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Cogito

AP Calculus AB · Chapter 7 · Lesson 2

Practice Worksheet

Slope Fields

Seeing a solution before finding one.

12 problems · about 21 minutes · FUN-7.C

NameDate

Figure — use these to answer the problems

Change the equation and watch the segments reorganise. Follow them from any point to trace a solution.
  • The equationdy/dx = a·y
  • Through(0, 1)
k, the growth constant1
Starting x0
Starting y1
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. dy/dx = 4y. What is the slope where y = 3?

    Answer

  2. What does an initial condition do?

    • It selects one curve from the family of solutions.
    • It makes the equation solvable.

  3. dy/dx = x. What is the slope at x = 4?

    Answer

  4. dy/dx = 3y. Slope where y = 2?

    Answer

  5. dy/dx = y. At which y is there an equilibrium solution?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. dy/dx = y − 5. At which y is there an equilibrium?

    Answer

  2. How many solutions does a differential equation have without an initial condition? Enter 0 for infinitely many.

    Answer

  3. dy/dx = x + y. Slope at (1, 2)?

    Answer

Stretch Yourself

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

4 problems

  1. Sort each field description by what dy/dx depends on.

    Write each item under the heading it belongs to: Every row looks alike · The slope changes as you move sideways · Every column looks alike · The slope changes as you move up

    Depends on x only

    Depends on y only

  2. dy/dx = 2x. Slope at x = 5?

    Answer

  3. The Pattern

    Every segment in a column is identical. Does dy/dx depend on x or y? 1 x only, 2 y only.

    Answer

  4. The Slope

    dy/dx = 2y. What is the slope at the point where y = 3?

    Answer