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Cogito

Differential Equations · Chapter 1 · Lesson 2

Practice Worksheet

Slope Fields

The shape of every solution, before solving anything.

12 problems · about 23 minutes · F-IF.B.4, F-IF.B.6

NameDate

Figures — use these to answer the problems

Here dy/dx = −y. Release a solution from anywhere and watch it settle toward zero, whichever side it starts on.
  • The equationdy/dx = −a·y
  • Through(-3, 3)
the decay rate1
starting x-3
starting y3
Figure 1
A field that depends on both variables. Follow the dashes from different starting points and compare where they lead.
  • The equationdy/dx = a·(x + y)
  • Through(0, 0)
starting x0
starting y0
Figure 2

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dx = x + y. What is the slope at (1, 6)?

    Answer

  2. Why is a slope field useful?

    • It shows how solutions behave even when no formula can be found.
    • It gives the exact formula for the solution.

  3. dy/dx = x. What is the slope at (4, 9)?

    Answer

  4. dy/dx = y. What is the slope at (4, 9)?

    Answer

  5. dy/dx = xy. What is the slope at (3, 2)?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. dy/dx = −y. What is the slope at (0, 5)?

    Answer

  2. dy/dx = 3. Are all the dashes parallel? 1 yes, 0 no.

    Answer

  3. dy/dx = y − 4. At what value of y is the equilibrium?

    Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

  1. Match each equation to what its field looks like.

    Draw a line from each item on the left to its match on the right.

    • dy/dx = 2
    • dy/dx = y
    • dy/dx = x
    • Every dash has the same slope
    • Dashes steepen as you move up
    • Dashes steepen as you move right

  2. A solution starting exactly at an equilibrium. How far does it move?

    Answer

  3. The Slope

    dy/dx = x + y. What is the slope of the field at the point (2, 3)?

    Answer

  4. The Flat Line

    dy/dx = 2y. At what value of y is the slope zero?

    Answer