Question

Find the slope of the tangent line to the graph at the indicated point. Cissoid: \((4-x) y^{2}=x^{3}\) Point: (2,2)

Short answer

The slope of the tangent line to the graph at the point (2,2) is 2.25.

Step-by-step solution

Differentiate the given equation implicitly

Differentiate both sides of the equation with respect to x, treating y as a function of x. This gives: \(-y^{2} - 2xy\frac{dy}{dx} = 3x^{2}\)

Solve for dy/dx

Isolate \(\frac{dy}{dx}\) to get the derivative of y with respect to x. This gives: \(\frac{dy}{dx} = \frac{y^{2} + 3x^{2}}{2xy}\)

Substitute the coordinates of the point into the derivative formula

Substitute x = 2 and y = 2 into \(\frac{dy}{dx} = \frac{y^{2} + 3x^{2}}{2xy}\) to find the slope of the tangent line at the given point. This gives: \(\frac{dy}{dx} = \frac{2^{2} + 3*2^{2}}{2*2*2} = 2.25\)