Cogito
Differential Equations · Chapter 1 · Lesson 2
Slope Fields
The shape of every solution, before solving anything.
12 problems · about 23 minutes · F-IF.B.4, F-IF.B.6
What this lesson teaches
The student sketches and reads slope fields and traces solution curves through them.
- A slope field draws the required slope at every point of the plane.
- Solution curves must run tangent to the dashes they pass through.
- Growth, decay and equilibria are all visible without solving anything.
Warm Up
Straightforward practice. Get the method working first.
5 problemsdy/dx = x + y. What is the slope at (1, 6)?
Answer 7
Why 7.
Why is a slope field useful?
Answer It shows how solutions behave even when no formula can be found.
Why It works without solving.
dy/dx = x. What is the slope at (4, 9)?
Answer 4
Why Only x matters.
dy/dx = y. What is the slope at (4, 9)?
Answer 9
Why Only y matters.
dy/dx = xy. What is the slope at (3, 2)?
Answer 6
Why Multiply the coordinates.
Build It Up
The same ideas with more to keep track of.
3 problemsdy/dx = −y. What is the slope at (0, 5)?
Answer -5
Why Negative y.
dy/dx = 3. Are all the dashes parallel? 1 yes, 0 no.
Answer 1
Why The slope never varies.
dy/dx = y − 4. At what value of y is the equilibrium?
Answer 4
Why Set the rate to zero.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsMatch each equation to what its field looks like.
Answer dy/dx = 2 → Every dash has the same slope; dy/dx = y → Dashes steepen as you move up; dy/dx = x → Dashes steepen as you move right
Why Ask which variable the slope depends on.
A solution starting exactly at an equilibrium. How far does it move?
Answer 0
Why The rate is zero there forever.
The Slope: dy/dx = x + y. What is the slope of the field at the point (2, 3)?
Answer 5
Why 5.
The Flat Line: dy/dx = 2y. At what value of y is the slope zero?
Answer 0
Why 0.