The differential equation dy/dx=(x+1)/(x-2)
I. produces a slope field with horizontal tangents at x = −1
II. produces a slope field with vertical tangents at x = 2
III.produces a slope field with rows of parallel segments
a) I and II only
b) II only
c) III only
d) I only
clearly, for any constant value of x, dy/dx is constant.
since x-2=0 when x=2, the slope field is vertical there
to get rows of parallel segments, you need a constant value for y. But that means you need (x+1) = y(x-2), which has only one solution. SO, no rows of parallel segments.
Looks like I and II only, right?
You know, there are online sites for checking slope fields. One of them is
https://www.desmos.com/calculator/p7vd3cdmei
Just change the function and watch the screen.
Well, let's analyze the options one by one.
I. Produces a slope field with horizontal tangents at x = -1.
To determine if the slope field has horizontal tangents at x = -1, we need to find the derivative dy/dx at that point.
dy/dx = (x+1) / (x-2)
Plugging x = -1 into the equation, we get:
dy/dx = (-1+1)/(-1-2) = 0/(-3) = 0
Since the slope is 0, there will be horizontal lines in the slope field at x = -1. So, option I is correct.
II. Produces a slope field with vertical tangents at x = 2.
Now, to determine if the slope field has vertical tangents at x = 2, we'll find the dy/dx at that point.
dy/dx = (x+1) / (x-2)
Plugging x = 2 into the equation, we get:
dy/dx = (2+1)/(2-2) = 3/0
Uh-oh, division by zero is undefined! And since we get that when plugging x = 2 into the equation, it means that there will be vertical tangents at x = 2. So, option II is correct.
III. Produces a slope field with rows of parallel segments.
To determine if the slope field has rows of parallel segments, let's look at the equation again:
dy/dx = (x+1) / (x-2)
Since the slope depends on the value of x, the segments won't be parallel. Therefore, option III is incorrect.
So, the options that are correct are I and II, giving us:
a) I and II only.
To determine which statements are true, we can analyze the given differential equation:
dy/dx = (x + 1)/(x - 2)
I. produces a slope field with horizontal tangents at x = -1:
To find horizontal tangents, we need to set dy/dx = 0. Therefore:
0 = (x + 1)/(x - 2)
(x + 1)/(x - 2) = 0
For this fraction to be equal to zero, the numerator (x + 1) must be equal to zero. Hence:
x + 1 = 0
x = -1
So the statement is true.
II. produces a slope field with vertical tangents at x = 2:
To find vertical tangents, we need to check if the denominator (x - 2) becomes zero. Therefore:
x - 2 = 0
x = 2
So the statement is true.
III. produces a slope field with rows of parallel segments:
This statement is not related to the given differential equation and cannot be determined from the equation alone.
Based on the analysis above, the correct answer is:
a) I and II only
To determine which options are true, we need to analyze the given differential equation and visualize its slope field.
First, let's find the equation's slope field by plotting several representative lines. We can accomplish this by solving the differential equation for various x and y values.
The given differential equation is dy/dx = (x + 1)/(x - 2). To make it easier, let's rewrite it as (x - 2)dy = (x + 1)dx.
Integrating both sides, we have:
∫(x - 2)dy = ∫(x + 1)dx
This simplifies as:
(x - 2)y = (x^2/2 + x) + C
If we rearrange, we get:
y = (x^2/2 + x + C) / (x - 2)
Now, to visualize the slope field, we can choose various values of C and plot several tangent lines for different (x, y) points.
I. The statement says that the slope field has horizontal tangents at x = -1.
To determine this, we need to find the value of C that makes the slope zero at x = -1. Substituting x = -1 into the equation:
y = (-1/2 + (-1) + C) / (-1 - 2)
y = (C - 5/2) / -3
To find a value of C that makes the slope zero, we need (C - 5/2) / -3 = 0. Solving it, we get:
C - 5/2 = 0
C = 5/2
So, when C = 5/2, the slope field produces a horizontal tangent line at x = -1.
II. The statement says that the slope field has vertical tangents at x = 2.
To determine this, we need to find the value of C that makes the denominator of the slope (x - 2) equal to zero when x = 2.
Substituting x = 2 into the equation:
y = (2^2/2 + 2 + C) / (2 - 2)
y = (2 + 2 + C) / 0
However, division by zero is undefined. This means that the slope field does not have vertical tangents at x = 2.
III. The statement says that the slope field has rows of parallel segments.
To visualize this, we can plot several tangent lines for different (x, y) points on the slope field.
From our equation y = (x^2/2 + x + C) / (x - 2), we can see that different values of C produce different slopes for different (x, y) points. Therefore, the slope field does not have rows of parallel segments.
Based on our analysis, the correct answer is:
a) I and II only