Cogito
AP Calculus AB · Chapter 7 · Lesson 2
Slope Fields
Seeing a solution before finding one.
12 problems · about 21 minutes · FUN-7.C
What this lesson teaches
The student sketches and interprets slope fields and matches them to differential equations.
- A slope field draws the prescribed slope at each point.
- Following the segments traces a solution curve.
- An initial condition selects one curve from the family.
Warm Up
Straightforward practice. Get the method working first.
5 problemsdy/dx = 4y. What is the slope where y = 3?
Answer 12
Why 12.
What does an initial condition do?
Answer It selects one curve from the family of solutions.
Why It picks a single solution.
dy/dx = x. What is the slope at x = 4?
Answer 4
Why It equals x.
dy/dx = 3y. Slope where y = 2?
Answer 6
Why 3 × 2.
dy/dx = y. At which y is there an equilibrium solution?
Answer 0
Why Where the slope is zero.
Build It Up
The same ideas with more to keep track of.
3 problemsdy/dx = y − 5. At which y is there an equilibrium?
Answer 5
Why y − 5 = 0.
How many solutions does a differential equation have without an initial condition? Enter 0 for infinitely many.
Answer 0
Why A whole family.
dy/dx = x + y. Slope at (1, 2)?
Answer 3
Why 1 + 2.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each field description by what dy/dx depends on.
Answer Depends on x only: Every row looks alike, The slope changes as you move sideways · Depends on y only: Every column looks alike, The slope changes as you move up
Why A row holds y fixed while x varies.
dy/dx = 2x. Slope at x = 5?
Answer 10
Why 2 × 5.
The Pattern: Every segment in a column is identical. Does dy/dx depend on x or y? 1 x only, 2 y only.
Answer 2
Why It depends only on y.
The Slope: dy/dx = 2y. What is the slope at the point where y = 3?
Answer 6
Why 6.