Help me with math please

Question

The graph of the equation y^4=y^2-x^2 is given.

a) verify that dy/dx = (2x)/(2y-(4y^3)). Show why.

b) Find the point(s) (x,y) at which the graph has a horizontal tangent line. Show why/work.

c) Find the y-value(s) at which the graph has a vertical tangent line. Show why/work.

d) While the graph exists at (0,0), the slope does not. Using dy/dx, explain why this is so.

2.Consider the curve given by 6+x^3y=xy^2

a) show that dy/dx =(y^2-3x^2y)/(x^3-2xy)

b) find every point(s) on the graph of the curve that has an x-coordinate of 1, then write an equation for the tangent line at every/each of these point(s).

c) The graph of the curve has vertical tangent lines. Find the x-coordinate of each of these vertical tangent lines. Show why/work.

Answer 1

Use implicit differentiation like this:

To find the points where the graph has a horizontal tangent set dy/dx equal to zero. Note that the ONLY way a rational expression can equal zero is if the numerator equals zero. So 2x = 0 or x = 0. Then since y⁴ = y² - x², then:

y⁴ = y² - 0

y⁴ - y² = 0

y²(y² - 1) = 0

y²(y + 1)(y - 1) = 0

y = 0, y = -1, y = 1 but notice that y = 0 will give you a derivative with the denominator equal to zero so that cannot be a solution. That means (0, 1) and (0, -1) have horizontal tangent lines.

Vertical tangent lines will occur when the derivative is undefined. To find those point, set the denominator equal to zero.

Do the same process on the other equation