Math: Slope Fields and Direction Fields Worksheet

Question 1

For the differential equation dy/dx = x + y, find the slope of the solution curve at the point (2, -1).

Accepted Answer

The slope is 1 because dy/dx = x + y = 2 + (-1) = 1 at the point (2, -1).

Question 2

For the differential equation dy/dx = y - 2, describe the slope field along the horizontal line y = 2.

Accepted Answer

Along the line y = 2, the slope is 0 because dy/dx = 2 - 2 = 0. The slope field has horizontal line segments everywhere on y = 2.

Question 3

For the differential equation dy/dx = x, explain why all points with the same x-coordinate have the same slope in the slope field.

Accepted Answer

All points with the same x-coordinate have the same slope because the differential equation depends only on x, not on y. For example, every point with x = 3 has slope 3.

Question 4

A slope field is made for dy/dx = -y. Describe what the small line segments look like above the x-axis, on the x-axis, and below the x-axis.

Accepted Answer

Above the x-axis, y is positive, so the slopes are negative. On the x-axis, y = 0, so the slopes are 0 and the segments are horizontal. Below the x-axis, y is negative, so the slopes are positive.

Question 5

For the differential equation dy/dx = x - y, calculate the slopes at the points (0, 0), (1, 0), (1, 1), and (0, 2).

Accepted Answer

At (0, 0), the slope is 0 - 0 = 0. At (1, 0), the slope is 1 - 0 = 1. At (1, 1), the slope is 1 - 1 = 0. At (0, 2), the slope is 0 - 2 = -2.

Question 6

The differential equation dy/dx = 3 has a constant slope field. Describe the slope field and the general shape of its solution curves.

Accepted Answer

The slope field has the same positive slope of 3 at every point. The solution curves are straight lines with slope 3, so they have the form y = 3x + C.

Question 7

For dy/dx = y(4 - y), identify the equilibrium solutions and explain how they appear in the slope field.

Accepted Answer

The equilibrium solutions are y = 0 and y = 4 because y(4 - y) = 0 at those values. In the slope field, these appear as horizontal line segments along the lines y = 0 and y = 4.

Question 8

A solution curve to a differential equation passes through (0, 1). The slope field near that point has positive slopes that get steeper as x increases. Describe the likely behavior of the solution curve just to the right of x = 0.

Accepted Answer

Just to the right of x = 0, the solution curve should rise. Since the slopes get steeper as x increases, the curve should increase at a faster rate as it moves to the right.

Question 9

For the differential equation dy/dx = x + 1, sketching is not required. State where the slope field has horizontal segments and explain your reasoning.

Accepted Answer

The slope field has horizontal segments where dy/dx = 0. Since x + 1 = 0 when x = -1, the horizontal segments occur along the vertical line x = -1.

Question 10

The slope field for a differential equation shows horizontal segments along y = 1. Above y = 1, the slopes are positive. Below y = 1, the slopes are negative. Which differential equation best matches this description: dy/dx = y - 1, dy/dx = 1 - y, or dy/dx = x - 1? Explain your choice.

Accepted Answer

The best match is dy/dx = y - 1. This equation gives slope 0 when y = 1, positive slopes when y is greater than 1, and negative slopes when y is less than 1.

Math: Slope Fields and Direction Fields

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