4. Consider the differential equation Let y = x) be the particular solution to the differential equation with initial condition f (0) = 1. 1 (a) Find lim & x40 (3233 1 (b) Use the tangent line to the graph of y : r) at a: = 0 to estimate f(0.5). (o) Is the graph of f is concave up or concave down at (0, 1)? Is the approximation in part b) an overestimate or underestimate? (d) Use Euler's method, starting with a: = 0 with two steps of equal size, to approximate f(1). (e) Find y = f (33) the particular solution to the differential equation with initial condition f (O) = 1 5. A population is modeled by the function P that satises the logistic differential equation dP P P *(1 ) 55 _E (a) If P(0) = 3, what is slim P05)? If P(0) = 20, what is tlim P(t)? ace (b) If P (0) = 3, for what value of P is the population growing the fastest? If P (O) = 7, for what value of P is the population growing the fastest? (c) A different population is modeled by a function Y that satises the separable differential equation = X (1 _ i) dt 5 12 Find Y(t) if Y(0) = 3. (d) For the function Y found in part (c), what is lim Y(t)? tAoo