To find the slope of the curve at the point where y = 2, we first need to find the corresponding x-value by substituting y into the equation of the curve.
The curve is given by:
y³ + xy² = 4
Substituting y = 2:
(2)³ + x(2)² = 4
8 + 4x = 4
4x = 4 – 8
4x = -4
x = -1
Now, we have the point (-1, 2). To find the slope at this point, we can use implicit differentiation.
Differentiating both sides of y³ + xy² = 4 with respect to x:
3y²(dy/dx) + (y²)(1) + (x)(2y)(dy/dx) = 0
Rearranging the equation:
3y²(dy/dx) + y² + 2xy(dy/dx) = 0
(3y² + 2xy)(dy/dx) + y² = 0
Solving for dy/dx:
(dy/dx) = -y² / (3y² + 2xy)
Substituting x = -1 and y = 2 into the equation:
(dy/dx) = -(2)² / (3(2)² + 2(-1)(2))
=( -4 ) / ( 12 – 4 )
= -4 / 8 = -1/2
Therefore, the slope of the curve at the point where y = 2 is -1/2.