Consider the differential equation dy/dx=2y
(A) At the point (-1.5,1), the direction field has a slope of?
(B) At the point (-0.5,-1), the direction field has a slope of ?
C) Along the x-axis the slope of the direction field is equal to which number or expression?
(D) Along the y-axis the slope of the direction field is equal to which number or expression?
A,B: Really? Plug in the coordinates!
C. when y=0, y'=0
D undefined
To find the slopes of the direction field for the given differential equation dy/dx = 2y, we can use the fact that the slope is equal to the value of dy/dx at any given point in the xy-plane.
(A) At the point (-1.5, 1), we can substitute the values into the differential equation:
dy/dx = 2y
2 = 2 * 1
2 = 2
Therefore, the slope of the direction field at the point (-1.5, 1) is 2.
(B) At the point (-0.5, -1), we can substitute the values into the differential equation:
dy/dx = 2y
2 = 2 * (-1)
2 = -2
Therefore, the slope of the direction field at the point (-0.5, -1) is -2.
(C) Along the x-axis, the value of y is always 0. Substituting this into the differential equation gives:
dy/dx = 2 * 0
dy/dx = 0
Therefore, the slope of the direction field along the x-axis is 0.
(D) Along the y-axis, the value of x is always 0. Substituting this into the differential equation gives:
dy/dx = 2y / 0
This is undefined.
Therefore, the slope of the direction field along the y-axis is undefined.
To find the slope of the direction field for a given point, we can use the given differential equation and substitute the coordinates of the point into it.
(A) At the point (-1.5,1), the direction field has a slope of:
We substitute x = -1.5 and y = 1 into the differential equation dy/dx = 2y:
dy/dx = 2y
dy/dx = 2(1) = 2
So, at the point (-1.5,1), the slope of the direction field is 2.
(B) At the point (-0.5,-1), the direction field has a slope of:
We substitute x = -0.5 and y = -1 into the differential equation dy/dx = 2y:
dy/dx = 2y
dy/dx = 2(-1) = -2
So, at the point (-0.5,-1), the slope of the direction field is -2.
(C) Along the x-axis, the slope of the direction field is equal to which number or expression?
Along the x-axis, y = 0. Substituting y = 0 into the differential equation dy/dx = 2y:
dy/dx = 2y
dy/dx = 2(0) = 0
Thus, along the x-axis, the slope of the direction field is 0.
(D) Along the y-axis, the slope of the direction field is equal to which number or expression?
Along the y-axis, x = 0. Substituting x = 0 into the differential equation dy/dx = 2y:
dy/dx = 2y
dy/dx = 2y = 2(0) = 0
Thus, along the y-axis, the slope of the direction field is also 0.