Final answer:
The point (1,2) lies on the curve (4-xy)^2=2y, as verified by substitution. Using implicit differentiation, the derivative dy/dx at that point is found to be -4/3.
Step-by-step explanation:
To verify that the point (1,2) lies on the curve (4-xy)^2=2y, substitute x=1 and y=2 into the equation:
(4 - (1)(2))^2 = 2(2)
(4 - 2)^2 = 4
2^2 = 4
4 = 4
This confirms that (1,2) does lie on the curve.
Next, to find dy/dx at the point (1,2), we first take the derivative of both sides of the equation (4-xy)^2=2y with respect to x using implicit differentiation:
2(4 - xy)(-y - x(dy/dx)) = 2(dy/dx)
Now, we plug in the x and y values of our point:
2(4 - (1)(2))(-2 - 1(dy/dx)) = 2(dy/dx)
2(2)(-2 - (dy/dx)) = 2(dy/dx)
-8 - 4(dy/dx) = 2(dy/dx)
-8 = 6(dy/dx)
(dy/dx) = -8/6 = -4/3.
Therefore, the slope of the tangent to the curve at the point (1,2) is -4/3.