Slope Fields

The Slope Fields exercise appears under the Differential equations Math section on Khan Academy. This exercise shows the slope fields of differential equations.

Types of Problems

There are six types of problems in this exercise:

1. Which of the following differential equations generates the slope field pictured below: The student is asked to find the differential equation that would create the given slope field.
2. Which of the following slope fields is generated by the differential equation: The student is asked to find the slope field that is created by the differential equation.
3. Which of the following is the general solution to the differential equation: The student is asked to determine the solution to the differential equation given the slope field.
4. Which of the following is the range of the solution curve: The student is asked to find the range of the solution curve given the initial condition.
5. What short segment of slope would you draw: The student is asked to determine what short segment of slope should be drawn at the given point.
6. Match each slope field on the right with its generating differential equation on the left: The student is asked to match the slope field with its differential equation.

Strategies

Knowledge of derivatives, differential equations, and slope fields are encouraged to ensure success on this exercise.

1. When finding the general solution to the differential equation that generated the given slope field, pick any segment on the slope field and, using the direction of the segment, draw a particular solution to the differential equation used to create this slope field.
2. The horizontal asymptotes on a slope field can help determine the range of the function at the initial condition by eliminating choices.
3. Finding the short segment of slope for a differential equation can be down by substitution of the given x and y-coordinates into ${\frac{dy}{dx}}$
4. When $\frac{dy}{dx} = 0$, a horizontal line is produced on a slope field.
5. When ${\frac{dy}{dx}}$ = undefined, a vertical line is produced on a slope field.

Real-life Applications

1. Computers and calculators use slope fields to numerically find graphical solutions.
2. Slope fields are used to make sure the wing-load of an airplane does not get too high.

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