Problem 1. Consider the differential equation given by  .

(a) On the axes provided below, sketch a slope field for the given differential equation at the nine points indicated.

(b) Let y = f(x) be the particular solution to the given differential equation with the initial condition f(0) = 3. Use Euler's method starting at x = 0, with a step size of 0.1 to approximate f(0.2). Show the work that leads to your answer.

(c) Find the particular solution  y = f(x) to the given differential equation with the initial condition f(0) = 3.  Use your solution to find f(0.2).

Problem 2. Consider the differential equation given by  .

(a) On the axes provided below, sketch a slope field for the given differential equation at the eleven points indicated.

(b) Use the slope field for the given differential equation to explain why a solution could not have the graph shown below.

(d) Find the range of the solution found in part (c).

Problem 3. Consider the differential equation given by  .

(a) The slope field for the given differential equation is provided below. Sketch the solution curve that passes through the point [0, 1] and sketch the solution curve that passes through the point [0, -1] .

(b) Let f be the function that satisfies the given differential equation with the initial condition f(0) = -1. Use Euler's method starting at x = 0, with a step size of 0.1 to approximate f(0.2). Show the work that leads to your answer.

(c) Find the value of b for which  is a solution to the given differential equation. Justify your answer.

(d) Let g be the function that satisfies the given differential equation with the initial condition g(0) = 0. Does the graph of g have a local extremum at the point [0, 0]?  If so, is the point a local maximum or a local minimum? Justify your answer.