Key takeaway: A slope field is a grid of short line segments, each with the slope given by a differential equation at that point, showing the direction in which every solution curve must travel.
Grade 12 – AP Calculus ABAP Calculus AB Unit 7Calculus
Definition of Slope Field
A slope field turns a differential equation into a picture without solving it. At each grid point, plug the coordinates into the formula for \(\frac{dy}{dx}\) and draw a tiny segment with that slope. The segments form a landscape of arrows that solutions follow.
To sketch one solution, start at a given point and move so that the curve stays tangent to the segments it passes. Different starting points give different curves, all flowing with the same field.
Formula and notation
For a differential equation of the form
\[ \frac{dy}{dx}=F(x,y), \]
the segment at the point \((a,b)\) has slope \(F(a,b)\). Slope 0 gives a flat segment, a large positive number a steep climb, and a negative number a descent. Points with equal slope often lie along a curve.
Key facts
- The segment’s slope depends only on the coordinates of the point where it is drawn.
- If the formula has no y, every vertical column of segments looks identical.
- Solution curves never cross, because each point carries only one slope.
- Flat segments mark possible peaks, valleys, or equilibrium lines.
Slope Field example
For \(\frac{dy}{dx}=2x-y\), find the slope of the field at \((1,3)\), \((2,1)\) and \((0,2)\), and say where the segments are horizontal.
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At \((1,3)\): \(2(1)-3=-1\), a gentle descent.
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At \((2,1)\): \(2(2)-1=3\), a steep climb.
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At \((0,2)\): \(2(0)-2=-2\), a moderate descent.
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Horizontal segments need \(2x-y=0\), which is the line \(y=2x\).
The slopes are \(-1\), \(3\) and \(-2\), and the field is flat exactly along the line \(y=2x\).
Common mistake
Watch out
A student computes the slope at \((2,1)\) by putting 1 in for \(x\) and 2 in for \(y\), getting \(2(1)-2=0\), and draws a flat segment.
Why it is wrong
The coordinates were swapped. A point written \((2,1)\) has \(x=2\) and \(y=1\), so the slope is \(2(2)-1=3\). Label the values of \(x\) and \(y\) from the ordered pair before substituting into any slope field formula.
Related terms and lessons
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