Answer:
y = 0
y =2x
Step-by-step explanation:
Given parametric equations:
x (t) = sin (t)
y (t) = sin (t + sin (t))
The slope of the curve at any given point is given by dy / dx we will use chain rule to find dy / dx
(dy / dx) * (dx / dt) = (dy / dt)
(dy / dx) = (dy / dt) / (dx / dt)
Evaluate dx / dt and dy / dt
dx / dt = cos (t)
dy / dt = cos (t + sin (t)) * (1+cos (t))
Hence,
dy / dx = (1+cos(t))*cos(t + sin (t))) / cos (t)
@Given point (x,y) = 0 we evaluate t
0 = sin (t)
t = 0 , pi
Input two values of t and compute dy / dx
@ t = 0
dy / dx = (1 + cos (0))*cos (0 + sin (0))) / cos (0)
dy / dx = (1+1)*(1) / (1) = 2 @ t = 0
@t = pi
dy / dx = ( 1 + cos (pi))* cos (pi + sin (pi)) / cos (pi)
dy / dx = (1-1) * (-1) / (-1) = 0 @ t = pi
The corresponding gradients are 0 and 2 in increasing order and their respective equations are:
y = 2x
y = 0
