Expert: Ahmed Salami - 1/1/2005

Question
Consider the curve defined by the equation y+cosy=x+1 for 0 i less then or equal to y which is less then or equal to 2pie.

(a) Find dy/dx in terms of y
(b) write an equation for each vertical tangent to the curve
(c) Find the second derivative

Answer
Hi Nicole,
Please forgive the delay.
For y + cos y = x + 1
differentiating with respect to x
(1 - sin y)dy/dx = 1
dy/dx = 1/(1 - sin y)
The vertical tangent occurs when dy/dx is infinite i.e
1 - sin y = 0
sin y = 1
y = #/2  (# = pi)

From (1 - sin y)dy/dx = 1
dy/dx - sin y (dy/dx) = 1
differentiating again,
d^2y/dx^2 - [cos y(dy/dx) + sin y (d^2y/dx^2)]
d^2y/dx^2 - cos y(dy/dx) - sin y (d^2y/dx^2)= 0
(1 - sin y)d^2y/dx^2 = cos y(dy/dx)
d^2y/dx^2 = cos y(dy/dx) / (1 - sin y)
         = cos y / (1 - sin y)^2

I hope its helpful.
Regards.