Final answer:
The slope of the curve x^(1/2) + y^(1/2) = 6 at the point (9,9) is determined by implicit differentiation and substituting the point into the derivative, resulting in a slope of -1.
Step-by-step explanation:
To determine the slope at a point on the curve x1/2 + y1/2 = 6, we need to differentiate the equation implicitly with respect to x. Let's find the derivative step by step:
- Starting with the original equation x1/2 + y1/2 = 6.
- Differentiate both sides with respect to x: d/dx(x1/2) + d/dx(y1/2) = d/dx(6).
- Applying the chain rule to y1/2, the derivative becomes 1/(2x1/2) + (1/(2y1/2))*dy/dx = 0
- Solve for dy/dx, which gives us dy/dx = -1/(2x1/2) * (2y1/2) or dy/dx = -y1/2 / x1/2.
- Now, substitute the given point (9,9) into this derivative to find the slope at that point: dy/dx = -91/2 / 91/2 = -1.
Thus, the slope of the curve at the point (9,9) is -1.