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How do you draw the slope field of the differential equation #dy/dx=1/2(4-y)(y-2)^(4/3)# ?

Answer 1

Samantha Adkison

If you are allowed to use a software, then a software called GeoGebra gives us the slope field below.

I hope that this was helpful.

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Answer 2

Ezekiel Alexander

To draw the slope field of the differential equation ( \frac{dy}{dx} = \frac{1}{2}(4-y)(y-2)^{\frac{4}{3}} ), follow these steps:

Determine critical points by setting ( \frac{dy}{dx} = 0 ). Solve for ( y ) to find the critical points.
Identify regions where ( \frac{dy}{dx} ) is positive and negative.
Choose representative ( (x, y) ) points in each region and calculate the slope using the given equation.
Draw short line segments with the calculated slopes at each chosen point.
Repeat steps 3 and 4 for a grid of points to create a complete slope field.

The slope field provides a visual representation of the behavior of solutions to the differential equation across the plane.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.