ASAP: Justify your answers through calculus workout

Question

Consider the curve y^3-xy=2

a) Show work to support that dy/dx=y/(3y^2-x)

b) Write an equation for the line tangent to the curve at the point (-1,1). Use your equation to approximate the value for y when x=-1.1

c) The variables x and y represent horizontal and vertical positions (in feet) for an object in motion at any time in seconds. Find dx/dt at the moment when x=3, y=2, and dy/dt=-5

Answer 1

a) We differentiate this equation by x: 3y²dy/dx - y - xdy/dx = 0; (3y² - x)dy/dx = y; dy/dx = y/(3y² - x).

b)Slope of tangent line m = 1/(3 + 1) = 1/4.

Equation of tangent line y = 1 + 1/4(x + 1); y = 1/4x + 5/4.

If x = - 1.1 then y ≈ 1/4·(-1.1) + 5/4 = 0,975

c) We differentiate given equation by time t: 3y²dy/dt - ydx/dt - xdy/dt = 0; 3·2²·(- 5) - 2dx/dt - 3(- 5) = 0;

-2dx/dt = 45; dx/dt = - 22.5 ft/s