Sure, I can help you with that! Let's go step by step.
A) To find the general expression for the slope of the curve, we need to differentiate the equation of the curve with respect to x. So let's start by differentiating both sides of the equation x^2 - xy + y^2 = 9 with respect to x.
d/dx (x^2 - xy + y^2) = d/dx (9)
By using the chain rule, we can differentiate each term separately:
2x - (x * dy/dx + y) + 2y * dy/dx = 0
Now, let's isolate dy/dx (the slope) on one side of the equation:
2y * dy/dx - x * dy/dx = y - 2x
Factoring out dy/dx:
dy/dx (2y - x) = y - 2x
Finally, we can express the slope of the curve as:
dy/dx = (y - 2x) / (2y - x)
B) Now, let's find the coordinates of the points on the curve where the tangents are vertical. For the tangent to be vertical, the slope has to be undefined or infinite (since division by zero is undefined). In this case, the slope is given by the expression dy/dx = (y - 2x) / (2y - x).
To find the points where the tangent is vertical, we need to set the denominator (2y - x) equal to zero and solve for x and y:
2y - x = 0
Simplifying for x, we get:
x = 2y
Substituting this value of x back into the original equation x^2 - xy + y^2 = 9, we have:
(2y)^2 - (2y)y + y^2 = 9
Simplifying further:
4y^2 - 2y^2 + y^2 = 9
3y^2 = 9
y^2 = 3
Taking the square root of both sides, we get:
y = ± √3
Now, substitute the values of y back into x = 2y:
For y = √3: x = 2√3
For y = -√3: x = -2√3
So, the coordinates of the points on the curve where the tangents are vertical are (2√3, √3) and (-2√3, -√3).
C) To find the rate of change in the slope of the curve with respect to x at the point (0,3), we need to find the derivative of the slope expression dy/dx = (y - 2x) / (2y - x) with respect to x, and then substitute the coordinates (x, y) = (0, 3).
Taking the derivative of dy/dx:
d^2y/dx^2 = [(2y - x)(2dy/dx) - (y - 2x)(2dy/dx)] / (2y - x)^2
Substituting the coordinates (x, y) = (0, 3):
d^2y/dx^2 = [(6 - 0)(2dy/dx) - (3 - 2(0))(2dy/dx)] / (6 - 0)^2
Simplifying further:
d^2y/dx^2 = (12dy/dx - 3dy/dx) / 36
d^2y/dx^2 = 9dy/dx / 36
d^2y/dx^2 = dy/dx / 4
Since dy/dx at the point (0, 3) can be found by substituting the coordinates into the initial slope expression, we can calculate dy/dx = (y - 2x) / (2y - x) at (x, y) = (0, 3):
dy/dx = (3 - 2(0)) / (2(3) - 0) = 3/6 = 1/2
Substituting this value into d^2y/dx^2 = dy/dx / 4:
d^2y/dx^2 = (1/2) / 4 = 1/8
Therefore, at the point (0, 3), the rate of change in the slope of the curve with respect to x is 1/8.