Slope fields

About this lesson

Calculus AB (AP-aligned) › Differential equations

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Slope fields

Warm-up

Three quick questions from earlier lessons.

Discover

For dydx=x−y\dfrac{dy}{dx} = x - y, a slope field draws a short segment at each grid point with that slope. What is the slope at (3,1)(3, 1)?

  1. 2
  2. −2

Key idea

A slope field shows the shape of all solutions without solving.

At each point (x,y)(x, y) draw a segment of slope f(x,y)f(x, y). Solution curves follow the segments. If ff depends only on xx, segments are parallel along vertical lines; only on yy, along horizontal lines; where f=0f = 0, segments are horizontal.

Worked example

Describe the field of dydx=y−2\dfrac{dy}{dx} = y - 2.

Same slope along each horizontal line
Depends only on y.
Horizontal segments along y=2y = 2
y − 2 = 0.
Rising above y=2y = 2, falling below
Signs.

Your turn, guided

Your turn: dydx=xy\dfrac{dy}{dx} = xy at (2,−3)(2, -3).

slope −6-6
2 × (−3).

Practice ★

  1. In the slope field of dydx=x+3y\dfrac{dy}{dx} = x + 3y, what is the slope of the segment at (−3,1)(-3, 1)?
  2. In the slope field of dydx=x+2y\dfrac{dy}{dx} = x + 2y, what is the slope of the segment at (−1,3)(-1, 3)?
  3. In the slope field of dydx=x+2y\dfrac{dy}{dx} = x + 2y, what is the slope of the segment at (0,−1)(0, -1)?

Watch out!

Practice ★★

  1. In the slope field of dydx=9x2\dfrac{dy}{dx} = 9x^2, which segments are all parallel?
    1. Those along any vertical line x=cx = c
    2. Those along any horizontal line y=cy = c
    3. None: every segment has its own slope
  1. Slope field of dydx=y(3−y)\dfrac{dy}{dx} = y(3 - y). Tick all the true statements.
    1. Segments are horizontal along y=0y = 0 and y=3y = 3
    2. Slopes are positive for 0<y<30 < y < 3
    3. Slopes are negative for y>3y > 3
    4. Slopes are positive for y>3y > 3
    5. Segments are horizontal along x=3x = 3
  2. Slope field of dydx=y(5−y)\dfrac{dy}{dx} = y(5 - y). Tick all the true statements.
    1. Segments are horizontal along y=0y = 0 and y=5y = 5
    2. Slopes are positive for 0<y<50 < y < 5
    3. Slopes are negative for y>5y > 5
    4. Slopes are positive for y>5y > 5
    5. Segments are horizontal along x=5x = 5

Exit ticket

Three questions to check you have got it.

  1. In the slope field of dydx=x+3y\dfrac{dy}{dx} = x + 3y, what is the slope of the segment at (−3,1)(-3, 1)?
  2. In the slope field of dydx=x+2y\dfrac{dy}{dx} = x + 2y, what is the slope of the segment at (−1,3)(-1, 3)?
  3. In the slope field of dydx=x+2y\dfrac{dy}{dx} = x + 2y, what is the slope of the segment at (0,−1)(0, -1)?

Summary card

Slope fields

  • Slope at (x,y)(x, y): substitute into f(x,y)f(x, y).
  • Only xx: parallel along verticals. Only yy: along horizontals.
  • f=0f = 0: horizontal segments (equilibria if constant in yy).

Watch out: (x,y)(x, y) in that order.

End of the lesson

Next lesson: Separation of variables and initial conditions Back to the chapter

Answers: Slope fields

A

  1. 00
  2. 55
  3. −2-2

B

  1. Those along any vertical line x=cx = c

C

  1. Segments are horizontal along y=0y = 0 and y=3y = 3 · Slopes are positive for 0<y<30 < y < 3 · Slopes are negative for y>3y > 3
  2. Segments are horizontal along y=0y = 0 and y=5y = 5 · Slopes are positive for 0<y<50 < y < 5 · Slopes are negative for y>5y > 5

D

  1. 00
  2. 55
  3. −2-2

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