please do it fast

2. Thanos claims that the universe has grown too rapidly to sustain itself. He makes his argument mathematically by conjecturing that the universal population at a given time t, given by p : [0,00) (0,00), is continuous and differentiable and satisfies p'(t) = rp(t) where r (0,00) is a fixed growth rate parameter. That is to say, the population grows proportional to itself relative to a rate parameter r. Let p(0) = po. (a) Thanos claims that the growth rate of humans is exponential. Compute SPG dt two ways. One by u substitution and the other using the relation between p' and p. Use your answers to find an explicit form for p. Is Thanos' claim accurate? (b) Use the fundamental theorem of calculus to show that the only function that can model population given the growth pattern and continuity assumption must be the answer you got from a). (c) Bruce Banner claims Thanos is wrong as human population is self correcting. He argues that population growth instead satisfies the equation p' (t) = rp(t) ep(t)?, (0,1). Intuitively, as p gets large, the rate of growth should be controlled by the dampening factor ep(t)2. Compute p(t) p' (t) by considering S dt. rp(t)-ep(t)? (d) Compare the Thanos and Banner population growth models. How do they compare as t+00