Slope Field
In short: Slope field is a grid of short line segments, each drawn with the slope that a differential equation assigns to that point, showing the direction a solution curve must take.
AP Calculus ABCalculusAP Calculus AB Unit 7
What is Slope Field?
A slope field shows a differential equation without solving it. At each grid point \((x, y)\), plug the coordinates into \(\frac{dy}{dx}\) and draw a tiny segment with that slope.
A solution curve placed in the field must be tangent to the segments it passes. Start at an initial point and follow the marks like a boat drifting along a current. Where segments are flat the slope is 0; where they are steep the slope is large in size; where they tilt downward it is negative.
How to spot it
- A picture of many tiny line segments on a grid is shown, with a differential equation next to it.
- The problem asks which equation matches a given field, or which field matches an equation.
- You must sketch a solution curve through a given point, such as (0, 1).
- Segments in a whole row or column are identical, which hints that the equation depends on only one variable.
The method
- Pick a grid point and substitute its \(x\) and \(y\) into the formula for \(\frac{dy}{dx}\).
- Draw a short segment through the point with that slope: flat for 0, rising for positive, falling for negative.
- Repeat across the grid; look for places where the slope is 0 or the same along a line.
- To sketch a solution, begin at the initial point and trace a smooth curve that stays tangent to the nearby segments.
Slope Field: solved example
Problem
For \(\frac{dy}{dx} = x – y\), find the slope mark at \((2, 3)\), at \((1, 1)\) and at \((0, 2)\), and describe a solution that passes through \((0, 2)\).
Solution
At \((2, 3)\): \(2 – 3 = -1\), a falling mark. At \((1, 1)\): \(1 – 1 = 0\), a flat mark. At \((0, 2)\): \(0 – 2 = -2\), steeply falling.
The slope is zero wherever \(y = x\), so flat marks line up along that diagonal. The solution through \((0, 2)\) starts out falling with slope \(-2\) and keeps bending upward as it approaches the line \(y = x\).
Try it
Question 1
For \(\frac{dy}{dx} = 0.5y\), what is the slope mark at \((3, -4)\)?
Show the answer
\(0.5(-4) = -2\), so the segment tilts downward steeply.
Question 2
Every row of a field has identical segments. Which equation fits better: \(\frac{dy}{dx} = x\) or \(\frac{dy}{dx} = y^2\)?
Show the answer
\(\frac{dy}{dx} = y^2\). Its slope depends only on \(y\), so it stays the same along a horizontal row, while \(x\) would change the marks within a row.
Slope Field vs. Differential Equation
The differential equation is the formula that rules the slopes. The slope field is its picture on a grid. From the picture you can see the shape of solutions, find constant solutions where the marks are flat, and estimate values, but you do not obtain an exact formula.
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