The differential equation dy/dx = (y-2)/(y-1)
I.produces a slope field with horizontal tangents at y = 2
II.produces a slope field with vertical tangents at y = -1
III.produces a slope field with columns of parallel segments
A. I only
B. II only
C. I and II
D. III only
I. yes
II. No - vertical segments at y=+1
III. No. For any x, the slopes change with y.
In fact, you will get rows of parallel segments, since for any y, the slope does not change with x.
you can graph slope fields at desmos.com
Thank you
To analyze the given differential equation dy/dx = (y-2)/(y-1) and determine the properties of its slope field, we can rewrite it as (y-1)dy = (y-2)dx.
Now, let's evaluate each statement one by one:
I. Produces a slope field with horizontal tangents at y = 2
To find the horizontal tangents, we set dy/dx = 0. In this case, we have (y-2)/(y-1) = 0. However, the value of y that makes the numerator equal to zero is y = 2. Therefore, it does not produce horizontal tangents at y = 2.
II. Produces a slope field with vertical tangents at y = -1
To find the vertical tangents, we need to find the values of x and y at the points where dy/dx is undefined. In this case, dy/dx is undefined when the denominator (y-1) is equal to zero, which happens when y = 1. Therefore, it does not produce vertical tangents at y = -1.
III. Produces a slope field with columns of parallel segments
To determine if the slope field has columns of parallel segments, we examine the behavior of dy/dx for different values of y. We can see that the numerator and denominator will be equal (y-2)/(y-1) = 1, except when y = 1. This means that except at y = 1, dy/dx will always be equal to 1, resulting in parallel segments in the slope field. Therefore, it produces a slope field with columns of parallel segments.
Considering all of these evaluations, the correct answer is:
D. III only
To determine which statements are true, we need to analyze the given differential equation and its slope field.
First, let's simplify the differential equation by multiplying each side by (y-1):
(y-1)dy = (y-2)dx
Now, let's integrate both sides:
∫(y-1)dy = ∫(y-2)dx
Integrating the left side gives:
(1/2)(y^2 - y) = ∫(y-2)dx
(1/2)(y^2 - y) = xy - 2x + C, where C is the constant of integration
Now, let's analyze the slope field. The slope field represents the values of dy/dx at different points on the xy-plane. To do this, we can plot a few points on the plane and calculate the slope at each point.
Consider the point (x, y) = (0, 1). Plugging these values into the differential equation, we get:
dy/dx = (1-2)/(1-1) = -1/0
Since division by zero is undefined, the slope at this point is undefined, indicating a vertical tangent.
Consider the point (x, y) = (0, 2). Plugging these values into the differential equation, we get:
dy/dx = (2-2)/(2-1) = 0/1
The slope at this point is zero, indicating a horizontal tangent.
Based on this information, we can now evaluate the statements:
I. The differential equation produces a slope field with horizontal tangents at y = 2. This is true, as seen from the analysis of the point (0, 2).
II. The differential equation produces a slope field with vertical tangents at y = -1. This is true, as seen from the analysis of the point (0, 1).
III. The differential equation produces a slope field with columns of parallel segments. This statement does not have enough information to determine its veracity.
Therefore, the correct answer is C. I and II.