A weight hangs from a spring in a damping medium and oscillates up and down. It's height (h) at time (t) is given by h(t) = 1 +(e^-t)*cos(t) for t>=0.

a) Show that the extrema of h(t) occur at times t_n = (-pi/4) +(n*pi) where n= 1,2,3,......

b) Show that 1-(e^-t) <= h(t) <= 1 +(e^-t).

c) Sketch h(t) for t>=0.

d) Show that the weight drops a total distance of 1 + ((e^-3*pi/4)/sqrt(2)(1-(e^-pi)))as t approaches infinity. Recall that the sum of an infinite geometric series with first term a and geometric ratio r <1 is given by a+ar+ar^2+..... = a/(1-r)