Slope fields practice questions.

A slope field is drawn for the differential equation dy/dx = x − y. At the point (2, 2), which statement about the slope of the field is correct?

• The slope is 0, so the line segment is horizontal
• The slope is 4, so the line segment is steep and rising
• The slope is −4, so the line segment is steep and falling
• The slope is undefined (vertical segment)

Answer: The slope is 0, so the line segment is horizontal

At (2, 2), dy/dx = 2 − 2 = 0, so the segment is horizontal. Along the line y = x every segment of this field is horizontal.

The slope field for dy/dx = y shows segments whose slope depends only on y. Which description of the field is correct?

• On any horizontal line, segments are parallel
• On any vertical line, the segments are parallel
• All segments are horizontal
• Segments get steeper as x increases

Answer: On any horizontal line, segments are parallel

Because dy/dx = y is independent of x, the slope is fixed once y is fixed; so along a horizontal line (constant y) every segment has the same slope and is parallel.

A slope field is given for dy/dx = −x/y. Along which set of points are the segments horizontal?

• Where x = 0 (the y-axis)
• Where y = 0 (the x-axis)
• Where x = y
• Everywhere the segments are horizontal

Answer: Where x = 0 (the y-axis)

The slope is 0 when the numerator −x = 0, i.e. x = 0. (On y = 0 the slope is undefined, giving vertical segments.) These curves are circles x² + y² = c.

The differential equation dy/dx = y(1 − y) models limited growth. On its slope field, which horizontal lines consist entirely of horizontal segments (equilibrium solutions)?

• y = 0 and y = 1
• y = 0 and y = −1
• x = 0 and x = 1
• y = 1 only

Answer: y = 0 and y = 1

dy/dx = 0 when y(1 − y) = 0, i.e. y = 0 or y = 1. These are the equilibrium (constant) solutions where every segment is horizontal.

For the slope field of dy/dx = 2 − y, describe the long-term behaviour of the solution curve that starts at (0, 5) as x increases.

• It decreases towards y = 2
• It increases without bound
• It approaches the line y = 0
• It oscillates about y = 2

Answer: It decreases towards y = 2

For y > 2, dy/dx = 2 − y < 0 so the curve falls; as y approaches 2 the slope approaches 0. Thus the solution tends towards the equilibrium y = 2.

In the slope field for dy/dx = x, which statement is correct about how the segments look?

• Along any vertical line (x constant) all segments have the same slope
• Along any horizontal line (y constant) all segments have the same slope
• All segments are horizontal
• The slope depends on y but not x

Answer: Along any vertical line (x constant) all segments have the same slope

Since dy/dx = x depends only on x, the slope is constant along each vertical line; the field looks the same at every height for a given x.