Find all points on the curve ln(xy) = x^2 where the tangent lines are vertical.?

Question

a. None of the tangent lines are vertical.
b. (0,0)
c. (0,1)
d. (1,0)

Answer 1

a. None of the tangent lines are vertical.
Reason: Solve for y,
y = (1/x)e^(x^2) -> ∞, as x->0
Therefore, there is no point on the curve where the tangent lines are vertical. However, as x->0 from either side, there is a vertical asymptote x = 0.

Answer 2

The function, x^2 ? xy ? y^2 = 5, is a hyperbola, which means there could be 2 factors the place it curves around and for that reason has 2 vertical tangent lines. (once you're allowed to graph, as in AP Calc, you will see this.) for that reason, basically B or E is actual. we are going to examine the values for B. to locate the vertical tangent lines, you will locate the place the shrink of the derivative techniques infinity... Differentiate implicitly, (in fact multiply the two factors by utilising d/dx) x^2 ? xy ? y^2 = 5 and you get 2x - y - (dy/dx)x - 2(dy/dx)y = 0, or dy/dx = (2x - y) / (x + 2y). This techniques infinity if (2x - y) is non-0 and (x + 2y) = 0. for this reason, x = -2y on the tangent line. on account that -5y isn't inevitably 0, we are going to proceed. pondering option B, (2, -a million) seems to artwork, on account that -2y = x and -2(-a million) = 2. despite the fact that, -2(-2) =/= a million. the different surely answer is (-2, a million). for that reason, the respond is E.

Answer 3

implicit derivative
1/(xy) *(y+xy´)=2x so 1/x+1/y*y´= 2x
so y´= (2x-1/x)*y = (2x^2-1)y/x
as x can´t be 0 there is no vertical tangent