\text { 2. (36 pts) Let } u(t)=\frac{1}{1+(2 \pi t)^{2}} \text { and } v(t)=e^{-|t|} \text {. Express all answers with both an exact analytic expression } and a decimal

approximation. (a) Find the smallest frequency bands that contain 90%, 99%, and 99.9% of the energy of u. Do the same for v. (b) Find the frequency bands where |U(f)|² is at least c times the maximum value of |U(f)|² force {0.1,0.01,0.001}. Do the same for V.