AP® Calculus AB review sheet from Aim for Five (aimforfive.com/calc-ab/units/7/7-3)
Unit 7 · Topic 7.3
7.3 Sketching Slope Fields
A slope field is a picture of a differential equation: at each point on a grid, a short segment shows the slope dy/dx would have there. You sketch one by plugging each grid point into the equation.
Key terms
- slope field
- direction field
- dy/dx
- tangent segment
What a slope field shows
A first-order differential equation dy/dx = f(x, y) gives a slope at every point. A slope field (also called a direction field) draws a short line segment at each grid point with exactly that slope. The segments are tiny pieces of tangent lines to the solution curves.
Solution curves flow along the segments, like iron filings lining up around a magnet. You don't need to solve the equation to see their general shape.
How to sketch one
- List the grid points you're asked about, usually a small grid like x = −1, 0, 1 and y = −1, 0, 1.
- Plug each point into dy/dx to get a number.
- Draw a short segment centered at that point: horizontal for slope 0, rising left to right for positive slopes, falling for negative slopes, steeper for bigger absolute values.
- Keep segments short so they don't run into each other. A slope of 1 is a 45° segment; a slope of 2 is steeper; a slope of ½ is flatter.
Patterns that save time
- If dy/dx depends only on x (like dy/dx = x²), every segment in the same vertical column is parallel.
- If dy/dx depends only on y (like dy/dx = y − 2), every segment in the same horizontal row is parallel. These are called autonomous equations.
- Find where dy/dx = 0. Those points get horizontal segments. For dy/dx = x − y, that's along the line y = x.
- If dy/dx = 0 along a horizontal line y = c (as in dy/dx = y − 2 at y = 2), then y = c is itself a solution. It's called an equilibrium solution.
- If dy/dx is undefined at a point (like dy/dx = x/y when y = 0), leave that point blank or draw a vertical segment where the slope is infinite.
Why the exam likes slope fields
Slope fields connect a formula (the differential equation) with a picture (the field) and with solution curves. Sketching them by hand checks that you can evaluate the equation correctly and translate numbers into slopes. The partner topic, 7.4, uses the picture to reason about solutions.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Sketching on a 3 × 3 grid
For dy/dx = x − y, find the slope at each point with x = −1, 0, 1 and y = −1, 0, 1, and describe the slope field.
Show the solutionHide the solution
- Step 1: Compute x − y at each point. Row y = 1: at x = −1, 0, 1 the slopes are −2, −1, 0.
- Step 2: Row y = 0: slopes −1, 0, 1.
- Step 3: Row y = −1: slopes 0, 1, 2.
- Step 4: Slopes are 0 along the diagonal y = x: at (−1, −1), (0, 0) and (1, 1). Those segments are horizontal.
- Step 5: Above the line y = x (where y > x), slopes are negative, so segments fall. Below it, slopes are positive, so segments rise. Slopes get steeper the farther you go from the line y = x.
Answer: Slopes (rows y = 1, 0, −1; columns x = −1, 0, 1): −2, −1, 0 / −1, 0, 1 / 0, 1, 2. Horizontal segments lie on y = x; segments fall above that line and rise below it.
- Example 2
Reading features without plotting every point
Describe the slope field for dy/dx = y² − 4.
Show the solutionHide the solution
- Step 1: The equation depends only on y, so all segments in a horizontal row have the same slope.
- Step 2: Slope 0 where y² − 4 = 0: along y = 2 and y = −2. These rows are horizontal, so y = 2 and y = −2 are equilibrium solutions.
- Step 3: For |y| > 2, y² − 4 > 0: segments rise.
- Step 4: For −2 < y < 2, y² − 4 < 0: segments fall. The steepest fall is at y = 0, where the slope is −4.
Answer: Horizontal segments along y = ±2; rising segments above y = 2 and below y = −2; falling segments between, steepest (slope −4) along y = 0. Every row's segments are parallel.
Common mistakes
- Mixing up x and y when evaluating dy/dx at a point. Write the ordered pair before plugging in.
- Drawing long segments that look like a solution curve. Each mark should be short and centered on its point.
- Making all positive slopes look the same. Show that slope 2 is steeper than slope 1.
- Assuming the field is the same in each column when the equation depends on y.
On the exam
- Free-response questions often give a grid of points and ask you to sketch the slope field there. Accuracy at each point is what's graded: sign, zero and relative steepness.
- Before sketching, find where the slope is zero and where it's undefined. That gives you the skeleton of the picture.
Connected topics
Videos
Check yourself
4 questions on 7.3 Sketching Slope Fields. Pick an answer to see if you got it, and why.
In the slope field for the differential equation dy/dx = x − 2y, all of the segments are horizontal along which line?
Consider the slope field for the differential equation dy/dx = x² − y. At which of the following points does the segment in the slope field have the greatest slope?
In the slope field for the differential equation dy/dx = y(2 − y), for which values of y do the segments have negative slope?
In the slope field for the differential equation dy/dx = (x + 1)(y − 2), along which lines are all the segments horizontal?
0 of 4 answered