=-4A\pi t\cos(4\pi bt)\frac{d}{db}(4\pi bt)=-4A\pi t\cos(4\pi bt)\times4\pi t$$ $$=-16A\pi^2t^2\cos(4\pi bt)$$ - Jerk: $$\frac{d^3s}{db^3}=-16A\pi^2t^2(-\sin(4\pi bt))\frac{d}{db}(4\pi bt)=16A\pi^2 t^2\sin(4\pi bt)\times4\pi t$$ $$=64A\pi^3t^3\sin(4\pi bt)$$ So, from the calculations, doubling the frequency multiples the velocity by $2$, the acceleration by $4$ and the jerk by $8$ times.