Calc FRQ 2 AP Practice Question

Question

Consider the curve given by the equation y 2 – 2x 2y = 3. Show your work.

a) Determine 𝑑𝑦/𝑑𝑥.

b) Write an equation for the line tangent to the curve at the point (1, –1).

c) Determine the coordinates of all points on the curve at which tangent line is horizontal.

d) Evaluate 𝑑^2𝑦/𝑑𝑥^2 at the point (1, –1).

Answer 1

y² - 2x²y = 3;

a) 2ydy/dx - 4xy - 2x²dy/dx = 0; dy/dx = 2xy/(y - x²).

b) Point (1, - 1) belong to this curve.

At this point dy/dx = - 2/(-1 - 1) = 1. So, equation of tangent line y = - 1 + 1(x - 1); y = x - 2

c) dy/dx = 2xy/(y - x²) = 0; 2xy = 0 and y - x² ≠ 0; x = 0; y² = 3; y = ±√3; (0, √3), (0, -√3)

y = 0; we have not such a points on this curve (0 = 3 contradiction)

d) d²y/dx² = 2[(y + xdy/dx)(y - x²) - xy(dy/dx - 2x)]/(y - x²)²; at (1, - 1) d²y/dx² = 2(1·(1 - 2))/4 = - 1/2