IXL - Slope fields (Calculus practice)

Select the slope field for this differential equation.

′

–

ref_doc_title.

Learn with an example

question

Select the slope field for this differential equation.

′

key idea

The slope field of a differential equation

′

(

)

is made of many short line segments, where the slope of the segment at a point

(

)

is equal to

(

)

Slope fields can be used to visualize the solutions of a differential equation.

solution

Find the slope field for the differential equation

′

The slope of each line segment in the slope field must match the value of

′

at that point, so choose a few points and calculate the value of

′

For example, you can find the value of

′

(

–

)

by substituting

–

and

into the differential equation.

′

(

–

)

(

)

′

On the slope field, this value of

′

corresponds to the slope of the segment at the point

(

–

)

To find the slopes of more line segments in the slope field, substitute more points into the equation for

′

It can help to make a table to stay organized.

y ′ = x 2 y 2
	x = – 2	x = – 1	x = 0	x = 1	x = 2
y = 2	1	1 4	0	1 4	1
y = 1	4	1	0	1	4
y = 0	undefined	undefined	undefined	undefined	undefined
y = – 1	4	1	0	1	4
y = – 2	1	1 4	0	1 4	1

Notice that the value of

′

is undefined along the

-axis,

is 0 along the

-axis

except at the origin, and is positive everywhere else.

The slope of each line segment in the graph below matches the value of

′

for the corresponding point listed in the table. So, this is the slope field for

′

$Confetti$

Excellent!

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V.New!Verify solutions to differential equations

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