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In most speaker designs, woofers handle most of the program material (from about 60 Hz up to around 2000 Hz). Larger woofers tend to excel at low frequency performance, and smaller woofers are usually better on the higher end. Because they are designed to handle lots of low frequency sounds, their motor assembly (magnet, voice coil, etc.) are usually overbuilt to allow for greater power handing and less chance for overheating.
Modern woofers' cones are made of paper, carbon fiber, plastic, and other materials. The goal is to achieve a smooth response of the desired frequency range and a smooth transition to the higher frequency driver(s).