Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- A differential equation dy/dx = f(x, y) gives the slope of a solution curve at every point it passes through.
- A slope field shows this on a finite grid: at each point, a short segment with slope f(x, y).
- Each segment is a piece of a tangent line to the solution through that point. The segments are not the solution itself.
- Drawing one needs only substitution, no solving. That is why a field lets you estimate how solutions behave.
- Draw segments short, equal in length and centred on the points. Only the tilt carries information.
Key relationships
| What you see in the field | What it tells you about dy/dx |
|---|---|
| Every column identical | dy/dx depends on x only |
| Every row identical | dy/dx depends on y only |
| Neither repeats | dy/dx depends on both x and y |
| Horizontal segment at (a, b) | dy/dx = 0 at (a, b) |
| Rising to the right | dy/dx > 0 |
| Falling to the right | dy/dx < 0 |
| Steeper segment | Larger |dy/dx| |
Method for sketching: table → structure → draw. Method for matching: rows or columns → zero slopes → sign at a test point.
Assumptions behind the picture
- The axes use the same scale, so a slope of 1 looks like 45°. On unequal scales, only the relative steepness can be trusted.
- The field shows directions at the grid points only. Between the points you blend the directions smoothly.
Mistakes to avoid
- Swapping coordinates. In (1, −2), x = 1 and y = −2.
- Long lines that run into the next point.
- Blank points where the slope is 0. Draw a horizontal segment.
- Mixing up rows and columns. x only gives identical columns.
- Matching on zero slopes alone. 1 − y² and y² − 1 have the same flat rows but opposite signs.
- Inconsistent steepness, for example drawing slopes 1/2 and 2 at the same angle.
Quick self-check
- For dy/dx = xy − 2, what is the slope of the segment at (2, 3)? (2 × 3 − 2 = 4: a steep rise)
- For dy/dx = 3 − y, which segments are horizontal, and how do the segments look along y = 5? (All segments on the row y = 3. Along y = 5 every segment has slope −2, so each row is identical and falls.)
- A field has identical columns and horizontal segments only along the y-axis. Does dy/dx depend on y? (No. Identical columns mean dy/dx depends on x only.)
Next: practice questions.