3. Solve the IVP y' = e¯¹ (2x − 4), y(5)=0. Then, compute y(5.25). Remark. In this problem, we can easily write y explicitly as a function of x, i.e., y = o(x).

2. Assume that the population of a colony of Brazilian fire ants, P, is described by the function P(t)= \t+1·e-0.5t is measured in days (t=0 when one begins monitoring the population). (a) Find the initial population, P(0) (include units with your answer). +10t +200. The population here is measured in thousands of individuals, and time, t, %3D %3D (b) Find P (t). (c) Evaluate P (0) (include units with your answer).

5. Consider the equations: det f(x) = A(1+)™* g(x) = A· ex and %D %3D For x =1 (a one year investment), and A = 1 (a $1 investment) we have: %3D n.1 f(1) = 1(1+)" g(1) = 1· e"1 and Simplified: f(1) = (1 + #)" and g(1) = e™

11. INSTRUCTIONS: Two cars start moving from the same point. One travels south at 60 miles per hour. The other car travels west at 25 miles per hour. At what rate is the distance between the cars increasing 2 hours later? 12. 5 pu of: INSTRUCTIONS: State L(x) (the linearization) at a = ㅠ 6 f(x) = sin(x)

Part A and B

3. Use the Laplace transform method to solve the initial value problem x₁ = 6x₁ + 11x2, x2 = −4x1 - 9x2, with x₁(0) = 7 and x₂(0) = -1.

1. Suppose the function V represents the number of people vaccinated for COVID-19 in Kalamazoo country. (a) What can you conclude about the first and second derivatives of V from the statement "The number of people vaccinated is increasing faster and faster"? For your conclusion, choose from among: positive, negative, zero, or other. If other, please explain. i. V'(t) is ii. V"(t) is (b) What can you conclude about the first and second derivatives of V from the statement "The number of people vaccinated is close to reaching an all time high"? For your conclusion, choose from among: positive, negative, zero, or other. If other, please explain. i. V'(t) is ii. V"(t) is

Questions No.3 and No.4 refer to the following: Let f(z) = 5-2z. If g(z) is a function with derivative given by g'(x) = f'(1) · ƒ(1) · (z + 3). (3) On what interval(s) is g(x) increasing? (A) (-∞, -3]U[,00) (B) [-3, (C), ∞) (4) On the interval (-oo, oo), the function g(z) has (A) one local maximum, no local minimum (C) one local maxima, one local minimum (D) cannot be determined (B) no local maximum, one local minima (D) no local maximum, no local minima

Suppose that h(x) = f(g(x)), and that we have the following in- formation about these functions: f(3) 2 g(3) = -1 f'(3) = 4 g'(3) = 6 f(-1)%3D7/5 f'(-1) 33/2 g'(-1) = 0 g(-1) = 2/3 f(6) = 1/3 g(6) = 3 f'(6) = 4 g'(6) = 10 What's the value of h'(3)? a) 6 b) 9 novo