Two function questions

Eliot Rosewater

Hey! I've two questions relating to functions; any help would be appreciated.

Question 1 - Polar Coordinates

Consider the curve [LATEX]r=1-\cos{\theta}[/LATEX]. What is the equation in rectilinear coordinates?

So I think you must solve for x in terms of y. Two equations relating rectilinear to polar coordinates are:

[LATEX]x=r\cos{\theta}[/LATEX] and [LATEX]y=r\sin{\theta}[/LATEX]

I've been crunching this for a while but I can't eliminate both r and theta. Any advise? Or is my method wrong?

Question 2 - Implicit differenciation

I think I've this one solved, I'd just like to verify my method.

Find [LATEX]\frac{dy}{dx}[/LATEX] by implicitly differentiating

[LATEX]\displaystyle \cos{(x-y)}=\frac{x}{1+y^2}[/LATEX]

So differenciating both sides. Chain rule on the left; quotient on the right.

[LATEX]\displaystyle -\sin{(x-y)}\frac{d}{dx}(x-y)=\frac{(1)(1+y^2)-(x)(\frac{d}{dx}(1+y^2))}{(1+y^2)^2}[/LATEX]

Aside,

[LATEX]\displaystyle \frac{d}{dx}(x-y)=1-\frac{dy}{dx}[/LATEX]

and

[LATEX]\displaystyle \frac{d}{dx}(1+y^2)=0+2y^1(\frac{dy}{dx})[/LATEX]

Sub back in and solve for dy/dx. Is this ok?

Thanks again!

MathsManiac

For the first one, try using r=sqrt(x^2+y^2) and theta = arctan(y/x).

That should allow you to get yout equation in terms of x and y. Then see if it can be simplified at all.

Second one looks ok to me.

(First one is a cardioid, by the way:
http://www.wolframalpha.com/input/?i=r%3D1-cos%28theta%29)

Eliot Rosewater

Thanks MathsManiac; that's great!

ray giraffe

Re Question 1

Use cos(theta) = x/r first, then replace all instances of r with sqrt(x^2+y^2).

Square roots can be removed if desired.