2008-01-17

Today's nonsense
- Massive alien sighting conspiracy in Stevensville.
- Double angle formula.
- Testing Alyssa's ability to increase her intelligence with the mangerene



Slope and Tangent Lines of Polar Graphs

Slope in Polar Form

If f is a differentiable function of theta, then the slope of the tangent line to the graph of r = f(theta) at the point (r, theta) is:
dy/dx = ( (dy/dtheta) / (dx/dtheta) ) = f(theta) cos (theta) + f ' (theta)sin(theta) all over f(theta)sin(theta) + f ' (theta)cos(theta)
provided that dx/dtheta does not equal zero at (r, theta)


So they are calling r=f(theta) and they are using the product rule in the numerator and the product rule in the denominator. So this is simply dy/dtheta over dx/dtheta and so dy/dtheta is just the derivative with respect to theta of rsin(theta)


Note:
1. Solutions to dy/dtheta = 0 yield horizontal tangents of dx/dtheta not= zero.
2. Solutions to dx/dtheta = 0 yield vertical tangents, if dy/dtheta not=0 zero.
If dy/dtheta and dx/dtheta are simultaneously zero, then no conclusion can be drawn about the tangent lines.


ex 1) Find the horizontal and vertical tangent lines of r=sin(theta), 0 < = theta < = pi.
Our dy/dx is going to be our derivative of y over the derivative of x. So what's our y? Our y = rsin(theta) so therefore = sin(theta)sin(theta) = sin^2 (theta)
x = rcos(theta) = sin(theta)cos(theta)
This is because the r is defined as sin(theta).


dy/dx = (2sin(theta)cos(theta)) / (cos^2 (theta) - sin^2 (theta) )
sin(2theta) = 2sin(theta)cos(theta)
So this dy/dx is tan(2theta)

H: When is the numerator zero? 2sincos = 0. So it's 0, pi/2, pi (not 3pi/2 or 2pi, because of the restricted domain)
V: When is the denominator zero? (cos^2 = sin^2). So pi/4 ? So it's pi/4, 3pi/4, etc.



ex 2) r = 2(1-cos(theta))
x = rcos(theta) = >> 2(1-cos(theta)) cos(theta) >>
y = rsin(theta) = >> 2(1-cos(theta)) sin(theta) >>
dy/dx = 2sin^2 + 2(1-cos)cos all over 2sincos - 2(1-cos)sin
V: ... at theta = 0, pi, 2 pi, pi / 3, 5 pi / 3,


Rectangular is to origin as polar is to pole.


Tangent lines at the pole - if f(theta) = 0 and f prime theta does not equal zero, then the line theta = alpha is tangent to the apole of graph r=f(theta) ex a) r =2 cos3theta