SOLUTION: 1. For the curve x + y^2 -y =1, find equations of all lines tangent to the curve at x=1

We want the equations of the two green lines below



The y-coordinates of the two points at x=1 are found by setting x=1 and
solving for y:



So the two points of tangency are the points (1,0) and (1,1)

We take the derivative implicitly (i.e., without solving for y),
to find the slope m of each of the green tangent lines:



So the slope m of the tangent line at (1,0) is 
found by setting y equal to the y-coordinate of
(1,0), which is 0, in the equation for the 
derivative above:



So we are looking for the equation of the green line through 
(1,0) with slope m = 1



That's the equation of the tangent line at (1,0).

Similarly the slope m of the green tangent line at (1,1) is 
found by setting y equal to the y-coordinate of
(1,1), which is 1, in the equation for the 
derivative above:



So we are looking for the equation of the green line through 
(1,1) with slope m = -1



That's the equation of the green tangent line at (1,1).

Edwin