Derivatives with Polar Coordinates

The basic idea of finding the tangent line of a graph that is using the polar coordinate system as opposed to the Cartesian system is basically by finding a point and a slope like you would with normal equations. However instead of simply taking the derivative of an equation you have to do a little bit more work.




This is the most important equation when trying to find the tangent line. This equation is basically using the derivatives of the equations for crossing between the polar and cartesian planes, the equations being

and



so in order to find an equation at a point you will be given an equation in polar form and a point that you want to find it at.
at



All you have to do is take the derivative of that equation with respect to theta and use that to plug it into the master formula. Once you simplify it down to a more manageable equation you then plug in
and solve to get your slope. Then to find a point you will need to plug in

into the original equation that you had. This will give you an "r" value that then finally you just need to take the crossing over equations and put in the r value and the initial condition in and solve which will give you an x and y point.

and



this will give you a Cartesian set of points and a slope and all you have to do from there is plug it into the equation


That is all you have to do.
For more help on this subject you can go to this site.
For more help on polar equations in general you can go to here.

Hyperbolic Functions!

A quick review -- Hyperbolic functions are analogs of trigonometric functions. Cos(x) and sin(x) relate to points corresponding with unit circle, while cosh(x) and sinh(x) relate to points corresponding with the right half of the hyperbola.

Three main functions:

And their graphs:

There are certain hyperbolic identities that seem fairly similar to trigonometric identities...

To solve hyperbolic function can be solved my just plugging x into the main functions mentioned earlier. You can also solve hyperbolic integrals using non-hyperbolic integral rules. Check out this example we used on our powerpoint: