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### Course: AP®︎ Calculus AB (2017 edition)>Unit 10

Lesson 4: Slope fields

# Slope fields introduction

Slope fields allow us to analyze differential equations graphically. Learn how to draw them and use them to find particular solutions.

## Want to join the conversation?

• Are there no practice exercises, or leveling up or any such thing for the differential equations mission? Is it all only videos? How can I make sure that I retain what the video's teach me?
• If you want good practice with differential equations, I would recommend going to MIT OpenCourseWare online. Navigate to their differential equations classes and you will find course notes, exams, and practice questions with solutions. While it's not as fun as Khan Academy, it will definitely give you a chance to test your abilities.
• I have two questions:
1) Couldn't Sal have just multiplied the right hand side by dx and the left hand side by y, took the integral and solved it that way or was a slope field really necessary?
2) Is there an abbreviation for undefined?
• i think it was just to get the idea of slope fields across. some functions are hard to figure out by hand, but if you draw a slope field and then if the existence uniqueness theorem applies, you can draw out an aprox solution by hand. RK4 method is what programs like maple uses to aprox a solution to differential equations. Sorry i couldnt really explain existence uniqueness theorem, but i think google would be your friend
• A small confusion , while estimating the solution using slope field you give arbitrary values to x and y then we find dy/dx at that point .So my doubt is how can y take different values for a particular value of x ( no more a function ), also, that (x,y) might not satisfy our function so how can it still give a correct result
• The thing is, differential equations don't only have one function as their solution. They can have an infinite number of solutions. In this case, the solution in general may be written as:
y^2 + x^2 = c;
where c is any constant. So by adjusting c, we can make an indefinite number of functions that satisfy the solution. In fact, square root of c represents the radius of the circle. (More about that in conic sections)
• so he is plugging in random numbers to the derivative?
• What if the equation were dy/dx = 2x? And there isn't a y?
• Then the slope field will be independent of y. It will look like a lot of "columns" of lines all with the same slope. So on the x-axis the lines will be horizontal, for x=1/2 they'll be diagonal lines, etc.
We can solve dy/dx = 2x directly (by integration), giving y = x² + C.
The result is a family of parabolas - a different one for each value of C.
• Hi, i'm just wondering, other than visualizing solutions, are there any other uses for slope fields?
• i dont think so. i think slope fields are just a way to visualize the antiderivative.
• how do i make a slope feild
• First, draw your axes. Then pick a bunch of points and draw lines with the slope at each point.
• I always face a difficulty when drawing solution curves through slope field.How do I know which points my curve goes through?The slope field gives a visual idea about what my solution may look like but i can not figure out how to draw my exact solution curves by making the curves tangential to the slope field always.:) would be grateful if i get a reply.
• at , why did Sal not draw the solution passing through the points that he drew?