1. Given the curve x² - xy + y² = 9
(a) Write a general expression for the slope of the curve
(b) find the coordinates of the points on the curve where the tangents are vertical
(c) at the point (0,3) find the rate of change in the slope of the curve with respect to x.
The Attempt at a Solution
No problems with a or b i believe:
(a)2x - x(dy/dx) - y + 2y (dy/dx) = 0
Put all the terms containing dy/dx to one side and everything else on the other:
(2y-x) (dy/dx) = y-2x
dy/dx = (y-2x) / (2y-x)
(b) (2y)^2 - (2y)y + y^2 = 9
4y^2 - 2y^2 + y^2 = 9
y^2 = 3
y = +- sqrt(3) so x = +- 2sqrt(3)
So the points are (2sqrt(3),sqrt(3)) and (-2sqrt(3),-sqrt(3)).
(c) i'm not exactly sure how to do this one, im thinking to just plug (0,3) into the first derivative? or do i need to take the 2nd derivative?
Help is appreciated