Drifting Along the Ocean

Background:

In this application, the direction field graphs derivatives of the underlying functions that are solutions to the differential equation y'(x) = f(x,y), where f(x,y) is the function chosen under the problem selection in the application. The boat is placed along the lefthand edge of the graph. You can alter the vertical position of the boat. The dock is placed in the water along another edge of the graph. You will determine a route for your ship having only the vectors in the direction field to guide you. Choose wisely as once you click the "Let the ship sail" button, the ship's route follows the path of the underlying solution to the differential equation y'(x) = f(x,y) as computed by the application. Can you guide the ship safely to port?

Directions

First choose an equation under problem selection.
Next set the detail of the grid to the default value of 4. Observe that with such a level of accuracy it can be hard to predict where the boat will float from its starting location.
Move the boat to a desired start location.
Let the ship sail!
Now adjust the detail of the grid and the starting location of the boat to get a feel for how the boat traverses the slope field.
Repeat this process for the various problems and then discuss the questions below.

Thought Questions

1. How did you use the arrows to determine where to place the ship and let it sail? How did the number of arrows change the difficulty of the game?

2. The black line is our numerical estimate to the true underlying solution. Given your work in this module, what ideas do you have regarding an algorithm that might produce such a numerical estimate?

After completing the tasks above, move the dock by dragging it to another location in the ocean.