Final answer:
To find the equation of the tangent line at the point (0, -3) on the devil's curve, we use implicit differentiation, solve for the slope dy/dx, and then utilize the point-slope equation with the calculated slope and given point.
Step-by-step explanation:
To find the equation of the tangent line to the given curve y^2(y^2-9) = x^2(x^2-11) at the point (0, -3), we first need to use implicit differentiation to find the slope of the tangent line. Differentiating both sides of the equation with respect to x, we get:
- 2y(dy/dx)(y^2 - 9) + y^2(2y)(dy/dx) = 2x(x^2 - 11) + x^2(2x), since d/dx [y^2] = 2y(dy/dx).
- Now solve for dy/dx, which represents the slope of the tangent line at any point on the curve.
After solving for dy/dx, substitute the given point (0, -3) into the derivative to find the slope at that point.
Finally, use the point-slope form of the equation for a line, which is y-y1 = m(x-x1) to write the equation of the tangent line, where (x1, y1) is the given point, and m is the slope we just found.
The slope of the curve at a point is always equal to the slope of the tangent line at that same point. This relationship is fundamental in calculus and is used to determine the direction in which a curve is increasing or decreasing at a specific point.