IXL - Solutions to differential equations using slope fields (Calculus practice)

Each graph shows the slope field for the same differential equation

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(

)

Which graph shows a family of solutions to the differential equation?

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question

Each graph shows the slope field for the same differential equation

′

(

)

Which graph shows a family of solutions to the differential equation?

key idea

The slope field of a differential equation

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

(

)

is equal to

(

)

A slope field can be used to visualize all curves that satisfy the differential equation

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(

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called the solutions of the equation. When graphed, a solution will follow the direction of the slope field so that the slope of the tangent line at a point will match the slope of the segment at that point.

solution

You were asked to find the graph that shows a family of solutions to the differential equation

′

(

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Any solution to the differential equation must follow the direction of the slope field.

green checkmark, All of the curves in this graph follow the direction of the slope field. The slopes of the tangent lines of each curve match the slopes of the line segments in the slope field. So, each curve is a solution to the differential equation. This graph shows a family of solutions.

The other answer choices are not correct because their curves do not follow the direction of the slope field. For example, the line segment at

(

–

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has a slope of zero, but in this graph, the tangent line to the curve has a negative slope near that point.

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