Tangents and Implicit Differentiation

Question

1. Tangents and Implicit Differentiation. Analyze the graph of 

y^3 - 3y = x^2 -2x - 1

     a) Algebraically find the coordinates of two points on the graph with y =1

    b) Use implicit differentiation to find the derivative.

   c) Find the coordinates of all points on the graph where the tangent is horizontal.

      

d) Find all points on the graph where the tangent is vertical.

Answer 1

a) - 2 = x² - 2x - 1

x² - 2x + 1 = 0

(x - 1)² = 0 ; x = 1 ; (1 , 1)

b) 3y²·y^' - 3y^' = 2x - 2

y^' · (3y² - 3) = 2x - 2

y^' = [2x - 2] / [3y² - 3]

c) y^' = 0 when x = 1 ; y³ - 3y + 2 = 0 ; (y - 1)(y² + y - 2) = 0 ; (y -1)²(y + 2) = 0 ; y = 1 or -2 ; but y = 1 makes y^' undefined so only (1 , -2)

d) y^' undefined for y = ± 1 ; (-1 , -1) or (3 , -1)