1) The acceleration function is found by differentiating the velocity function.

2) The location function is found by integrating the velocity function, and using the initial condition to find the constant of integration. Details follow:

1) Taking derivative of v(t) gives, by the Chain Rule, a(t)=(-2sin(pi*t)*pi= -2pi*cos(pi*t).

2) Taking anti-derivative of v(t) gives s(t)= (2/pi)*sin(pi*t) + C, which can be checked by differentiation.

Fitting the initial value s(0)=2 gives

2=(2/pi)*sin(0) + C; but sin (0)=0. So C=2

The final result is then s(t) = (2/pi)*sin(pi*t) + 2

Letty B.

good math24d

Jeffrey M.

04/16/20