I. Foundations/Comprehension (No technology allowed, show all work!) 1. Given a 1st order differential equation, Euler's method will provide an overestimate when the solution curve is concave (3 points). 2. Compute the exact sum of the following infinite series (5 points): 13 iM: + 12 WIN n(n + 1) 3. The following sequence is convergent. Find the limit (5 points). ON = cos(2 ) + 2 arctan (N) - sin 6TIN-7 12N+5.4. Suppose you are given the following 1st order differential equation: dy dx - + 2xy = y a) Using separation of variables, find the general solution to this equation, written in the form y = f(x) (5 points). b) Besides the result of the previous part (a family of solutions), another solution not part of the family is y = _ (3 points). c) Find the specific solution (part of the family) which satisfies the initial condition y (0) = 5 (4 points).II. Applications/Analysis (Technology allowed, show all work!) 1. Suppose you are given the following differential equation: dP dt = 3P2 -66P + 171 a) Find the equilibrium solutions for this equation (4 points). b) Using the 2"d derivative, determine the value of P where there is a change in concavity; that is, a point of inflection (4 points). c) Sketch the solution curves for this differential equation having initial points (0, 0), (0, 5), (0, 17), and (0, 20), along with the equilibrium solutions already found (6 points).2. Suppose you are given . 0416, a repeating decimal. By viewing the repeating part of this decimal as an infinite geometric series, convert the decimal into a completely simplified fraction (5 points). 3. Set up, and solve, a 1" order differential equation for the following situation: The time rate of change of an alligator population P in a swamp is directly proportional to the square of P. The swamp contained 1 dozen alligators in 2008 and 2 dozen alligators in 2018. In what year will there be 4 dozen alligators in the swamp (6 points)