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AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB>Unit 7

Lesson 3: Sketching slope fields

Worked example: forming a slope field

Given a differential equation in x and y, we can draw a segment with dy/dx as slope at any point (x,y). That's the slope field of the equation. See how we determine the slopes of a few segments in the slope field of an equation.

Want to join the conversation?

• What computer software would create a slope field for us?
• What are slope fields actually used for - i.e. what are some applications of slope fields?

Also, what exactly is a slope field telling us or describing?
• Slope fields are used in environmental engineering to monitor (or remediate) an aquifer, in biology to understand predator/prey populations over a geographical area, in oceanography to measure currents, salinity, temp., etc.
• so in this case we have a set of solutions not one?am i right?
• Yes, If you solve it, you would get
`y(x) = c_1 e^x + 2 x + 2` where c_1 is a constant, so in fact we have infinitely many solutions.Also, if you use a computer to graph the slope field you would find a family of curves.
Hope this helps
• Hi, can we use the Slope Fields for Higher Order Differential Eqns? Also how are Slope Fields helping us to solve Differential Eqns? :D
• A first-order differential equation is basically an equation in three variables: x, y, and y'. Because we have three pieces of information to compare, we essentially have three "dimensions" of information to graph: two dimensions as axes of the plane, and one represented by the slope lines.

If we try to use slope fields with higher-order differential equations, we need to pack more dimensions into our graph, for instance by creating 3D graphs or including colors. This makes the graph much harder to create and read, so we only use slope fields for first-order differential equations in practice.

Also, slope fields don't help us solve differential equations so much as they help us visualize the solutions. At best, if the slope field suggests a very simple-looking function, we may be able to guess a simple solution to the equation. But in most cases, we still have to do the hard work of solving the equation algebraically.
• how accurate do the little lines need to be? Is it enough to estimate the slopes?