¹ Notice that if k = 0, x′(t) will also = 0. But x′(t) is just the change in temperature, so the temperature does not change (which would have alien archeologists digging through the rubble of our failed civilization to find a still-warm cup of coffee). In more mathematical terms, if x′(t) = 0, the only functions whose derivatives are 0 are constant functions. So the temperature of the coffee is constant always and forever, which would undoubtedly astound the alien archeologists even more.