. (12 points) Another example of a differential equation is the following: the population, P(t), of a certain species native to Houston - Instructorus Mathematicae - changes depending on time, t. Specifically, the rate of change in population is proportional to the population level with constant of proportionality given by 3. In other words, P'(t) = 3P(t). (A) (5 points) Show that the function P(t) = Set satisfies the differential equation above with initial condition P(0) = 8. (B) (2 points) What is the initial rate of change in the population level of Instructorus Mathematicae (i.e. what is the rate of change when t = 0?) (C) (3 points) What is the rate of change in the population level of Instructorus Mathematicae at the point in time when the population is P = 33?7. (7 points) Lets compute a derivative in reverse! Given the function f(x) = e + sinha + 1+ + VI-, find a function F(x) that satisfies F'(x) = f(x) and F(0) = 3. (Be sure to explain why F(x) = f(x).) 8. (7 points) Find the equation of the line tangent to the graph of y = arctan (sinh(x)) at r = 0.9. (7 points) The function f(x) = sinh(x) is one-to-one and so is invertible. Using our rule / formula for relating the derivative of an inverse to the derivative of the original function, find the slope of the line tangent to the graph of y = f (x) = aresinh(x) when r = 0