# The graph of the function (x^2 + y^2)^2 = 4x^2y is a double folium as shown below. (a) Find, algebraically, all points on the curve with y = 1? (b) Verify that the slopes of tangent lines to both points with y = 1 is equal to 0?

We have:

Part (A)

Part (B) Differentiating implicitly using the chain rule and product rule we get:

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(a) The points on the curve with y = 1 can be found by substituting y = 1 into the equation (x^2 + y^2)^2 = 4x^2y. Simplifying the equation, we get (x^2 + 1)^2 = 4x^2. Expanding and rearranging, we have x^4 + 2x^2 + 1 = 4x^2. Combining like terms, we get x^4 - 2x^2 - 3 = 0. This is a quadratic equation in terms of x^2. Solving this equation, we find that x^2 = -1 or x^2 = 3. Taking the square root of both sides, we get x = ±√(-1) or x = ±√(3). Therefore, the points on the curve with y = 1 are (√(-1), 1), (-√(-1), 1), (√(3), 1), and (-√(3), 1).

(b) To verify that the slopes of tangent lines to both points with y = 1 are equal to 0, we need to find the derivative of the function (x^2 + y^2)^2 = 4x^2y with respect to x. Differentiating both sides of the equation implicitly, we get 2(x^2 + y^2)(2x) = 8xy + 4x^2(dy/dx). Simplifying, we have 4x^3 + 4x^2(dy/dx) = 8xy. Rearranging and solving for dy/dx, we get dy/dx = (4x^3 - 8xy) / (4x^2). Substituting the x-values of the points we found in part (a), we can calculate the slopes of the tangent lines. For each point, substitute the x-value into the equation dy/dx = (4x^3 - 8xy) / (4x^2) and simplify to find the slope. If the slope is equal to 0, then the verification is complete.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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