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โ™พ๏ธย ap calc

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๐Ÿ’Žย Unit 7

ย ย โ€ขย ย โฑ๏ธ1 min read

7.3 Sketching Slope Fields

Jacob Jeffries

jacob jeffries

โฑ๏ธ April 26, 2020

๐Ÿ“…โŒจ๏ธ

Slope fields allow us to visualize a solution to a differential equation without actually solving the differential equation. Letโ€™s construct a slope field to solidify this idea. ๐Ÿง 

Slope fields essentially draw the slopes of line segments that go through certain points. Letโ€™s consider the following differential equation:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(931).png?alt=media&token=7639fd09-c6ef-42fb-aaea-50fdba7cb745

The slope (m) at point (x, y), in this case, is just x + y, which we can put into a table for various coordinates:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(932).png?alt=media&token=3c8b9a1e-545c-443e-a283-f65932e751e4

We can use this data to draw an approximate solution to the differential equation by drawing short line segments through each point that have the corresponding slope: ๐ŸŒ„

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(933).png?alt=media&token=c1fa6f93-c2c6-45be-bce8-4798f4bdba14

To be continued in 7.4 Reasoning Using Slope Fields.

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